# Mass concentration in rescaled first order integral functionals

Antonin Monteil, Paul Pegon

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Antonin Monteil, Paul Pegon. Mass concentration in rescaled first order integral functionals. Journal de l'École polytechnique - Mathématiques, 2024, 11, pp.42. 10.5802/jep.257. hal-03517074v4

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Distributed under a Creative Commons Attribution 4.0 International License
Journal de l'École polytechnique
Mathématiques

Antonin MONTEIL & Paul PEGON

Mass concentration in rescaled first order integral functionals

Tome II (2024), p. 431-472.

https://doi.org/10.5802/jep.257

© Les auteurs, 2024.

<figure><img src="image_1.png" /></figure>

Cet article est mis à disposition selon les termes de la licence
LICENCE INTERNATIONALE D'ATTRIBUTION CREATIVE COMMONS BY 4.0.
https://creativecommons.org/licenses/by/4.0/

Publié avec le soutien
du Centre National de la Recherche Scientifique
Journal de l'École polytechnique
Mathématiques

Antonin MONTEIL & Paul PEGON

Mass concentration in rescaled first order integral functionals

Tome II (2024), p. 431-472.

https://doi.org/10.5802/jep.257

© Les auteurs, 2024.

<figure><img src="image_3.png" /></figure>

Cet article est mis à disposition selon les termes de la licence
LICENCE INTERNATIONALE D'ATTRIBUTION CREATIVE COMMONS BY 4.0.
https://creativecommons.org/licenses/by/4.0/

Publié avec le soutien
du Centre National de la Recherche Scientifique
# MASS CONCENTRATION IN RESCALED FIRST ORDER INTEGRAL FUNCTIONALS

BY ANTONIN MONTEIL & PAUL PEGON

**ABSTRACT.** — We consider first order local minimization problems of the form $\min \int_{\mathbb{R}^N} f(u, \nabla u)$ under a mass constraint $\int_{\mathbb{R}^N} u = m$. We prove that the minimal energy function $H(m)$ is always concave, and that relevant rescalings of the energy, depending on a small parameter $\epsilon$, $\Gamma$-converge towards the $H$-mass, defined for atomic measures $\sum_i m_i \delta_{x_i}$ as $\sum_i H(m_i)$. We also consider Lagrangians depending on $\epsilon$, as well as space-inhomogeneous Lagrangians and $H$-masses. Our result holds under mild assumptions on $f$, and covers in particular $\alpha$-masses in any dimension $N \ge 2$ for exponents $\alpha$ above a critical threshold, and all concave $H$-masses in dimension $N = 1$. Our result yields in particular the concentration of Cahn-Hilliard fluids into droplets, and is related to the approximation of branched transport by elliptic energies.

**RÉSUMÉ** (Concentration de masse dans des fonctionnelles intégrales d'ordre 1 rééchelonnées)

Nous considérons des problèmes de minimisation locaux d'ordre 1 de la forme $\min \int_{\mathbb{R}^N} f(u, \nabla u)$ sous contrainte de masse $\int_{\mathbb{R}^N} u = m$. Nous prouvons que la fonction d'énergie minimale $H(m)$ est toujours concave, et que des rééchelonnements appropriés de l'énergie, dépendant d'un petit paramètre $\epsilon$, $\Gamma$-convergent vers la $H$-masse, définie pour les mesures atomiques $\sum_i m_i \delta_{x_i}$ par $\sum_i H(m_i)$. Nous considérons également des lagrangiens dépendant de $\epsilon$, et des lagrangiens et $H$-masses spatialement inhomogènes. Notre résultat est valable sous de faibles hypothèses sur $f$, et couvre les $\alpha$-masses en toute dimension $N \ge 2$ pour des exposants $\alpha$ au-dessus d'un seuil critique, et toutes les $H$-masses concaves en dimension $N = 1$. Notre résultat donne en particulier la concentration des fluides de Cahn-Hilliard en gouttelettes, et est lié à l'approximation du transport branché par des énergies elliptiques.

## CONTENTS

<table><tr><td>Notation</td><td>432</td></tr><tr><td>1. Introduction</td><td>432</td></tr><tr><td>2. Minimal cost function and H-mass</td><td>437</td></tr><tr><td>3. Lower bound for the energy and existence of optimal profiles</td><td>447</td></tr><tr><td>4. Γ-convergence of the rescaled energies towards the H-mass</td><td>458</td></tr><tr><td>5. Examples, counterexamples and applications</td><td>463</td></tr><tr><td>References</td><td>471</td></tr></table>

MATHEMATICAL SUBJECT CLASSIFICATION (2020). — 28A33, 49J45, 46E35, 49Q20, 76T99, 49Q22, 49J10.
KEYWORDS. — Γ-convergence, semicontinuity, integral functionals, convergence of measures, concentration-compactness, Cahn-Hilliard fluids, branched transport.

A.M. acknowledges support by Leverhulme grant RPG-2018-438.
MASS CONCENTRATION IN
RESCALED FIRST ORDER INTEGRAL FUNCTIONALS

BY ANTONIN MONTEIL & PAUL PEGON

ABSTRACT. — We consider first order local minimization problems of the form $\min \int_{\mathbb{R}^N} f(u, \nabla u)$ under a mass constraint $\int_{\mathbb{R}^N} u = m$. We prove that the minimal energy function $H(m)$ is always concave, and that relevant rescalings of the energy, depending on a small parameter $\epsilon$, $\Gamma$-converge towards the $H$-mass, defined for atomic measures $\sum_i m_i \delta_{x_i}$ as $\sum_i H(m_i)$. We also consider Lagrangians depending on $\epsilon$, as well as space-inhomogeneous Lagrangians and $H$-masses. Our result holds under mild assumptions on $f$, and covers in particular $\alpha$-masses in any dimension $N \ge 2$ for exponents $\alpha$ above a critical threshold, and all concave $H$-masses in dimension $N = 1$. Our result yields in particular the concentration of Cahn-Hilliard fluids into droplets, and is related to the approximation of branched transport by elliptic energies.

RÉSUMÉ (Concentration de masse dans des fonctionnelles intégrales d'ordre 1 rééchelonnées)

Nous considérons des problèmes de minimisation locaux d'ordre 1 de la forme $\min \int_{\mathbb{R}^N} f(u, \nabla u)$ sous contrainte de masse $\int_{\mathbb{R}^N} u = m$. Nous prouvons que la fonction d'énergie minimale $H(m)$ est toujours concave, et que des rééchelonnements appropriés de l'énergie, dépendant d'un petit paramètre $\epsilon$, $\Gamma$-convergent vers la $H$-masse, définie pour les mesures atomiques $\sum_i m_i \delta_{x_i}$ par $\sum_i H(m_i)$. Nous considérons également des lagrangiens dépendant de $\epsilon$, et des lagrangiens et $H$-masses spatialement inhomogènes. Notre résultat est valable sous de faibles hypothèses sur $f$, et couvre les $\alpha$-masses en toute dimension $N \ge 2$ pour des exposants $\alpha$ au-dessus d'un seuil critique, et toutes les $H$-masses concaves en dimension $N = 1$. Notre résultat donne en particulier la concentration des fluides de Cahn-Hilliard en gouttelettes, et est lié à l'approximation du transport branché par des énergies elliptiques.

CONTENTS

<table><tr><td>Notation</td><td>432</td></tr><tr><td>1. Introduction</td><td>432</td></tr><tr><td>2. Minimal cost function and H-mass</td><td>437</td></tr><tr><td>3. Lower bound for the energy and existence of optimal profiles</td><td>447</td></tr><tr><td>4. Γ-convergence of the rescaled energies towards the H-mass</td><td>458</td></tr><tr><td>5. Examples, counterexamples and applications</td><td>463</td></tr><tr><td>References</td><td>471</td></tr></table>

MATHEMATICAL SUBJECT CLASSIFICATION (2020). — 28A33, 49J45, 46E35, 49Q20, 76T99, 49Q22, 49J10.
KEYWORDS. — Γ-convergence, semicontinuity, integral functionals, convergence of measures, concentration-compactness, Cahn-Hilliard fluids, branched transport.

A.M. acknowledges support by Leverhulme grant RPG-2018-438.
## NOTATION

$B_r(x)$ open ball of radius $r$ centered at $x$;

$B_r$ open ball $B_r(0)$;

$\mathcal{M}(\mathbb{R}^N)$ set of finite signed Borel measures on $\mathbb{R}^N$;

$\mathcal{M}_+(\mathbb{R}^N)$ set of finite positive Borel measures on $\mathbb{R}^N$;

$\Phi_2\mu$ pushforward of a measure $\mu \in \mathcal{M}(\mathbb{R}^N)$ by a map $\Phi : \mathbb{R}^N \to \mathbb{R}^k$, defined as $A \mapsto \mu(\Phi^{-1}(A))$;

$\tau_x\mu$ Borel measure $A \mapsto \mu(A - x)$ if $\mu \in \mathcal{M}(\mathbb{R}^N)$ and $x \in \mathbb{R}^N$;

$c_B\mu$ Borel measure $\tau_{-x}(\mu \llcorner B)$ if $B$ is the ball $B_r(x)$;

$\mu_\ell \xrightarrow{\mathcal{C}'_0} \mu$ weak convergence of measures, i.e., weak-⋆ convergence in duality with the space $\mathcal{C}_0(\mathbb{R}^N)$ of continuous functions vanishing at infinity;

$\mu_\ell \xrightarrow{\mathcal{C}'_b} \mu$ narrow convergence of measures, i.e., weak-⋆ convergence in duality with the space $\mathcal{C}_b(\mathbb{R}^N)$ of continuous and bounded function;

$\Sigma$ set of increasing maps $\sigma : \mathbb{N} \to \mathbb{N}$;

$\sigma_1 \preceq \sigma_2$ $\sigma_1, \sigma_2 \in \Sigma$ are such that $\sigma_1([n, +\infty]) \subseteq \sigma_2(\mathbb{N})$ for some $n \in \mathbb{N}$;

$\pm$ fixed to either $+$ or $-$ in the whole statement or proof, and $\mp = -(\pm)$.

# 1. INTRODUCTION

## 1.1. SETTING.
Let $N \in \mathbb{N}^*$ and let $f : \mathbb{R}^N \times \mathbb{R} \times \mathbb{R}^N \to [0, +\infty)$ be a Borel function such that $f(\cdot, 0, 0) \equiv 0$. Consider the following energy functional, defined for any fixed $x \in \mathbb{R}^N$ on the set of finite Borel measures $\mathcal{M}(\mathbb{R}^N)$ on $\mathbb{R}^N$ by

$$ (1.1) \quad \mathcal{E}_f^x(\mu) = \begin{cases} \int_{\mathbb{R}^N} f(x, u(y), \nabla u(y)) \, dy & \text{if } \mu = u \mathcal{L}^N, u \in W_{\text{loc}}^{1,1}(\mathbb{R}^N), \\ +\infty & \text{otherwise.} \end{cases} $$

The minimization of this energy under a mass constraint gives rise to the notion of minimal cost function, valued in $[0, +\infty]$ and defined by

$$ (1.2) \quad H_f(x, m) := \inf \left\{ \mathcal{E}_f^x(u \mathcal{L}^N) : u \in W_{\text{loc}}^{1,1} \cap L^1(\mathbb{R}^N) \text{ such that } \int_{\mathbb{R}^N} u = m \right\}. $$

As a preliminary result, which applies to (1.1) and deserves interest on its own, we will establish the following:

**THEOREM 1.1.** — Let $f : \mathbb{R} \times \mathbb{R}^N \to [0, +\infty]$ be Borel measurable such that $f(0, 0) = 0$. The function defined for every $m \in \mathbb{R}$ by

$$ (1.3) \quad H_f(m) := \inf \left\{ \int_{\mathbb{R}^N} f(u, \nabla u) : u \in W_{\text{loc}}^{1,1} \cap L^1(\mathbb{R}^N, \mathbb{R}), \int_{\mathbb{R}^N} u = m \right\} $$

vanishes at 0 and it is either identically $+\infty$ on $(0, +\infty)$, or it is everywhere finite, continuous, concave and non-decreasing on $[0, +\infty)$. The symmetric statement on $(-\infty, 0]$ holds as well.
## NOTATION

$B_r(x)$ open ball of radius $r$ centered at $x$;

$B_r$ open ball $B_r(0)$;

$\mathcal{M}(\mathbb{R}^N)$ set of finite signed Borel measures on $\mathbb{R}^N$;

$\mathcal{M}_+(\mathbb{R}^N)$ set of finite positive Borel measures on $\mathbb{R}^N$;

$\Phi_2\mu$ pushforward of a measure $\mu \in \mathcal{M}(\mathbb{R}^N)$ by a map $\Phi : \mathbb{R}^N \to \mathbb{R}^k$, defined as $A \mapsto \mu(\Phi^{-1}(A))$;

$\tau_x\mu$ Borel measure $A \mapsto \mu(A - x)$ if $\mu \in \mathcal{M}(\mathbb{R}^N)$ and $x \in \mathbb{R}^N$;

$c_B\mu$ Borel measure $\tau_{-x}(\mu \llcorner B)$ if $B$ is the ball $B_r(x)$;

$\mu_\ell \xrightarrow{\mathcal{C}'_0} \mu$ weak convergence of measures, i.e., weak-⋆ convergence in duality with the space $\mathcal{C}_0(\mathbb{R}^N)$ of continuous functions vanishing at infinity;

$\mu_\ell \xrightarrow{\mathcal{C}'_b} \mu$ narrow convergence of measures, i.e., weak-⋆ convergence in duality with the space $\mathcal{C}_b(\mathbb{R}^N)$ of continuous and bounded function;

$\Sigma$ set of increasing maps $\sigma : \mathbb{N} \to \mathbb{N}$;

$\sigma_1 \preceq \sigma_2$ $\sigma_1, \sigma_2 \in \Sigma$ are such that $\sigma_1([n, +\infty]) \subseteq \sigma_2(\mathbb{N})$ for some $n \in \mathbb{N}$;

$\pm$ fixed to either $+$ or $-$ in the whole statement or proof, and $\mp = -(\pm)$.

# 1. INTRODUCTION

## 1.1. SETTING.
Let $N \in \mathbb{N}^*$ and let $f : \mathbb{R}^N \times \mathbb{R} \times \mathbb{R}^N \to [0, +\infty)$ be a Borel function such that $f(\cdot, 0, 0) \equiv 0$. Consider the following energy functional, defined for any fixed $x \in \mathbb{R}^N$ on the set of finite Borel measures $\mathcal{M}(\mathbb{R}^N)$ on $\mathbb{R}^N$ by

$$ (1.1) \qquad \mathcal{E}_f^x(\mu) = \begin{cases} \int_{\mathbb{R}^N} f(x, u(y), \nabla u(y)) \, dy & \text{if } \mu = u \mathcal{L}^N, u \in W_{\text{loc}}^{1,1}(\mathbb{R}^N), \\ +\infty & \text{otherwise.} \end{cases} $$

The minimization of this energy under a mass constraint gives rise to the notion of minimal cost function, valued in $[0, +\infty]$ and defined by

$$ (1.2) \qquad H_f(x, m) := \inf \left\{ \mathcal{E}_f^x(u \mathcal{L}^N) : u \in W_{\text{loc}}^{1,1} \cap L^1(\mathbb{R}^N) \text{ such that } \int_{\mathbb{R}^N} u = m \right\}. $$

As a preliminary result, which applies to (1.1) and deserves interest on its own, we will establish the following:

**THEOREM 1.1.** — Let $f : \mathbb{R} \times \mathbb{R}^N \to [0, +\infty]$ be Borel measurable such that $f(0, 0) = 0$. The function defined for every $m \in \mathbb{R}$ by

$$ (1.3) \qquad H_f(m) := \inf \left\{ \int_{\mathbb{R}^N} f(u, \nabla u) : u \in W_{\text{loc}}^{1,1} \cap L^1(\mathbb{R}^N, \mathbb{R}), \int_{\mathbb{R}^N} u = m \right\} $$

vanishes at 0 and it is either identically $+\infty$ on $(0, +\infty)$, or it is everywhere finite, continuous, concave and non-decreasing on $[0, +\infty)$. The symmetric statement on $(-\infty, 0]$ holds as well.
The proof is very simple and works with no further assumptions on $f$.

Our main purpose is to prove that if $(f_\epsilon)_{\epsilon>0}$ is a family of functions

$$f_\epsilon : \mathbb{R}^N \times \mathbb{R} \times \mathbb{R}^N \to [0, +\infty]$$

which converges pointwise to $f$ as $\epsilon \to 0$ and satisfies some conditions, then the rescaled energy functionals $\mathcal{E}_\epsilon$, defined for each $\epsilon > 0$ on $\mathcal{M}(\mathbb{R}^N)$ by (1.4)

$$\mathcal{E}_\epsilon(\mu) = \begin{cases} \int_{\mathbb{R}^N} f_\epsilon(x, \epsilon^N u(x), \epsilon^{N+1} \nabla u(x)) \epsilon^{-N} dx & \text{if } \mu = u \mathcal{L}^N, u \in W_{\text{loc}}^{1,1}(\mathbb{R}^N), \\ +\infty & \text{otherwise,} \end{cases}$$

$\Gamma$-converge as $\epsilon \to 0$, for the narrow or weak convergence of measures, to the $H_f$-mass, defined on $\mathcal{M}(\mathbb{R}^N)$ by (see Definition 2.2):

$$\mathcal{M}^{H_f}(\mu) := \int_{\mathbb{R}^N} H_f(x, \mu(\{x\})) d\mathcal{H}^0(x) + \int_{\mathbb{R}^N} H'_f(x, 0^+) d\mu_+^d(x) + \int_{\mathbb{R}^N} H'_f(x, 0^-) d\mu_-^d(x).$$

where $\mu = \mu^a + \mu^d$ is the decomposition of $\mu$ into its atomic part $\mu^a$ and its diffuse part $\mu^d$, $\mu^d = \mu_+^d - \mu_-^d$ is the Jordan decomposition of $\mu^d$ into positive and negative parts, and $H'_f(x, 0^\pm)$ is the recession at 0, that is

$$H'_f(x, 0^\pm) := \lim_{m \to 0^\pm} \frac{H_f(x, m)}{|m|} \in [0, +\infty].$$

Notice that the exponents over $\epsilon$ in the definition of $\mathcal{E}_\epsilon$ are tuned so that if $B_r(x_0) \subseteq \mathbb{R}^N$ and $u_\epsilon$ is a mass-preserving rescaling of $v_\epsilon$ given by $u_\epsilon(x) := \epsilon^{-N} v_\epsilon(\epsilon^{-1}(x - x_0))$, then

$$\int_{B_r(x_0)} f_\epsilon(x, \epsilon^N u_\epsilon(x), \epsilon^{N+1} \nabla u_\epsilon(x)) \epsilon^{-N} dx = \int_{B_{r/\epsilon}} f_\epsilon(x_0 + \epsilon y, v_\epsilon(y), \nabla v_\epsilon(y)) dy,$$

so that the energy contribution of a mass $m \ge 0$ contained in a ball $B_r(x_0)$ should be of the order of $H_f(x_0, m)$. This explains why $\mathcal{M}^{H_f}$ is expected to be the $\Gamma$-limit of $\mathcal{E}_\epsilon$. Nevertheless, it is not true in general (see Section 1.3 below), and we will need a couple of assumptions on $f$ and $f_\epsilon$ detailed in the next section.

This kind of singular limit of integral functionals is reminiscent of several variational models with physical relevance which have been the object of intensive mathematical analysis, such as Cahn-Hilliard fluids with concentration on droplets [BDS96] (which we recover in Section 5.5) or on singular interfaces [MM77], toy models for micromagnetism and liquid crystals like Aviles-Giga [AG99] and Landau-de Gennes [BPP12], or Ginzburg-Landau theory of superconductivity [BBH17].

## 1.2. ASSUMPTIONS AND MAIN RESULT. — Our first two assumptions are rather standard and guarantee the sequential lower semicontinuity of the functionals $\mathcal{E}_f^x$,

(H1) $f: \mathbb{R}^N \times \mathbb{R} \times \mathbb{R}^N \to [0, +\infty]$ is lower semicontinuous,

(H2) $f(x, u, \cdot)$ is convex for every $x \in \mathbb{R}^N, u \in \mathbb{R}$.
The proof is very simple and works with no further assumptions on $f$.

Our main purpose is to prove that if $(f_\varepsilon)_{\varepsilon>0}$ is a family of functions

$$f_\varepsilon : \mathbb{R}^N \times \mathbb{R} \times \mathbb{R}^N \to [0, +\infty]$$

which converges pointwise to $f$ as $\varepsilon \to 0$ and satisfies some conditions, then the rescaled energy functionals $\mathcal{E}_\varepsilon$, defined for each $\varepsilon > 0$ on $\mathcal{M}(\mathbb{R}^N)$ by (1.4)

$$\mathcal{E}_\varepsilon(\mu) = \begin{cases} \int_{\mathbb{R}^N} f_\varepsilon(x, \varepsilon^N u(x), \varepsilon^{N+1} \nabla u(x)) \varepsilon^{-N} dx & \text{if } \mu = u \mathcal{L}^N, u \in W_{\text{loc}}^{1,1}(\mathbb{R}^N), \\ +\infty & \text{otherwise,} \end{cases}$$

$\Gamma$-converge as $\varepsilon \to 0$, for the narrow or weak convergence of measures, to the $H_f$-mass, defined on $\mathcal{M}(\mathbb{R}^N)$ by (see Definition 2.2):

$$\mathcal{M}^{H_f}(\mu) := \int_{\mathbb{R}^N} H_f(x, \mu(\{x\})) d\mathcal{H}^0(x) \\ + \int_{\mathbb{R}^N} H'_f(x, 0^+) d\mu_+^d(x) + \int_{\mathbb{R}^N} H'_f(x, 0^-) d\mu_-^d(x).$$

where $\mu = \mu^a + \mu^d$ is the decomposition of $\mu$ into its atomic part $\mu^a$ and its diffuse part $\mu^d$, $\mu^d = \mu_+^d - \mu_-^d$ is the Jordan decomposition of $\mu^d$ into positive and negative parts, and $H'_f(x, 0^\pm)$ is the recession at 0, that is

$$H'_f(x, 0^\pm) := \lim_{m \to 0^\pm} \frac{H_f(x, m)}{|m|} \in [0, +\infty].$$

Notice that the exponents over $\varepsilon$ in the definition of $\mathcal{E}_\varepsilon$ are tuned so that if $B_r(x_0) \subseteq \mathbb{R}^N$ and $u_\varepsilon$ is a mass-preserving rescaling of $v_\varepsilon$ given by $u_\varepsilon(x) := \varepsilon^{-N} v_\varepsilon(\varepsilon^{-1}(x - x_0))$, then

$$\int_{B_r(x_0)} f_\varepsilon(x, \varepsilon^N u_\varepsilon(x), \varepsilon^{N+1} \nabla u_\varepsilon(x)) \varepsilon^{-N} dx = \int_{B_{r/\varepsilon}} f_\varepsilon(x_0 + \varepsilon y, v_\varepsilon(y), \nabla v_\varepsilon(y)) dy,$$

so that the energy contribution of a mass $m \ge 0$ contained in a ball $B_r(x_0)$ should be of the order of $H_f(x_0, m)$. This explains why $\mathcal{M}^{H_f}$ is expected to be the $\Gamma$-limit of $\mathcal{E}_\varepsilon$. Nevertheless, it is not true in general (see Section 1.3 below), and we will need a couple of assumptions on $f$ and $f_\varepsilon$ detailed in the next section.

This kind of singular limit of integral functionals is reminiscent of several variational models with physical relevance which have been the object of intensive mathematical analysis, such as Cahn-Hilliard fluids with concentration on droplets [BDS96] (which we recover in Section 5.5) or on singular interfaces [MM77], toy models for micromagnetism and liquid crystals like Aviles-Giga [AG99] and Landau-de Gennes [BPP12], or Ginzburg-Landau theory of superconductivity [BBH17].

### 1.2. ASSUMPTIONS AND MAIN RESULT.
Our first two assumptions are rather standard and guarantee the sequential lower semicontinuity of the functionals $\mathcal{E}_f^x$,

(H1) $f: \mathbb{R}^N \times \mathbb{R} \times \mathbb{R}^N \to [0, +\infty]$ is lower semicontinuous,

(H2) $f(x, u, \cdot)$ is convex for every $x \in \mathbb{R}^N, u \in \mathbb{R}$.
In order for vanishing parts to have no energetic contribution, we will impose

(H3) $f(x, 0, 0) = 0$ for every $x \in \mathbb{R}^N$.

We also need continuity in the spatial variable,

(H4) $f(\cdot, u, \xi)$ is continuous for every $u \in \mathbb{R}, \xi \in \mathbb{R}^N$.

Next, we need a compactness assumption which ensures relative compactness in the weak topology of $W_{\text{loc}}^{1,p}(\mathbb{R}^N)$ for sequences of bounded energy $\mathcal{E}_f^x$ and bounded mass; it will also be needed in obtaining lower bounds for the energy (see Proposition 3.8):

(H5) there exist $\alpha, \beta \in (0, +\infty), p \in (1, +\infty)$ such that for all $(x, u, \xi) \in \mathbb{R}^N \times \mathbb{R} \times \mathbb{R}^N$,

$$f(x, u, \xi) \geq \alpha |\xi|^p - \beta u.$$

We also impose a comparison condition on the slopes of $f(x, \cdot, \xi)$ and $H_f(x, \cdot)$ at the origin, which will be needed in order to show that the $\Gamma$-lim inf is bounded from below by the $H_f$-mass, and which rules out some non-trivial scale invariant Lagrangians for which the expected $\Gamma$-convergence result fails (see Section 1.3):

(H6) for every $x_0 \in \mathbb{R}^N$,

$$(1.5) \qquad H'_{f}(x_0, 0^{\pm}) \le f'_{-}(x_0, 0^{\pm}, 0) := \liminf_{(x,u,\xi) \to (x_0,0^{\pm},0)} \frac{f(x, u, \xi)}{|u|}.$$

We give a general assumption (simple in dimension one but quite technical in dimension $N \ge 2$) depending only on the Lagrangian $f$ so as to guarantee such a condition (see (S) and Corollary 2.7 in Section 2.3).

Since our aim is not to care much about the dependence on $x$, we shall impose a spatial quasi-homogeneity condition:

(H7) there exists $C < +\infty$ such that for every $x, y \in \mathbb{R}^N, u \in \mathbb{R}, \xi \in \mathbb{R}^N$,

$$f(y, u, \xi) \le C(f(x, u, \xi) + u).$$

Last of all, we need the family of functions $f_\epsilon : \mathbb{R}^N \times \mathbb{R} \times \mathbb{R}^N \to [0, +\infty]$ to converge towards $f$ in a suitable sense, namely, we assume

(H8) $f_\epsilon \uparrow f$ and $f'_{\epsilon, -}(\cdot, 0^\pm, 0) \uparrow f'_{-}(\cdot, 0^\pm, 0)$ as $\epsilon \to 0$.

Notice that this assumption is empty if $f_\epsilon$ does not depend on $\epsilon$.

Our main result is the following:

**THEOREM 1.2.** — If $(f_\epsilon)_{\epsilon>0}$ satisfies (H8) with each $f_\epsilon$ satisfying (H1)–(H5) and the limit $f$ satisfying (H6)–(H7), then $M^{H_f}$ is the $\Gamma$-limit as $\epsilon \to 0$ of the functionals $\mathcal{E}_\epsilon$, defined in (1.4), for both the weak convergence and the narrow convergence of measures.

In particular, as a $\Gamma$-limit, the functional $M^{H_f}$ must be lower semicontinuous for the weak convergence of measures (and so for the narrow convergence as well). This implies that $H_f$ is lower semicontinuous on $\mathbb{R}^N \times \mathbb{R}$ (see Proposition 2.4).

We point out that for the $\Gamma$-lim sup, we need weaker assumptions on $f_\epsilon$ and $f$ (see Proposition 4.2), which will be useful for some applications (see Section 5.5).
In order for vanishing parts to have no energetic contribution, we will impose

(H3) $f(x, 0, 0) = 0$ for every $x \in \mathbb{R}^N$.

We also need continuity in the spatial variable,

(H4) $f(\cdot, u, \xi)$ is continuous for every $u \in \mathbb{R}, \xi \in \mathbb{R}^N$.

Next, we need a compactness assumption which ensures relative compactness in the weak topology of $W_{\text{loc}}^{1,p}(\mathbb{R}^N)$ for sequences of bounded energy $\mathcal{E}_f^x$ and bounded mass; it will also be needed in obtaining lower bounds for the energy (see Proposition 3.8):

(H5) there exist $\alpha, \beta \in (0, +\infty), p \in (1, +\infty)$ such that for all $(x, u, \xi) \in \mathbb{R}^N \times \mathbb{R} \times \mathbb{R}^N$,

$$f(x, u, \xi) \ge \alpha |\xi|^p - \beta u.$$

We also impose a comparison condition on the slopes of $f(x, \cdot, \xi)$ and $H_f(x, \cdot)$ at the origin, which will be needed in order to show that the $\Gamma$-lim inf is bounded from below by the $H_f$-mass, and which rules out some non-trivial scale invariant Lagrangians for which the expected $\Gamma$-convergence result fails (see Section 1.3):

(H6) for every $x_0 \in \mathbb{R}^N$,

$$(1.5) \qquad H'_{f}(x_0, 0^{\pm}) \le f'_{-}(x_0, 0^{\pm}, 0) := \liminf_{(x,u,\xi) \to (x_0,0^{\pm},0)} \frac{f(x, u, \xi)}{|u|}.$$

We give a general assumption (simple in dimension one but quite technical in dimension $N \ge 2$) depending only on the Lagrangian $f$ so as to guarantee such a condition (see (S) and Corollary 2.7 in Section 2.3).

Since our aim is not to care much about the dependence on $x$, we shall impose a spatial quasi-homogeneity condition:

(H7) there exists $C < +\infty$ such that for every $x, y \in \mathbb{R}^N, u \in \mathbb{R}, \xi \in \mathbb{R}^N$,

$$f(y, u, \xi) \le C(f(x, u, \xi) + u).$$

Last of all, we need the family of functions $f_\epsilon : \mathbb{R}^N \times \mathbb{R} \times \mathbb{R}^N \to [0, +\infty]$ to converge towards $f$ in a suitable sense, namely, we assume

(H8) $f_\epsilon \uparrow f$ and $f'_{\epsilon, -}(\cdot, 0^\pm, 0) \uparrow f'_{-}(\cdot, 0^\pm, 0)$ as $\epsilon \to 0$.

Notice that this assumption is empty if $f_\epsilon$ does not depend on $\epsilon$.

Our main result is the following:

**THEOREM 1.2.** — If $(f_\epsilon)_{\epsilon>0}$ satisfies (H8) with each $f_\epsilon$ satisfying (H1)–(H5) and the limit $f$ satisfying (H6)–(H7), then $M^{H_f}$ is the $\Gamma$-limit as $\epsilon \to 0$ of the functionals $\mathcal{E}_\epsilon$, defined in (1.4), for both the weak convergence and the narrow convergence of measures.

In particular, as a $\Gamma$-limit, the functional $M^{H_f}$ must be lower semicontinuous for the weak convergence of measures (and so for the narrow convergence as well). This implies that $H_f$ is lower semicontinuous on $\mathbb{R}^N \times \mathbb{R}$ (see Proposition 2.4).

We point out that for the $\Gamma$-lim sup, we need weaker assumptions on $f_\epsilon$ and $f$ (see Proposition 4.2), which will be useful for some applications (see Section 5.5).
We will allow ourselves slight abuses of notation. First of all, we will sometimes consider Lagrangians defined on $\mathbb{R} \times \mathbb{R}^N$ which do not depend on $x$ and still refer to hypotheses $(H_1)-(H_8)$; we will use the notation $\mathcal{E}_f$ instead of $\mathcal{E}_f^x$ in (1.1) and consider $H_f$ as a function of $m$ only in (1.2). Besides, we will often identify functions $u \in L_{\text{loc}}^1$ with the measures $\mu = u\mathcal{L}^N$, so as to concisely write $\mathcal{E}_f(u)$ instead of $\mathcal{E}_f(u\mathcal{L}^N)$. Finally, we will also consider Lagrangians defined only for $u \in \mathbb{R}_+$, which may be thought as defined for $u \in \mathbb{R}$, set to $+\infty$ when $u$ is negative. (1) The resulting minimal cost function $H_f$ and its associated $H_f$-mass may be thought as defined on $\mathbb{R}_+$ and $M_+(\mathbb{R}^N)$ respectively, as they will be infinite on the respective complements.

### 1.3. REMARKS AND APPLICATIONS.
We start with two situations where the expected $\Gamma$-convergence fails and which justify the importance of $(H_3)$ and $(H_6)$, then we provide examples and applications of our result, as a short summary of Section 5, where full details are provided. We restrict our attention to positive measures and Lagrangians defined for $u \ge 0$.

**Vanishing parts do not contribute to energy.** — By assumption $(H_3)$ no energy is given to any set where a function $u$ vanishes. It is a necessary condition for $M^{H_f}$ to be lower semicontinuous (a necessary condition to be a $\Gamma$-limit) and not identically $+\infty$. Indeed if $M^{H_f}$ is lower semicontinuous and finite for some measure $\mu \in \mathcal{M}(\mathbb{R}^N)$ then (see Remark 2.1) $M^{H_f}(t\mu) \le M^{H_f}(\mu)$ for every $t \in (0,1)$ hence $M^{H_f}(0) \le \liminf_{t \to 0+} M^{H_f}(t\mu) \le M^{H_f}(\mu) < +\infty$. Thus $M^{H_f}$ is not identically $+\infty$ if and only if $M^{H_f}(0) < +\infty$, i.e., $\int_{\mathbb{R}^N} H_f(x,0) d\mathcal{H}^0(x) < +\infty$. But since $H_f(x,0) = (+\infty) \times f(x,0,0)$ this can only happen if $f(\cdot, 0, 0) \equiv 0$. This justifies imposing $(H_3)$.

**Scale invariant Lagrangians.** — In Section 5.1, we will see that in the particular case $f_\varepsilon(x, u, \xi) = u^{-p(1-1/p^*)}|\xi|^p$ for $p \in (1, N)$ and $p^* = pN/(N-p)$, our $\Gamma$-convergence result does not hold as a consequence of the fact that $\mathcal{E}_\varepsilon$ does not depend on $\varepsilon$. Note that in this case, the slope assumption $(H_6)$ does not hold, and we also provide a simple variant of such energies which satisfies all our assumptions except $(H_6)$ where the $\Gamma$-convergence towards $M^{H_f}$ also fails, thus justifying the need for such a slope condition.

**Concave $H$-masses in dimension one.** — Consider the energy given for $\mu = u\mathcal{L}^N$ by

$$ \mathcal{E}_f(\mu) = \int_{\mathbb{R}^N} |\nabla u|^2 + c(u) \quad \text{with Lagrangian} \quad f(x, u, \xi) = |\xi|^2 + c(u). $$

In dimension $N=1$, it is shown in [Wir19] that for any concave continuous function $H$ with $H(0)=0$, there exists a suitable $c \ge 0$ such that $H_f = H$. As explained in Section 5.2, Theorem 1.2 implies that the rescaled energies

$$ (1.6) \qquad \mathcal{E}_{\varepsilon}(\mu) = \int_{\mathbb{R}^N} f(\varepsilon^N u, \varepsilon^{N+1} \nabla u) \varepsilon^{-N} = \int_{\mathbb{R}^N} \varepsilon^{N+2} |\nabla u|^2 + \varepsilon^{-N} c(\varepsilon^N u). $$

<footnote>(1) Notice that if any of our assumptions is satisfied for a Lagrangian defined for $u \in \mathbb{R}_+$, then it holds also for the Lagrangian extended to $\mathbb{R}$ in this way.</footnote>
We will allow ourselves slight abuses of notation. First of all, we will sometimes consider Lagrangians defined on $\mathbb{R} \times \mathbb{R}^N$ which do not depend on $x$ and still refer to hypotheses $(H_1)-(H_8)$; we will use the notation $\mathcal{E}_f$ instead of $\mathcal{E}_f^x$ in (1.1) and consider $H_f$ as a function of $m$ only in (1.2). Besides, we will often identify functions $u \in L_{\text{loc}}^1$ with the measures $\mu = u\mathcal{L}^N$, so as to concisely write $\mathcal{E}_f(u)$ instead of $\mathcal{E}_f(u\mathcal{L}^N)$. Finally, we will also consider Lagrangians defined only for $u \in \mathbb{R}_+$, which may be thought as defined for $u \in \mathbb{R}$, set to $+\infty$ when $u$ is negative. (1) The resulting minimal cost function $H_f$ and its associated $H_f$-mass may be thought as defined on $\mathbb{R}_+$ and $\mathcal{M}_+(\mathbb{R}^N)$ respectively, as they will be infinite on the respective complements.

### 1.3. REMARKS AND APPLICATIONS.
We start with two situations where the expected $\Gamma$-convergence fails and which justify the importance of $(H_3)$ and $(H_6)$, then we provide examples and applications of our result, as a short summary of Section 5, where full details are provided. We restrict our attention to positive measures and Lagrangians defined for $u \ge 0$.

**Vanishing parts do not contribute to energy.** — By assumption $(H_3)$ no energy is given to any set where a function $u$ vanishes. It is a necessary condition for $M^{H_f}$ to be lower semicontinuous (a necessary condition to be a $\Gamma$-limit) and not identically $+\infty$. Indeed if $M^{H_f}$ is lower semicontinuous and finite for some measure $\mu \in \mathcal{M}(\mathbb{R}^N)$ then (see Remark 2.1) $M^{H_f}(t\mu) \le M^{H_f}(\mu)$ for every $t \in (0,1)$ hence $M^{H_f}(0) \le \liminf_{t \to 0+} M^{H_f}(t\mu) \le M^{H_f}(\mu) < +\infty$. Thus $M^{H_f}$ is not identically $+\infty$ if and only if $M^{H_f}(0) < +\infty$, i.e., $\int_{\mathbb{R}^N} H_f(x,0) d\mathcal{H}^0(x) < +\infty$. But since $H_f(x,0) = (+\infty) \times f(x,0,0)$ this can only happen if $f(\cdot, 0, 0) \equiv 0$. This justifies imposing $(H_3)$.

**Scale invariant Lagrangians.** — In Section 5.1, we will see that in the particular case $f_\varepsilon(x, u, \xi) = u^{-p(1-1/p^*)}|\xi|^p$ for $p \in (1, N)$ and $p^* = pN/(N-p)$, our $\Gamma$-convergence result does not hold as a consequence of the fact that $\mathcal{E}_\varepsilon$ does not depend on $\varepsilon$. Note that in this case, the slope assumption $(H_6)$ does not hold, and we also provide a simple variant of such energies which satisfies all our assumptions except $(H_6)$ where the $\Gamma$-convergence towards $M^{H_f}$ also fails, thus justifying the need for such a slope condition.

**Concave $H$-masses in dimension one.** — Consider the energy given for $\mu = u\mathcal{L}^N$ by

$$ \mathcal{E}_f(\mu) = \int_{\mathbb{R}^N} |\nabla u|^2 + c(u) \quad \text{with Lagrangian } f(x, u, \xi) = |\xi|^2 + c(u). $$

In dimension $N=1$, it is shown in [Wir19] that for any concave continuous function $H$ with $H(0)=0$, there exists a suitable $c \ge 0$ such that $H_f = H$. As explained in Section 5.2, Theorem 1.2 implies that the rescaled energies

$$ (1.6) \qquad \mathcal{E}_{\varepsilon}(\mu) = \int_{\mathbb{R}^N} f(\varepsilon^N u, \varepsilon^{N+1} \nabla u) \varepsilon^{-N} = \int_{\mathbb{R}^N} \varepsilon^{N+2} |\nabla u|^2 + \varepsilon^{-N} c(\varepsilon^N u). $$

<footnote>(1) Notice that if any of our assumptions is satisfied for a Lagrangian defined for $u \in \mathbb{R}_+$, then it holds also for the Lagrangian extended to $\mathbb{R}$ in this way.</footnote>
Γ-converge to $M^H$, leading to an elliptic approximation of any concave $H$-mass in dimension one. In dimension $N \ge 2$, we will show that $H_f$ must be concave on $[0, +\infty)$, and strictly concave after the possible initial interval where it is linear (see Proposition 2.8); however, we have no solution to the inverse problem, consisting in characterizing the class of attainable minimal cost functions $H = H_f$ for Lagrangians $f$ satisfying our assumptions.

*Homogeneous H-masses in any dimension.* — We consider the functional given for $\mu = u \mathcal{L}^N$ by

$$ (1.7) \qquad \mathcal{E}_f(\mu) = \int_{\mathbb{R}^N} f(u, \nabla u) = \int_{\mathbb{R}^N} |\nabla u|^p + u^s, \quad p > 1, s \in (-p', 1]. $$

Then, the rescaled energies

$$ \mathcal{E}_{\varepsilon}(\mu) = \int_{\mathbb{R}^N} f(\varepsilon^N u, \varepsilon^{N+1} \nabla u) \varepsilon^{-N} = \int_{\mathbb{R}^N} \varepsilon^{pN+p-N} |\nabla u|^p + \varepsilon^{-(1-s)N} u^s $$

Γ-converge to a non-trivial multiple of some $\alpha$-mass $M^\alpha := M^{t\to t^\alpha}$ where the exponent $\alpha = (1-s/p+s/N)(1-s/p+1/N)^{-1}$ ranges over $(1-2/N+1, 1]$ when $(s,p)$ varies in its range and $N \ge 1$. More details are given in Section 5.3.

*Space-inhomogeneous H-masses.* — We may consider for example functionals given, for $\mu = u \mathcal{L}^N$, by

$$ \mathcal{E}_f(\mu) = \int_{\mathbb{R}^N} a(x)g(u(x), \nabla u(x)) \, dx, $$

where $a: \mathbb{R}^N \to \mathbb{R}$ is a continuous function valued in $[M^{-1}, M]$ for some $M \in (0, +\infty)$ and $g: \mathbb{R} \times \mathbb{R}^N \to [0, +\infty]$ satisfies our assumptions $(H_1)-(H_6)$, e.g. $g(u, \xi) = |\xi|^p + u^s$ with $s \in (-p', 1]$ as above. In the latter case we obtain Γ-convergence towards a space-inhomogeneous $\alpha$-mass for some $\alpha \in (0, 1]$ given by

$$ M^{H_f}(\mu) = \begin{cases} C \sum_{1 \le i \le \ell} a(x_i) m_i^\alpha & \text{if } \mu = \sum_{i=1}^\ell m_i \delta_{x_i}, \\ +\infty & \text{otherwise,} \end{cases} $$

for some constant $C \in (0, +\infty)$.

*Cahn-Hilliard approximations of droplets models.* — Following the works of [BDS96, Dub98], we consider the functionals

$$ (1.8) \qquad W_{\varepsilon}(u) = \int_{\mathbb{R}^N} \varepsilon^{-p} (W(u) + \varepsilon |\nabla u|^2), $$

where $W(t) \sim_{t \to +\infty} t^s$ for some exponent $s \in (-2, 1)$. As shown in Section 5.5, we way rewrite these functionals to fit our general framework, and recover known Γ-convergence results, under slightly more general assumptions, as stated in Theorem 5.1. The Γ-limit is a non-trivial multiple of the $\alpha$-mass with $\alpha = \frac{1-s/2+s/N}{1-s/2+1/N}$.
Γ-converge to $M^H$, leading to an elliptic approximation of any concave $H$-mass in dimension one. In dimension $N \ge 2$, we will show that $H_f$ must be concave on $[0, +\infty)$, and strictly concave after the possible initial interval where it is linear (see Proposition 2.8) ; however, we have no solution to the inverse problem, consisting in characterizing the class of attainable minimal cost functions $H = H_f$ for Lagrangians $f$ satisfying our assumptions.

*Homogeneous H-masses in any dimension.* — We consider the functional given for $\mu = u \mathcal{L}^N$ by

$$ (1.7) \qquad \mathcal{E}_f(\mu) = \int_{\mathbb{R}^N} f(u, \nabla u) = \int_{\mathbb{R}^N} |\nabla u|^p + u^s, \quad p > 1, s \in (-p', 1]. $$

Then, the rescaled energies

$$ \mathcal{E}_{\varepsilon}(\mu) = \int_{\mathbb{R}^N} f(\varepsilon^N u, \varepsilon^{N+1} \nabla u) \varepsilon^{-N} = \int_{\mathbb{R}^N} \varepsilon^{pN+p-N} |\nabla u|^p + \varepsilon^{-(1-s)N} u^s $$

Γ-converge to a non-trivial multiple of some $\alpha$-mass $M^\alpha := M^{t\to t^\alpha}$ where the exponent $\alpha = (1 - s/p + s/N)(1 - s/p + 1/N)^{-1}$ ranges over $(1 - 2/N + 1, 1]$ when $(s, p)$ varies in its range and $N \ge 1$. More details are given in Section 5.3.

*Space-inhomogeneous H-masses.* — We may consider for example functionals given, for $\mu = u \mathcal{L}^N$, by

$$ \mathcal{E}_f(\mu) = \int_{\mathbb{R}^N} a(x)g(u(x), \nabla u(x)) \, dx, $$

where $a : \mathbb{R}^N \to \mathbb{R}$ is a continuous function valued in $[M^{-1}, M]$ for some $M \in (0, +\infty)$ and $g : \mathbb{R} \times \mathbb{R}^N \to [0, +\infty]$ satisfies our assumptions $(H_1)-(H_6)$, e.g. $g(u, \xi) = |\xi|^p + u^s$ with $s \in (-p', 1]$ as above. In the latter case we obtain Γ-convergence towards a space-inhomogeneous $\alpha$-mass for some $\alpha \in (0, 1]$ given by

$$ M^{H_f}(\mu) = \begin{cases} C \sum_{1 \le i \le \ell} a(x_i) m_i^\alpha & \text{if } \mu = \sum_{i=1}^\ell m_i \delta_{x_i}, \\ +\infty & \text{otherwise,} \end{cases} $$

for some constant $C \in (0, +\infty)$.

*Cahn-Hilliard approximations of droplets models.* — Following the works of [BDS96, Dub98], we consider the functionals

$$ (1.8) \qquad W_{\varepsilon}(u) = \int_{\mathbb{R}^N} \varepsilon^{-p} (W(u) + \varepsilon |\nabla u|^2), $$

where $W(t) \sim_{t \to +\infty} t^s$ for some exponent $s \in (-2, 1)$. As shown in Section 5.5, we way rewrite these functionals to fit our general framework, and recover known Γ-convergence results, under slightly more general assumptions, as stated in Theorem 5.1. The Γ-limit is a non-trivial multiple of the $\alpha$-mass with $\alpha = \frac{1-s/2+s/N}{1-s/2+1/N}$.
*Elliptic approximations of branched transport.* — The energy of branched transport (see [BCM09] for an account of the theory), in its Eulerian formulation, is an *H*-mass defined this time on vector measures *w* whose divergence is also a measure,

$$ (1.9) \qquad M_1^H(w) := \int_{\Sigma} H(x, \theta(x)) \, d\mathcal{H}^1(x) + \int_{\mathbb{R}^d} H'(x, 0^+) \, d|w^{\perp}|, $$

where $w = \theta\xi \cdot \mathcal{H}^1 \llcorner \Sigma + w^{\perp}$ is the decomposition of $w$ into its 1-rectifiable and 1-diffuse parts (see Section 5.4 for more details). An elliptic approximation of Modica-Mortola type has been introduced in [OS11] for $H(m) = m^{\alpha}, \alpha \in (0, 1)$, and their $\Gamma$-convergence result in dimension $d = 2$ has been extended to any dimension in [Mon15] by a slicing method which relates the energy of $w$ to the energy of its slicings. The same slicing method, together with Theorem 1.2, would allow to prove the $\Gamma$-convergence of the functionals

$$ (1.10) \qquad \mathcal{E}_{\varepsilon}(w) = \begin{cases} \int_{\mathbb{R}^d} f_{\varepsilon}(x, \varepsilon^{d-1}|v|(x), \varepsilon^d|\nabla v|(x)) \varepsilon^{1-d} \, dx & \text{if } w = v \mathcal{L}^d, v \in W_{\text{loc}}^{1,1}(\mathbb{R}^d, \mathbb{R}^d), \\ +\infty & \text{otherwise,} \end{cases} $$

toward $M_1^{H_f}$ for Lagrangians $f_{\varepsilon} \to f$ satisfying $(H_1)-(H_8)$, thus covering a wide range of concave $H$-masses over vector measures with divergence.

## 1.4. STRUCTURE OF THE PAPER.
In Section 2, we prove the concavity of the minimal cost function $H_f$ with respect to the mass variable $m$ in full generality (Theorem 1.1), we establish useful properties of general $H$-masses, and we identify the slope at the origin of $H_f$ in terms of $f$ under our assumption (Proposition 2.5 and Proposition 2.6). In Section 3, we apply a concentration-compactness principle to provide a profile decomposition theorem for sequences of positive measures (Theorem 3.2), which is used to obtain our main lower bound for the energy $\mathcal{E}_f$ (Proposition 3.10) and also yields an existence criterion for profiles with minimal energy under a mass constraint (Proposition 3.12). Section 4 is dedicated to proving lower and upper bounds on the rescaled energies $\mathcal{E}_{\varepsilon}$ (Proposition 4.1 and Proposition 4.2) that imply in particular our main $\Gamma$-convergence result (Theorem 1.2). Last of all, in Section 5, we provide counterexamples and several examples of energy functionals that fall into our framework, as summarized in the previous section.

## 2. MINIMAL COST FUNCTION AND $H$-MASS

In this section, we study the properties of general $H$-masses, of costs $H_f$ associated with general Lagrangians $f$, and we relate the slope of $H_f$ at $m = 0$ to that of $f$ at $(u, \xi) = (0, 0)$ in the variable $u$, under particular conditions.

### 2.1. CONCAVITY AND LOWER SEMicontINUITY OF THE MINIMAL COST FUNCTION.
Before proving Theorem 1.1, let us note that it covers the case where we have a constraint $(u, \nabla u) \in A$, where $A \subseteq \mathbb{R} \times \mathbb{R}^N$ is Borel measurable, by considering Lagrangians $f$ taking infinite values.
*Elliptic approximations of branched transport.* — The energy of branched transport (see [BCM09] for an account of the theory), in its Eulerian formulation, is an *H*-mass defined this time on vector measures *w* whose divergence is also a measure,

$$ (1.9) \qquad \mathbf{M}_1^H(w) := \int_{\Sigma} H(x, \theta(x)) \, d\mathcal{H}^1(x) + \int_{\mathbb{R}^d} H'(x, 0^+) \, d|w^{\perp}|, $$

where $w = \theta\xi \cdot \mathcal{H}^1 \llcorner \Sigma + w^{\perp}$ is the decomposition of $w$ into its 1-rectifiable and 1-diffuse parts (see Section 5.4 for more details). An elliptic approximation of Modica-Mortola type has been introduced in [OS11] for $H(m) = m^{\alpha}, \alpha \in (0,1)$, and their $\Gamma$-convergence result in dimension $d=2$ has been extended to any dimension in [Mon15] by a slicing method which relates the energy of $w$ to the energy of its slicings. The same slicing method, together with Theorem 1.2, would allow to prove the $\Gamma$-convergence of the functionals

(1.10)

$$ \mathcal{E}_{\varepsilon}(w) = \begin{cases} \int_{\mathbb{R}^d} f_{\varepsilon}(x, \varepsilon^{d-1}|v|(x), \varepsilon^d|\nabla v|(x)) \varepsilon^{1-d} \, dx & \text{if } w = v \mathcal{L}^d, v \in W_{\text{loc}}^{1,1}(\mathbb{R}^d, \mathbb{R}^d), \\ +\infty & \text{otherwise,} \end{cases} $$

toward $\mathbf{M}_1^{H_f}$ for Lagrangians $f_{\varepsilon} \to f$ satisfying $(H_1)-(H_8)$, thus covering a wide range of concave $H$-masses over vector measures with divergence.

## 1.4. STRUCTURE OF THE PAPER.
In Section 2, we prove the concavity of the minimal cost function $H_f$ with respect to the mass variable $m$ in full generality (Theorem 1.1), we establish useful properties of general $H$-masses, and we identify the slope at the origin of $H_f$ in terms of $f$ under our assumption (Proposition 2.5 and Proposition 2.6). In Section 3, we apply a concentration-compactness principle to provide a profile decomposition theorem for sequences of positive measures (Theorem 3.2), which is used to obtain our main lower bound for the energy $\mathcal{E}_f$ (Proposition 3.10) and also yields an existence criterion for profiles with minimal energy under a mass constraint (Proposition 3.12). Section 4 is dedicated to proving lower and upper bounds on the rescaled energies $\mathcal{E}_{\varepsilon}$ (Proposition 4.1 and Proposition 4.2) that imply in particular our main $\Gamma$-convergence result (Theorem 1.2). Last of all, in Section 5, we provide counterexamples and several examples of energy functionals that fall into our framework, as summarized in the previous section.

## 2. MINIMAL COST FUNCTION AND $H$-MASS

In this section, we study the properties of general $H$-masses, of costs $H_f$ associated with general Lagrangians $f$, and we relate the slope of $H_f$ at $m=0$ to that of $f$ at $(u, \xi) = (0,0)$ in the variable $u$, under particular conditions.

### 2.1. CONCAVITY AND LOWER SEMicontINUITY OF THE MINIMAL COST FUNCTION.
Before proving Theorem 1.1, let us note that it covers the case where we have a constraint $(u, \nabla u) \in A$, where $A \subseteq \mathbb{R} \times \mathbb{R}^N$ is Borel measurable, by considering Lagrangians $f$ taking infinite values.
*Proof of Theorem 1.1.* — Let us first assume that $f(u, \xi) = +\infty$ when $u < 0$, so as to restrict ourselves to non-negative functions. We first prove that $H_f$ is concave on $(0, +\infty)$. Let $m > 0$ and $u \in W_{\text{loc}}^{1,1} \cap L^1(\mathbb{R}^N, \mathbb{R}_+)$ such that $\int_{\mathbb{R}^N} u = m$. We pick a non-zero vector $v \in \mathbb{R}^N$ and for every $t \in \mathbb{R}$, we set $u^t(\cdot) = u(\cdot + tv)$ and

$$u \wedge u^t(\cdot) = \min\{u(\cdot), u^t(\cdot)\}, \quad u \vee u^t(\cdot) = \max\{u(\cdot), u^t(\cdot)\}.$$

We have $u \wedge u^t + u \vee u^t = u + u^t$. Hence

$$(2.1) \qquad \int_{\mathbb{R}^N} u \wedge u^t + \int_{\mathbb{R}^N} u \vee u^t = 2 \int_{\mathbb{R}^N} u = 2m.$$

Moreover, it is standard that $u \wedge u^t = u - (u^t - u)_- \in W_{\text{loc}}^{1,1}(\mathbb{R}^N)$ with $\nabla(u \wedge u^t) = \nabla u$ a.e. in $\{u \le u^t\}$ and $\nabla(u \wedge u^t) = \nabla u^t$ a.e. in $\{u > u^t\}$. Since $u \vee u^t = u + u^t - u \wedge u^t$, we have similar identities for $u \vee u^t$, and we obtain

$$(2.2) \qquad \mathcal{E}_f(u \wedge u^t) + \mathcal{E}_f(u \vee u^t) = \mathcal{E}_f(u) + \mathcal{E}_f(u^t) = 2\mathcal{E}_f(u).$$

Let $M: t \mapsto \int_{\mathbb{R}^N} u \wedge u^t$. In view of (2.1), (2.2), and by definition of $H$, we have proved

$$(2.3) \qquad H_f(M(t)) + H_f(2m - M(t)) \le 2\mathcal{E}_f(u).$$

By continuity of translations in $L^1$ and since the map $(x, y) \mapsto x \wedge y$ is Lipschitz on $\mathbb{R}^2$, we have that $M$ is continuous on $\mathbb{R}$ with $M(0) = m$. Moreover $\lim_{t \to +\infty} M(t) = 0$ by dominated convergence. So, by the intermediate value theorem $M(\mathbb{R}) \supseteq (0, m]$. Hence, we have proved $H_f(\theta) + H_f(2m - \theta) \le 2\mathcal{E}_f(u)$ for every $\theta \in (0, m]$. Taking the infimum over $u$ such that $\int_{\mathbb{R}^N} u = m$, we obtain

$$\frac{H_f(\theta) + H_f(2m - \theta)}{2} \le H_f(m), \quad \forall \theta \in (0, m],$$

that is, $H_f$ is midpoint concave on $(0, +\infty)$. Since $H_f$ is also bounded below (by 0), we can deduce that $H_f$ is concave $(0, +\infty)$ (see [RV73, §72]), and since $H_f \ge 0$, either $H_f$ is identically $+\infty$ on $(0, +\infty)$, or it is finite everywhere, continuous, concave and non-decreasing on $(0, +\infty)$.

We now justify that if $H_f(m) < +\infty$ for some $m > 0$ and $f(0, 0) = 0$, then $\lim_{m \to 0^+} H_f(m) = 0 = H_f(0)$. Let $u \in W_{\text{loc}}^{1,1}(\mathbb{R}^N, \mathbb{R}_+)$ such that $\int_{\mathbb{R}^N} u = m > 0$ and $\mathcal{E}_f(u) < +\infty$, and set

$$t_* := \sup\{t \ge 0 : M(t) > 0\} \in [0, +\infty], \quad \text{where} \quad M(t) = \int_{\mathbb{R}^N} u \wedge u^t.$$

Since $M$ is continuous with $M(0) = \int_{\mathbb{R}^N} u > 0$ and $\lim_{t \to +\infty} M(t) = 0$ as seen above, we have that $t_* \in (0, +\infty]$, $\lim_{t \to t_*} M(t) = 0$ and $M(t)$ does not vanish identically near $t_*$. Moreover, if $t_* = +\infty$, since $u^t \to 0$ locally in measure, by dominated convergence,

$$\limsup_{m \to 0^+} H_f(m) \le \limsup_{t \to +\infty} \mathcal{E}_f(u \wedge u^t) = \limsup_{t \to +\infty} \int_{\{u < u^t\}} f(u, \nabla u) + \int_{\{u^{-t} \ge u\}} f(u, \nabla u) \\ \le 2f(0, 0)|\{u = 0\}| = 0.$$
*Proof of Theorem 1.1.* — Let us first assume that $f(u, \xi) = +\infty$ when $u < 0$, so as to restrict ourselves to non-negative functions. We first prove that $H_f$ is concave on $(0, +\infty)$. Let $m > 0$ and $u \in W_{\text{loc}}^{1,1} \cap L^1(\mathbb{R}^N, \mathbb{R}_+)$ such that $\int_{\mathbb{R}^N} u = m$. We pick a non-zero vector $v \in \mathbb{R}^N$ and for every $t \in \mathbb{R}$, we set $u^t(\cdot) = u(\cdot + tv)$ and

$$u \wedge u^t(\cdot) = \min\{u(\cdot), u^t(\cdot)\}, \quad u \vee u^t(\cdot) = \max\{u(\cdot), u^t(\cdot)\}.$$

We have $u \wedge u^t + u \vee u^t = u + u^t$. Hence

$$(2.1) \qquad \int_{\mathbb{R}^N} u \wedge u^t + \int_{\mathbb{R}^N} u \vee u^t = 2 \int_{\mathbb{R}^N} u = 2m.$$

Moreover, it is standard that $u \wedge u^t = u - (u^t - u)_- \in W_{\text{loc}}^{1,1}(\mathbb{R}^N)$ with $\nabla(u \wedge u^t) = \nabla u$ a.e. in $\{u \le u^t\}$ and $\nabla(u \wedge u^t) = \nabla u^t$ a.e. in $\{u > u^t\}$. Since $u \vee u^t = u + u^t - u \wedge u^t$, we have similar identities for $u \vee u^t$, and we obtain

$$(2.2) \qquad \mathcal{E}_f(u \wedge u^t) + \mathcal{E}_f(u \vee u^t) = \mathcal{E}_f(u) + \mathcal{E}_f(u^t) = 2\mathcal{E}_f(u).$$

Let $M: t \mapsto \int_{\mathbb{R}^N} u \wedge u^t$. In view of (2.1), (2.2), and by definition of $H$, we have proved

$$(2.3) \qquad H_f(M(t)) + H_f(2m - M(t)) \le 2\mathcal{E}_f(u).$$

By continuity of translations in $L^1$ and since the map $(x, y) \mapsto x \wedge y$ is Lipschitz on $\mathbb{R}^2$, we have that $M$ is continuous on $\mathbb{R}$ with $M(0) = m$. Moreover $\lim_{t \to +\infty} M(t) = 0$ by dominated convergence. So, by the intermediate value theorem $M(\mathbb{R}) \supseteq (0, m]$. Hence, we have proved $H_f(\theta) + H_f(2m - \theta) \le 2\mathcal{E}_f(u)$ for every $\theta \in (0, m]$. Taking the infimum over $u$ such that $\int_{\mathbb{R}^N} u = m$, we obtain

$$\frac{H_f(\theta) + H_f(2m - \theta)}{2} \le H_f(m), \quad \forall \theta \in (0, m],$$

that is, $H_f$ is midpoint concave on $(0, +\infty)$. Since $H_f$ is also bounded below (by 0), we can deduce that $H_f$ is concave $(0, +\infty)$ (see [RV73, §72]), and since $H_f \ge 0$, either $H_f$ is identically $+\infty$ on $(0, +\infty)$, or it is finite everywhere, continuous, concave and non-decreasing on $(0, +\infty)$.

We now justify that if $H_f(m) < +\infty$ for some $m > 0$ and $f(0, 0) = 0$, then $\lim_{m \to 0^+} H_f(m) = 0 = H_f(0)$. Let $u \in W_{\text{loc}}^{1,1}(\mathbb{R}^N, \mathbb{R}_+)$ such that $\int_{\mathbb{R}^N} u = m > 0$ and $\mathcal{E}_f(u) < +\infty$, and set

$$t_* := \sup\{t \ge 0 : M(t) > 0\} \in [0, +\infty], \quad \text{where} \quad M(t) = \int_{\mathbb{R}^N} u \wedge u^t.$$

Since $M$ is continuous with $M(0) = \int_{\mathbb{R}^N} u > 0$ and $\lim_{t \to +\infty} M(t) = 0$ as seen above, we have that $t_* \in (0, +\infty]$, $\lim_{t \to t_*} M(t) = 0$ and $M(t)$ does not vanish identically near $t_*$. Moreover, if $t_* = +\infty$, since $u^t \to 0$ locally in measure, by dominated convergence,

$$\limsup_{m \to 0^+} H_f(m) \le \limsup_{t \to +\infty} \mathcal{E}_f(u \wedge u^t) = \limsup_{t \to +\infty} \int_{\{u < u^t\}} f(u, \nabla u) + \int_{\{u^{-t} \ge u\}} f(u, \nabla u) \\ \le 2f(0, 0)|\{u = 0\}| = 0.$$
If $t_* < +\infty$, we have $u \wedge u^{t_*} = 0$ a.e. and $u^t \to u^{t_*}$ locally in measure as $t \to t_*$ by continuity of translation in $L^1$. Thus using dominated convergence again,

$$
\begin{align*}
\limsup_{m \to 0^+} H_f(m) \le \limsup_{t \to (t_*)^-} \mathcal{E}_f(u \wedge u^t) &= \limsup_{t \to (t_*)^-} \int_{\{u < u^t\}} f(u, \nabla u) + \int_{\{u^{-t} \ge u\}} f(u, \nabla u) \\
&= \int_{\{u < u^{t_*}\}} f(u, \nabla u) + \int_{\{u^{-t_*} \ge u\}} f(u, \nabla u) \\
&= \mathcal{E}_f(u \wedge u^{t_*}) = f(0, 0) \times (+\infty) = 0.
\end{align*}
$$

Finally, we treat the general case (without assuming that $f(u, \xi) = +\infty$ if $u < 0$) For this, notice that for every $u \in W_{\text{loc}}^{1,1} \cap L^1(\mathbb{R}^N)$ such that $\int_{\mathbb{R}^N} u = m \ge 0$, denoting by $u_+$ the positive part of $u$ and $m_+ = \int_{\mathbb{R}^N} u_+$, since $f(0, 0) = 0$ we have $\int_{\mathbb{R}^N} f(u, \nabla u) \ge \int_{\mathbb{R}^N} f(u_+, \nabla u_+) \ge H_g(m_+)$, where $g$ is defined by $g(u, \xi) = f(u, \xi)$ if $u \ge 0$ and is set to $+\infty$ if $u < 0$. Since $H_g$ is non-decreasing and $m_+ \ge m$, it yields $H_f(m) \ge H_g(m_+) \ge H_g(m)$. The another inequality $H_f(m) \le H_g(m)$ being trivial, $H_f = H_g$ on $\mathbb{R}_+$.
$\square$

REMARK 2.1. — Let us remark two things about the proof.

First, we actually proved the concavity and monotonicity of

$$
m \mapsto H_f(m) := \inf \left\{ \int_{\mathbb{R}^N} f(u, \nabla u) : u \in W_{\text{loc}}^{1,1}(\mathbb{R}^N, \mathbb{R}_+), \int_{\mathbb{R}^N} u = m \right\}
$$

on $(0, +\infty)$ for $f$ Borel, without assuming $f(0, 0) = 0$.

Second, the end of the proof shows that under this extra assumption, minimizing among signed profiles or non-negative (resp. non-positive) profiles is equivalent when $m \ge 0$ (resp $m \le 0$).

2.2. DEFINITION AND RELAXATION OF THE H-MASS

**DEFINITION 2.2.** — Let $H : \mathbb{R}^N \times \mathbb{R} \to [0, +\infty]$ be a Borel measurable function having left/right slopes at the origin defined for each $x \in \mathbb{R}^N$ by

$$
(2.4) \qquad H'(x, 0^{\pm}) := \lim_{m \to 0^{\pm}} \frac{H(x, m)}{|m|} \in [0, +\infty].
$$

For every finite signed Borel measure $\mu \in \mathcal{M}(\mathbb{R}^N)$, we set

$$
H(\mu) := H(\cdot, \mu(\{\cdot\})) \mathcal{H}^0 + H'(\cdot, 0^+) \mu_+^d + H'(\cdot, 0^-) \mu_-^d,
$$

where $\mu = \mu^a + \mu^d$ is the decomposition of $u$ into its atomic part $\mu^a$ and its diffuse (or non-atomic) part $\mu^d$, $\mu^d = \mu_+^d - \mu_-^d$ is the Jordan decomposition of $\mu^d$ into positive and negative parts.

The H-mass of $\mu$ is then defined as the total variation of $H(\mu)$, that is:

$$
\mathcal{M}^H(\mu) := \int_{\mathbb{R}^N} H(x, \mu(\{x\})) \, d\mathcal{H}^0(x) \\
\qquad + \int_{\mathbb{R}^N} H'(x, 0^+) \, d\mu_+^d(x) + \int_{\mathbb{R}^N} H'(x, 0^-) \, d\mu_-^d(x).
$$
If $t_* < +\infty$, we have $u \wedge u^{t_*} = 0$ a.e. and $u^t \to u^{t_*}$ locally in measure as $t \to t_*$ by continuity of translation in $L^1$. Thus using dominated convergence again,

$$
\begin{align*}
\limsup_{m \to 0^+} H_f(m) \le \limsup_{t \to (t_*)^-} \mathcal{E}_f(u \wedge u^t) &= \limsup_{t \to (t_*)^-} \int_{\{u < u^t\}} f(u, \nabla u) + \int_{\{u^{-t} \ge u\}} f(u, \nabla u) \\
&= \int_{\{u < u^{t_*}\}} f(u, \nabla u) + \int_{\{u^{-t_*} \ge u\}} f(u, \nabla u) \\
&= \mathcal{E}_f(u \wedge u^{t_*}) = f(0, 0) \times (+\infty) = 0.
\end{align*}
$$

Finally, we treat the general case (without assuming that $f(u, \xi) = +\infty$ if $u < 0$) For this, notice that for every $u \in W_{\text{loc}}^{1,1} \cap L^1(\mathbb{R}^N)$ such that $\int_{\mathbb{R}^N} u = m \ge 0$, denoting by $u_+$ the positive part of $u$ and $m_+ = \int_{\mathbb{R}^N} u_+$, since $f(0, 0) = 0$ we have $\int_{\mathbb{R}^N} f(u, \nabla u) \ge \int_{\mathbb{R}^N} f(u_+, \nabla u_+) \ge H_g(m_+)$, where $g$ is defined by $g(u, \xi) = f(u, \xi)$ if $u \ge 0$ and is set to $+\infty$ if $u < 0$. Since $H_g$ is non-decreasing and $m_+ \ge m$, it yields $H_f(m) \ge H_g(m_+) \ge H_g(m)$. The another inequality $H_f(m) \le H_g(m)$ being trivial, $H_f = H_g$ on $\mathbb{R}_+$.
$\square$

REMARK 2.1. — Let us remark two things about the proof.

First, we actually proved the concavity and monotonicity of

$$
m \mapsto H_f(m) := \inf \left\{ \int_{\mathbb{R}^N} f(u, \nabla u) : u \in W_{\text{loc}}^{1,1}(\mathbb{R}^N, \mathbb{R}_+) , \int_{\mathbb{R}^N} u = m \right\}
$$

on $(0, +\infty)$ for $f$ Borel, without assuming $f(0, 0) = 0$.

Second, the end of the proof shows that under this extra assumption, minimizing among signed profiles or non-negative (resp. non-positive) profiles is equivalent when $m \ge 0$ (resp $m \le 0$).

2.2. DEFINITION AND RELAXATION OF THE H-MASS

**DEFINITION 2.2.** — Let $H : \mathbb{R}^N \times \mathbb{R} \to [0, +\infty]$ be a Borel measurable function having left/right slopes at the origin defined for each $x \in \mathbb{R}^N$ by

$$
(2.4) \qquad H'(x, 0^{\pm}) := \lim_{m \to 0^{\pm}} \frac{H(x, m)}{|m|} \in [0, +\infty].
$$

For every finite signed Borel measure $\mu \in \mathcal{M}(\mathbb{R}^N)$, we set

$$
H(\mu) := H(\cdot, \mu(\{\cdot\})) \mathcal{H}^0 + H'(\cdot, 0^+) \mu_+^d + H'(\cdot, 0^-) \mu_-^d,
$$

where $\mu = \mu^a + \mu^d$ is the decomposition of $u$ into its atomic part $\mu^a$ and its diffuse (or non-atomic) part $\mu^d$, $\mu^d = \mu_+^d - \mu_-^d$ is the Jordan decomposition of $\mu^d$ into positive and negative parts.

The H-mass of $\mu$ is then defined as the total variation of $H(\mu)$, that is:

$$
\mathcal{M}^H(\mu) := \int_{\mathbb{R}^N} H(x, \mu(\{x\})) \, d\mathcal{H}^0(x) \\
\qquad + \int_{\mathbb{R}^N} H'(x, 0^+) \, d\mu_+^d(x) + \int_{\mathbb{R}^N} H'(x, 0^-) \, d\mu_-^d(x).
$$
The previous definitions and notations extend in the obvious way to the case of functions $H : \mathbb{R} \to [0, +\infty]$ with no space variable $x$, interpreted as functions independent from $x$.

$M^H$ is a natural spatially non-homogeneous extension (depending on the position $x$) of the $H$-mass of $k$-dimensional flat currents$^{(2)}$ from Geometric Measure Theory, introduced by [Fle66] (see also the more recent works [DPH03, CDRMS17]).

We say that $H : \mathbb{R}^N \times \mathbb{R} \to [0, +\infty]$ is mass-concave if $m \mapsto H(x, m)$ is concave on $(0, +\infty)$ and $(-\infty, 0)$ for every $x \in \mathbb{R}^N$. From [BB90], we have the following result:$^{(3)}$

**PROPOSITION 2.3** ([BB90, Th. 3.3]). — Assume that $H : \mathbb{R}^N \times \mathbb{R} \to [0, +\infty]$ is lower semicontinuous, mass-concave and $H(\cdot, 0) \equiv 0$. Then $M^H$ is sequentially lower semicontinuous on $\mathcal{M}(\mathbb{R}^N)$ for the weak topology.

From another work from the same authors [BB93, Th. 3.2], we know that under some further assumptions on $H$, $M^H$ is the relaxation for the weak topology of the functional

$$ M_{\text{atom}}^{H}(\mu) = \begin{cases} \sum_{i=1}^{k} H(x_i, m_i) & \text{if } \mu = \sum_{i=1}^{k} m_i \delta_{x_i} \text{ with } k \in \mathbb{N}^{*}, x_i \in \mathbb{R}^{N}, m_i \in \mathbb{R}, \\ +\infty & \text{otherwise.} \end{cases} $$

We need a slightly different result,$^{(4)}$ namely that for any function $H : \mathbb{R}^N \times \mathbb{R} \to [0, +\infty]$ satisfying all the assumptions of Proposition 2.3 except the lower semicontinuity, the relaxation of $M_{\text{atom}}^H$ for the narrow sequential convergence is $M_{\text{lsc}}^{H}$, where $H_{\text{lsc}}$ is the lower semicontinuous envelope of $H$, which can be expressed as

$$ (2.5) \qquad H_{\text{lsc}}(x, m) = \sup\{G(x, m) : G \le H \text{ with } G \text{ lower semicontinuous}\}. $$

It is worth noticing that if $H(\cdot, 0) \equiv 0$ and $H$ is mass-concave, then these properties hold also for $H_{\text{lsc}}$.

**PROPOSITION 2.4.** — Let $H : \mathbb{R}^N \times \mathbb{R} \to [0, +\infty]$ be a function which is mass-concave and such that $H(\cdot, 0) \equiv 0$. Then, the sequentially lower semicontinuous envelope of $M_{\text{atom}}^H$ in the narrow topology of $\mathcal{M}(\mathbb{R}^N)$ is given by $M_{\text{lsc}}^H$, namely we have:

$$ (2.6) \qquad M_{\text{lsc}}^H = \sup\{F : F \le M_{\text{atom}}^H, F \text{ sequentially narrowly l.s.c. on } \mathcal{M}(\mathbb{R}^N)\}. $$

We point out that for a general $H$, for $M^H$ to be sequentially lower semicontinuous (for the narrow topology) it is necessary that $H$ is lsc on $\mathbb{R}^N \times (0, +\infty)$. However, neither the subadditivity of $H$ in $m$ nor its lower semicontinuity on $\mathbb{R}^N \times \mathbb{R}_+$ are necessary. Indeed, $M^H$ is sequentially lower semicontinuous if for instance $H(x, m) = +\infty$ when $x \ne 0, m > 0$, $H(x, 0) = 0$ when $x \ne 0$ and $H(0, \cdot)$ is any lower semicontinuous

<footnote>$^{(2)}$In the case $k=0$, since signed measures are merely 0-currents with finite mass.</footnote>

<footnote>$^{(3)}$In the notation of this paper, we take $\mu = 0$ and $f(x, s) = |s|^2$; we have $\varphi_f(x, 0) = 0$ and $\varphi_f(x, s) = +\infty$ if $s \ne 0$.</footnote>

<footnote>$^{(4)}$In [BB93, Th. 3.2], $H$ is assumed to be lower semicontinuous and the authors make a further coercivity assumption (assumption (3.5) in the paper) that we want to avoid.</footnote>
The previous definitions and notations extend in the obvious way to the case of functions $H : \mathbb{R} \to [0, +\infty]$ with no space variable $x$, interpreted as functions independent from $x$.

$M^H$ is a natural spatially non-homogeneous extension (depending on the position $x$) of the $H$-mass of $k$-dimensional flat currents$^{(2)}$ from Geometric Measure Theory, introduced by [Fle66] (see also the more recent works [DPH03, CDRMS17]).

We say that $H : \mathbb{R}^N \times \mathbb{R} \to [0, +\infty]$ is mass-concave if $m \mapsto H(x, m)$ is concave on $(0, +\infty)$ and $(-\infty, 0)$ for every $x \in \mathbb{R}^N$. From [BB90], we have the following result:$^{(3)}$

**PROPOSITION 2.3** ([BB90, Th. 3.3]). — Assume that $H : \mathbb{R}^N \times \mathbb{R} \to [0, +\infty]$ is lower semicontinuous, mass-concave and $H(\cdot, 0) \equiv 0$. Then $M^H$ is sequentially lower semicontinuous on $\mathcal{M}(\mathbb{R}^N)$ for the weak topology.

From another work from the same authors [BB93, Th. 3.2], we know that under some further assumptions on $H$, $M^H$ is the relaxation for the weak topology of the functional

$$ M_{\text{atom}}^{H}(\mu) = \begin{cases} \sum_{i=1}^{k} H(x_i, m_i) & \text{if } \mu = \sum_{i=1}^{k} m_i \delta_{x_i} \text{ with } k \in \mathbb{N}^{*}, x_i \in \mathbb{R}^{N}, m_i \in \mathbb{R}, \\ +\infty & \text{otherwise.} \end{cases} $$

We need a slightly different result,$^{(4)}$ namely that for any function $H : \mathbb{R}^N \times \mathbb{R} \to [0, +\infty]$ satisfying all the assumptions of Proposition 2.3 except the lower semicontinuity, the relaxation of $M_{\text{atom}}^H$ for the narrow sequential convergence is $M_{\text{lsc}}^{H}$, where $H_{\text{lsc}}$ is the lower semicontinuous envelope of $H$, which can be expressed as

$$ (2.5) \qquad H_{\text{lsc}}(x, m) = \sup\{G(x, m) : G \le H \text{ with } G \text{ lower semicontinuous}\}. $$

It is worth noticing that if $H(\cdot, 0) \equiv 0$ and $H$ is mass-concave, then these properties hold also for $H_{\text{lsc}}$.

**PROPOSITION 2.4.** — Let $H : \mathbb{R}^N \times \mathbb{R} \to [0, +\infty]$ be a function which is mass-concave and such that $H(\cdot, 0) \equiv 0$. Then, the sequentially lower semicontinuous envelope of $M_{\text{atom}}^H$ in the narrow topology of $\mathcal{M}(\mathbb{R}^N)$ is given by $M_{\text{lsc}}^H$, namely we have:

$$ (2.6) \qquad M_{\text{lsc}}^H = \sup\{F : F \le M_{\text{atom}}^H, F \text{ sequentially narrowly l.s.c. on } \mathcal{M}(\mathbb{R}^N)\}. $$

We point out that for a general $H$, for $M^H$ to be sequentially lower semicontinuous (for the narrow topology) it is necessary that $H$ is lsc on $\mathbb{R}^N \times (0, +\infty)$. However, neither the subadditivity of $H$ in $m$ nor its lower semicontinuity on $\mathbb{R}^N \times \mathbb{R}_+$ are necessary. Indeed, $M^H$ is sequentially lower semicontinuous if for instance $H(x, m) = +\infty$ when $x \ne 0, m > 0$, $H(x, 0) = 0$ when $x \ne 0$ and $H(0, \cdot)$ is any lower semicontinuous

<footnote>$^{(2)}$In the case $k=0$, since signed measures are merely 0-currents with finite mass.</footnote>

<footnote>$^{(3)}$In the notation of this paper, we take $\mu = 0$ and $f(x, s) = |s|^2$; we have $\varphi_f(x, 0) = 0$ and $\varphi_f(x, s) = +\infty$ if $s \ne 0$.</footnote>

<footnote>$^{(4)}$In [BB93, Th. 3.2], $H$ is assumed to be lower semicontinuous and the authors make a further coercivity assumption (assumption (3.5) in the paper) that we want to avoid.</footnote>
function. Nevertheless the subadditivity in the mass $m$ and the lower semicontinuity would be necessary if $H$ did not depend on $x$.

**Proof of Proposition 2.4.** — Since $H_{\text{lsc}}$ is lower semicontinuous and mass-concave, we know from Proposition 2.3 that $M^{H_{\text{lsc}}}$ is sequentially lower semicontinuous in the weak topology hence also in the narrow topology of $\mathcal{M}(\mathbb{R}^N)$. Since $M^{H_{\text{lsc}}} \le M_{\text{atom}}^H$, we deduce that $M^{H_{\text{lsc}}}$ is lower or equal than the sequentially lower semicontinuous envelope of $M_{\text{atom}}^H$ in the narrow topology, i.e., the right hand side in (2.6), which we denote by $F: \mathcal{M}(\mathbb{R}^N) \to \mathbb{R}_+$. We shall see that $F \le M^{H_{\text{lsc}}}$.

We first prove that

$$ (2.7) \qquad F \le M_{\text{atom}}^{H_{\text{lsc}}}. $$

For this, we let $\mu = \sum_{i=1}^k m_i \delta_{x_i}$ be a finitely atomic positive measure and we let $\mu_n := \sum_{i=1}^k m_{i,n} \delta_{x_{i,n}}$ where for each $i \in \{1, \dots, k\}$, $(x_{i,n})_{n \in \mathbb{N}}$ is a sequence of points converging to $x_i$ and $(m_{i,n})_{n \in \mathbb{N}}$ is a sequence converging to $m_i$ such that $H_{\text{lsc}}(x_i, m_i) = \lim_{n \to \infty} H(x_{i,n}, m_{i,n})$. Then $(\mu_n)_{n \in \mathbb{N}}$ converges narrowly to $\mu$ and, by lower semicontinuity,

$$ \begin{aligned} F(\mu) &\le \liminf_{n \to \infty} F(\mu_n) \le \liminf_{n \to \infty} M_{\text{atom}}^H(\mu_n) \\ &= \lim_{n \to \infty} \sum_{i=1}^k H(x_{i,n}, m_{i,n}) = \sum_{i=1}^k H_{\text{lsc}}(x_i, m_i), \end{aligned} $$

so that $F(\mu) \le M_{\text{atom}}^{H_{\text{lsc}}}(\mu)$ as wanted.

We now prove that $F \le M^{H_{\text{lsc}}}$, Let $\mu \in \mathcal{M}(\mathbb{R}^N)$ and $\mu = \mu^a + \mu^d$ be the decomposition of $\mu$ into its atomic part $\mu^a = \sum_{i=1}^k m_i \delta_{x_i}$, with $k \in \mathbb{N} \cup \{+\infty\}$ (here, $k=0$ if there is no atom), and its diffuse part $\mu^d$, and let $\mu^d = \mu_+^d - \mu_-^d$ be the Jordan decomposition of $\mu^d$ into positive and negative parts. We then discretize $\mu_\pm^d$ by taking $n \in \mathbb{N}^*$, a partition $(Q_i^n)_{i \in \{1, \dots, (n2^n)^N\}}$ of $[-n, n)^N$ by means of cubes of the form $Q_i^n = c_i^n + 2^{-n}[-1, 1)^N$ with $c_i^n \in \mathbb{R}^N$, and we define

$$ \mu_n := \sum_{i=1}^{n \wedge k} m_i \delta_{x_i} + \sum_{i=1}^{(n2^n)^N} \mu_+^d(Q_i^n) \delta_{x_i^n} - \sum_{i=1}^{(n2^n)^N} \mu_-^d(Q_i^n) \delta_{y_i^n}, $$

where for each $i \in \{1, \dots, (n2^n)^N\}$, $x_i^n, y_i^n \in \bar{Q}_i^n$ are some points such that

$$ (2.8) \qquad H_{\text{lsc}}'(x_i^n, 0^+) = \inf_{x \in \bar{Q}_i^n} H_{\text{lsc}}'(x, 0^+), \quad H_{\text{lsc}}'(y_i^n, 0^-) = \inf_{x \in \bar{Q}_i^n} H_{\text{lsc}}'(x, 0^-). $$

Such points exist since $\bar{Q}_i^n$ is compact and since by concavity,

$$ (2.9) \qquad H_{\text{lsc}}'(x, 0^{\pm}) = \sup_{\pm m > 0} \frac{H_{\text{lsc}}(x, m)}{|m|}, $$

so that $H'_{\text{lsc}}(\cdot, 0^{\pm})$ are lower semicontinuous as suprema of lower semicontinuous functions.
function. Nevertheless the subadditivity in the mass $m$ and the lower semicontinuity would be necessary if $H$ did not depend on $x$.

*Proof of Proposition 2.4.* — Since $H_{\text{lsc}}$ is lower semicontinuous and mass-concave, we know from Proposition 2.3 that $M^{H_{\text{lsc}}}$ is sequentially lower semicontinuous in the weak topology hence also in the narrow topology of $\mathcal{M}(\mathbb{R}^N)$. Since $M^{H_{\text{lsc}}} \le M_{\text{atom}}^H$, we deduce that $M^{H_{\text{lsc}}}$ is lower or equal than the sequentially lower semicontinuous envelope of $M_{\text{atom}}^H$ in the narrow topology, i.e., the right hand side in (2.6), which we denote by $F: \mathcal{M}(\mathbb{R}^N) \to \mathbb{R}_+$. We shall see that $F \le M^{H_{\text{lsc}}}$.

We first prove that

$$ (2.7) \qquad F \le M_{\text{atom}}^{H_{\text{lsc}}}. $$

For this, we let $\mu = \sum_{i=1}^k m_i \delta_{x_i}$ be a finitely atomic positive measure and we let $\mu_n := \sum_{i=1}^k m_{i,n} \delta_{x_{i,n}}$ where for each $i \in \{1, \dots, k\}$, $(x_{i,n})_{n \in \mathbb{N}}$ is a sequence of points converging to $x_i$ and $(m_{i,n})_{n \in \mathbb{N}}$ is a sequence converging to $m_i$ such that $H_{\text{lsc}}(x_i, m_i) = \lim_{n \to \infty} H(x_{i,n}, m_{i,n})$. Then $(\mu_n)_{n \in \mathbb{N}}$ converges narrowly to $\mu$ and, by lower semicontinuity,

$$ \begin{aligned} F(\mu) &\le \liminf_{n \to \infty} F(\mu_n) \le \liminf_{n \to \infty} M_{\text{atom}}^H(\mu_n) \\ &= \lim_{n \to \infty} \sum_{i=1}^k H(x_{i,n}, m_{i,n}) = \sum_{i=1}^k H_{\text{lsc}}(x_i, m_i), \end{aligned} $$

so that $F(\mu) \le M_{\text{atom}}^{H_{\text{lsc}}}(\mu)$ as wanted.

We now prove that $F \le M^{H_{\text{lsc}}}$, Let $\mu \in \mathcal{M}(\mathbb{R}^N)$ and $\mu = \mu^a + \mu^d$ be the decomposition of $\mu$ into its atomic part $\mu^a = \sum_{i=1}^k m_i \delta_{x_i}$, with $k \in \mathbb{N} \cup \{+\infty\}$ (here, $k=0$ if there is no atom), and its diffuse part $\mu^d$, and let $\mu^d = \mu_+^d - \mu_-^d$ be the Jordan decomposition of $\mu^d$ into positive and negative parts. We then discretize $\mu_\pm^d$ by taking $n \in \mathbb{N}^*$, a partition $(Q_i^n)_{i \in \{1, \dots, (n2^n)^N\}}$ of $[-n, n)^N$ by means of cubes of the form $Q_i^n = c_i^n + 2^{-n}[-1, 1)^N$ with $c_i^n \in \mathbb{R}^N$, and we define

$$ \mu_n := \sum_{i=1}^{n \wedge k} m_i \delta_{x_i} + \sum_{i=1}^{(n2^n)^N} \mu_+^d(Q_i^n) \delta_{x_i^n} - \sum_{i=1}^{(n2^n)^N} \mu_-^d(Q_i^n) \delta_{y_i^n}, $$

where for each $i \in \{1, \dots, (n2^n)^N\}$, $x_i^n, y_i^n \in \bar{Q}_i^n$ are some points such that

$$ (2.8) \qquad H_{\text{lsc}}'(x_i^n, 0^+) = \inf_{x \in \bar{Q}_i^n} H_{\text{lsc}}'(x, 0^+), \quad H_{\text{lsc}}'(y_i^n, 0^-) = \inf_{x \in \bar{Q}_i^n} H_{\text{lsc}}'(x, 0^-). $$

Such points exist since $\bar{Q}_i^n$ is compact and since by concavity,

$$ (2.9) \qquad H_{\text{lsc}}'(x, 0^{\pm}) = \sup_{\pm m > 0} \frac{H_{\text{lsc}}(x, m)}{|m|}, $$

so that $H'_{\text{lsc}}(\cdot, 0^{\pm})$ are lower semicontinuous as suprema of lower semicontinuous functions.
The sequence $(\mu_n)_{n \in \mathbb{N}}$ converges narrowly to $\mu$. We deduce by lower semicontinuity of $F$,

$$
\begin{align*}
F(\mu) &\le \liminf_{n \to \infty} F(\mu_n) \stackrel{(2.7)}{\le} \liminf_{n \to \infty} M_{\text{atom}}^{\text{lsc}}(\mu_n) \\
&= \liminf_{n \to \infty} \sum_{i=1}^{n \wedge k} H_{\text{lsc}}(x_i, m_i) + \sum_{i=1}^{(n2^n)^N} \left( H_{\text{lsc}}(x_i^n, \mu_+^d(Q_i^n)) + H_{\text{lsc}}(y_i^n, -\mu_-^d(Q_i^n)) \right), \\
&\stackrel{(2.9)}{\le} \liminf_{n \to \infty} \sum_{i=1}^{n \wedge k} H_{\text{lsc}}(x_i, m_i) + \sum_{i=1}^{(n2^n)^N} H_{\text{lsc}}'(x_i^n, 0^+) \mu_+^d(Q_i^n) \\
&\qquad + \sum_{i=1}^{(n2^n)^N} H_{\text{lsc}}'(y_i^n, 0^-) \mu_-^d(Q_i^n) \\
&\stackrel{(2.8)}{\le} \liminf_{n \to \infty} \sum_{i=1}^{n \wedge k} H_{\text{lsc}}(x_i, m_i) + \sum_{i=1}^{(n2^n)^N} \int_{Q_i^n} H_{\text{lsc}}'(\cdot, 0^+) \, d\mu_+^d + \int_{Q_i^n} H_{\text{lsc}}'(\cdot, 0^-) \, d\mu_-^d \\
&= \sum_{i=1}^k H_{\text{lsc}}(x_i, m_i) + \int_{\mathbb{R}^N} H_{\text{lsc}}'(\cdot, 0^+) \, d\mu_+^d + \int_{\mathbb{R}^N} H_{\text{lsc}}'(\cdot, 0^-) \, d\mu_-^d \\
&= M^{H_{\text{lsc}}}(\mu),
\end{align*}
$$

where we have used monotone convergence in the last but one equality. $\square$

<figure><img src="image_2.png" /></figure>

**2.3. SLOPE AT THE ORIGIN OF THE MINIMAL COST FUNCTION.** — In this section we provide a technical assumption, that is simple in dimension $N=1$, on the Lagrangian $f$ and which implies the comparison condition on the slopes $(H_6)$.

(S) for every $x_0 \in \mathbb{R}^N$,

$$
(2.10) \quad f'_{-}(x_0, 0^{\pm}, 0) = \liminf_{(x,u,\xi) \to (x_0, 0^{\pm}, 0)} \frac{f(x, u, \xi)}{|u|} \ge \limsup_{u \to 0^{\pm}} \sup_{|\xi|=1} \frac{f(x_0, u, \rho(|u|)\xi)}{|u|},
$$

with $\rho \equiv 0$ if $N=1$ and for some $\rho \in \mathcal{C}((0,1], (0, +\infty))$ satisfying

$$
\int_0^1 \left( \int_y^1 \frac{dt}{\rho(t)} \right)^N dy < +\infty \quad \text{if } N \ge 2.
$$

**PROPOSITION 2.5.** — Let $f: \mathbb{R}_+ \times \mathbb{R}^N \to [0, +\infty]$ be a lower semicontinuous function such that $f(0, 0) = 0$, with $N \ge 2$. For every function $\rho \in \mathcal{C}((0, 1], (0, +\infty))$ such that

$$
(2.11) \qquad \int_0^1 \left( \int_y^1 \frac{dt}{\rho(t)} \right)^N dy < +\infty,
$$

the function $H_f$ defined in (1.3) satisfies

$$
(2.12) \quad \lim_{m \to 0^+} \frac{H_f(m)}{m} \le \limsup_{u \to 0^+} \sup_{\xi \in \mathbb{S}^{N-1}} \frac{f(u, \rho(u)\xi)}{u}.
$$
The sequence $(\mu_n)_{n \in \mathbb{N}}$ converges narrowly to $\mu$. We deduce by lower semicontinuity of $F$,

$$
\begin{align*}
F(\mu) &\le \liminf_{n \to \infty} F(\mu_n) \stackrel{(2.7)}{\le} \liminf_{n \to \infty} M_{\text{atom}}^{\text{lsc}}(\mu_n) \\
&= \liminf_{n \to \infty} \sum_{i=1}^{n \wedge k} H_{\text{lsc}}(x_i, m_i) + \sum_{i=1}^{(n2^n)^N} (H_{\text{lsc}}(x_i^n, \mu_+^d(Q_i^n)) + H_{\text{lsc}}(y_i^n, -\mu_-^d(Q_i^n))), \\
&\stackrel{(2.9)}{\le} \liminf_{n \to \infty} \sum_{i=1}^{n \wedge k} H_{\text{lsc}}(x_i, m_i) + \sum_{i=1}^{(n2^n)^N} H_{\text{lsc}}'(x_i^n, 0^+) \mu_+^d(Q_i^n) \\
&\qquad + \sum_{i=1}^{(n2^n)^N} H_{\text{lsc}}'(y_i^n, 0^-) \mu_-^d(Q_i^n) \\
&\stackrel{(2.8)}{\le} \liminf_{n \to \infty} \sum_{i=1}^{n \wedge k} H_{\text{lsc}}(x_i, m_i) + \sum_{i=1}^{(n2^n)^N} \int_{Q_i^n} H_{\text{lsc}}'(\cdot, 0^+) \, d\mu_+^d + \int_{Q_i^n} H_{\text{lsc}}'(\cdot, 0^-) \, d\mu_-^d \\
&= \sum_{i=1}^k H_{\text{lsc}}(x_i, m_i) + \int_{\mathbb{R}^N} H_{\text{lsc}}'(\cdot, 0^+) \, d\mu_+^d + \int_{\mathbb{R}^N} H_{\text{lsc}}'(\cdot, 0^-) \, d\mu_-^d \\
&= M^{H_{\text{lsc}}}(\mu),
\end{align*}
$$

where we have used monotone convergence in the last but one equality. $\square$

<figure><img src="image_4.png" /></figure>

2.3. SLOPE AT THE ORIGIN OF THE MINIMAL COST FUNCTION. — In this section we provide a technical assumption, that is simple in dimension $N=1$, on the Lagrangian $f$ and which implies the comparison condition on the slopes $(H_6)$.

(S) for every $x_0 \in \mathbb{R}^N$,

$$
(2.10) \quad f'_{-}(x_0, 0^{\pm}, 0) = \liminf_{(x,u,\xi) \to (x_0, 0^{\pm}, 0)} \frac{f(x, u, \xi)}{|u|} \ge \limsup_{u \to 0^{\pm}} \sup_{|\xi|=1} \frac{f(x_0, u, \rho(|u|)\xi)}{|u|},
$$

with $\rho \equiv 0$ if $N=1$ and for some $\rho \in \mathcal{C}((0,1], (0, +\infty))$ satisfying

$$
\int_0^1 \left( \int_y^1 \frac{dt}{\rho(t)} \right)^N dy < +\infty \quad \text{if } N \ge 2.
$$

PROPOSITION 2.5. — Let $f : \mathbb{R}_+ \times \mathbb{R}^N \to [0, +\infty]$ be a lower semicontinuous function such that $f(0, 0) = 0$, with $N \ge 2$. For every function $\rho \in \mathcal{C}((0, 1], (0, +\infty))$ such that

$$
(2.11) \qquad \int_0^1 \left( \int_y^1 \frac{dt}{\rho(t)} \right)^N dy < +\infty,
$$

the function $H_f$ defined in (1.3) satisfies

$$
(2.12) \quad \lim_{m \to 0^+} \frac{H_f(m)}{m} \le \limsup_{u \to 0^+} \sup_{\xi \in \mathbb{S}^{N-1}} \frac{f(u, \rho(u)\xi)}{u}.
$$
Proof. — For every $y \ge 0$, we let

$$F(y) = \int_y^1 \frac{dt}{\rho(t)} \in [0, +\infty].$$

The function $F$ is decreasing, and belongs to $\mathscr{C}^1((0, 1])$ and $L^N((0, 1])$ by assumption. We now consider the solution of the ODE $v'_\epsilon = -\rho(v_\epsilon)$, with $v_\epsilon(0) = \epsilon$, given by

$$v_{\epsilon}(r) = \begin{cases} F^{-1}(F(\epsilon) + r), & \text{if } 0 \le r < F(0) - F(\epsilon), \\ 0, & \text{if } r \ge F(0) - F(\epsilon), \end{cases}$$

with $F(0)$ possibly equal to $+\infty$. Notice that $v_\epsilon \in W_{\text{loc}}^{1,1}(\mathbb{R}_+)$ because it is non-increasing and bounded, hence it has finite total variation, and because it is of class $\mathscr{C}^1$ except possibly at $r_\epsilon := F(0) - F(\epsilon)$, where it has no jump. As a consequence the radial profile defined by $u_\epsilon(x) := v_\epsilon(|x|)$ belongs to $W_{\text{loc}}^{1,1}(\mathbb{R}^N)$ and we compute, using the change of variables $s = v_\epsilon(r)$ (i.e., $r = F(s) - F(\epsilon)$) and an integration by parts combined with monotone convergence.

$$\begin{align*} m_{\epsilon} &:= \int_{\mathbb{R}^N} u_{\epsilon} = |\mathbb{S}^{N-1}| \int_0^{\infty} v_{\epsilon}(r) r^{N-1} dr \\ &= -|\mathbb{S}^{N-1}| \int_0^{\epsilon} s(F(s) - F(\epsilon))^{N-1} F'(s) ds \\ &= |\mathbb{S}^{N-1}| \lim_{t \downarrow 0} \left( \int_t^{\epsilon} \frac{(F(s) - F(\epsilon))^N}{N} ds - \left[ s \frac{(F(s) - F(\epsilon))^N}{N} \right]_t^{\epsilon} \right) \\ &= |\mathbb{S}^{N-1}| \int_0^{\epsilon} \frac{(F(s) - F(\epsilon))^N}{N} ds \xrightarrow{\epsilon \to 0} 0. \end{align*}$$

The equality on the last line holds because $\lim_{t \to 0+} \int_t^\epsilon (F - F(\epsilon))^N < +\infty$ (since $F \in L^N((0, 1]))$, hence $\lim_{t \to 0} t(F(t) - F(\epsilon))^N$ exists by existence of the limit in the previous line, and it must be zero (again, because $F \in L^N((0, 1]))$).

Moreover, since $\sup_{[0, +\infty)} v_\epsilon = \epsilon$,

$$\mathcal{E}(u_{\epsilon}) = \int_{0}^{\infty} \int_{\mathbb{S}^{N-1}} f(v_{\epsilon}(r), v'_{\epsilon}(r)\xi) r^{N-1} d\mathcal{H}^{N-1}(\xi) dr \le m_{\epsilon} \sup_{u \le \epsilon, |\xi|=1} \frac{f(u, \rho(u)\xi)}{u}.$$

By assumption, we deduce that

$$\limsup_{m \to 0^+} \frac{H_f(m)}{m} \le \limsup_{\epsilon \to 0^+} \frac{\mathcal{E}(u_\epsilon)}{m_\epsilon} \le \limsup_{u \to 0^+} \sup_{\xi \in \mathbb{S}^{N-1}} \frac{f(u, \rho(u)\xi)}{u}. \quad \square$$

In dimension $N = 1$, we need no other assumption than $H_f < +\infty$, as stated below.

**PROPOSITION 2.6.** — Let $f: \mathbb{R}_+ \times \mathbb{R}^N \to [0, +\infty)$ be Borel measurable with $N = 1$. The minimal cost function $H_f$ is either identically infinite on $(0, +\infty)$, or it satisfies (2.12) with $\rho \equiv 0$, i.e.,

$$\lim_{m \to 0^+} \frac{H_f(m)}{m} \le \limsup_{u \to 0^+} \frac{f(u, 0)}{u}.$$
Proof. — For every $y \ge 0$, we let

$$F(y) = \int_{y}^{1} \frac{dt}{\rho(t)} \in [0, +\infty].$$

The function $F$ is decreasing, and belongs to $\mathscr{C}^1((0, 1])$ and $L^N((0, 1])$ by assumption. We now consider the solution of the ODE $v'_\varepsilon = -\rho(v_\varepsilon)$, with $v_\varepsilon(0) = \varepsilon$, given by

$$v_{\varepsilon}(r) = \begin{cases} F^{-1}(F(\varepsilon) + r), & \text{if } 0 \le r < F(0) - F(\varepsilon), \\ 0, & \text{if } r \ge F(0) - F(\varepsilon), \end{cases}$$

with $F(0)$ possibly equal to $+\infty$. Notice that $v_\varepsilon \in W_{\text{loc}}^{1,1}(\mathbb{R}_+)$ because it is non-increasing and bounded, hence it has finite total variation, and because it is of class $\mathscr{C}^1$ except possibly at $r_\varepsilon := F(0) - F(\varepsilon)$, where it has no jump. As a consequence the radial profile defined by $u_\varepsilon(x) := v_\varepsilon(|x|)$ belongs to $W_{\text{loc}}^{1,1}(\mathbb{R}^N)$ and we compute, using the change of variables $s = v_\varepsilon(r)$ (i.e., $r = F(s) - F(\varepsilon)$) and an integration by parts combined with monotone convergence.

$$\begin{align*} m_{\varepsilon} &:= \int_{\mathbb{R}^N} u_{\varepsilon} = |\mathbb{S}^{N-1}| \int_0^{\infty} v_{\varepsilon}(r) r^{N-1} dr \\ &= -|\mathbb{S}^{N-1}| \int_0^{\varepsilon} s(F(s) - F(\varepsilon))^{N-1} F'(s) ds \\ &= |\mathbb{S}^{N-1}| \lim_{t \downarrow 0} \left( \int_t^{\varepsilon} \frac{(F(s) - F(\varepsilon))^N}{N} ds - \left[ s \frac{(F(s) - F(\varepsilon))^N}{N} \right]_t^{\varepsilon} \right) \\ &= |\mathbb{S}^{N-1}| \int_0^{\varepsilon} \frac{(F(s) - F(\varepsilon))^N}{N} ds \xrightarrow{\varepsilon \to 0} 0. \end{align*}$$

The equality on the last line holds because $\lim_{t \to 0+} \int_t^\varepsilon (F - F(\varepsilon))^N < +\infty$ (since $F \in L^N((0, 1]))$, hence $\lim_{t \to 0} t(F(t) - F(\varepsilon))^N$ exists by existence of the limit in the previous line, and it must be zero (again, because $F \in L^N((0, 1]))$).

Moreover, since $\sup_{[0, +\infty)} v_\varepsilon = \varepsilon$,

$$\mathcal{E}(u_{\varepsilon}) = \int_{0}^{\infty} \int_{\mathbb{S}^{N-1}} f(v_{\varepsilon}(r), v'_{\varepsilon}(r)\xi) r^{N-1} d\mathcal{H}^{N-1}(\xi) dr \le m_{\varepsilon} \sup_{u \le \varepsilon, |\xi|=1} \frac{f(u, \rho(u)\xi)}{u}.$$

By assumption, we deduce that

$$\limsup_{m \to 0^+} \frac{H_f(m)}{m} \le \limsup_{\varepsilon \to 0^+} \frac{\mathcal{E}(u_\varepsilon)}{m_\varepsilon} \le \limsup_{u \to 0^+} \sup_{\xi \in \mathbb{S}^{N-1}} \frac{f(u, \rho(u)\xi)}{u}. \quad \square$$

In dimension $N = 1$, we need no other assumption than $H_f < +\infty$, as stated below.

**PROPOSITION 2.6.** — Let $f: \mathbb{R}_+ \times \mathbb{R}^N \to [0, +\infty)$ be Borel measurable with $N = 1$. The minimal cost function $H_f$ is either identically infinite on $(0, +\infty)$, or it satisfies (2.12) with $\rho \equiv 0$, i.e.,

$$\lim_{m \to 0^+} \frac{H_f(m)}{m} \le \limsup_{u \to 0^+} \frac{f(u, 0)}{u}.$$
Proof. — One can assume that there exists $u \in W_{\text{loc}}^{1,1}(\mathbb{R}, \mathbb{R}_+)$ with $0 < \int_{\mathbb{R}} u < +\infty$ and $\mathcal{E}(u) < +\infty$. In particular, up to changing the value of $u$ on a negligible set, $u$ is continuous on $\mathbb{R}$. Let $\varepsilon \in (0, \sup_{\mathbb{R}} u)$, set $A_{\varepsilon} := \{x : u(x) = \varepsilon\}$ which is non-empty by the intermediate value theorem and integrability of $u$, and define

$$ 
\begin{aligned}
a_{\varepsilon} &= \begin{cases} \inf A_{\varepsilon} & \text{if } \inf A_{\varepsilon} > -\infty, \\ \inf_{A_{\varepsilon}} \inf_{(-\infty, -\varepsilon^{-1}) \cap A_{\varepsilon}} \inf_{A_{\varepsilon}} & \text{if } \inf_{A_{\varepsilon}} \inf_{(-\infty, -\varepsilon^{-1}) \cap A_{\varepsilon}} < +\infty, \end{cases} \\ 
b_{\varepsilon} &= \begin{cases} \sup A_{\varepsilon} & \text{if } \sup A_{\varepsilon} < +\infty, \\ \sup_{A_{\varepsilon}} \sup_{(\varepsilon^{-1}, +\infty) \cap A_{\varepsilon}} \sup_{A_{\varepsilon}} & \text{if } \sup_{A_{\varepsilon}} \sup_{(\varepsilon^{-1}, +\infty) \cap A_{\varepsilon}} > -\infty. \end{cases}
\end{aligned}
$$

By continuity and integrability of $u$, $u(a_{\varepsilon}) = u(b_{\varepsilon}) = \varepsilon$ and

$$ \sup_{x \in \mathbb{R} \setminus [a_{\varepsilon}, b_{\varepsilon}]} u(x) \le \varepsilon \lor \sup_{|x| > \varepsilon^{-1}} u(x) \xrightarrow{\varepsilon \to 0} 0. $$

Moreover $a_{\varepsilon}, b_{\varepsilon}$ converge to points $-\infty \le a \le b \le +\infty$, hence $u = 0$ on $\mathbb{R} \setminus (a, b)$ and by dominated convergence, since $\nabla u = 0$ a.e. on $\{u = 0\}$,

$$ +\infty > \lim_{\varepsilon \to 0^+} \int_{\mathbb{R} \setminus [a_{\varepsilon}, b_{\varepsilon}]} u + f(u, \nabla u) = f(0, 0) \mathcal{L}(\mathbb{R} \setminus (a, b)). $$

Notice that this limit is necessary zero. Let $m > 0$. If $\varepsilon$ is small enough, then $\int_{\mathbb{R} \setminus [a_{\varepsilon}, b_{\varepsilon}]} u < m$ so that we can take $R_{\varepsilon} > 0$ such that $\varepsilon R_{\varepsilon} = m - \int_{\mathbb{R} \setminus [a_{\varepsilon}, b_{\varepsilon}]} u$. We then define

$$ u_{\varepsilon}(x) = \begin{cases} u(x) & \text{if } x \le a_{\varepsilon}, \\ \varepsilon & \text{if } a_{\varepsilon} < x < a_{\varepsilon} + R_{\varepsilon}, \\ u(b_{\varepsilon} + x - (a_{\varepsilon} + R_{\varepsilon})) & \text{if } x \ge a_{\varepsilon} + R_{\varepsilon}, \end{cases} $$

so that $\int_{\mathbb{R}} v_{\varepsilon} = m$. Moreover,

$$ \mathcal{E}(v_{\varepsilon}) = \mathcal{E}(u, \mathbb{R} \setminus [a_{\varepsilon}, b_{\varepsilon}]) + R_{\varepsilon}f(\varepsilon, 0). $$

Hence, as $R_{\varepsilon} = (m + o(1))/\varepsilon$ as $\varepsilon \to 0$,

$$ H_f(m) \le \limsup_{\varepsilon \to 0^+} \mathcal{E}(v_{\varepsilon}) = m \limsup_{\varepsilon \to 0^+} \frac{f(\varepsilon, 0)}{\varepsilon}. \quad \square $$

From Proposition 2.5 and Proposition 2.6 we obtain the following corollary.

COROLLARY 2.7. — Let $f: \mathbb{R} \times \mathbb{R}^N \to [0, +\infty]$ satisfy $(H_1)$, $(H_3)$ and (S). Assume in addition that $H_f < +\infty$ if $N = 1$. Then $(H_6)$ holds.

Proof. — If $N = 1$ we apply Proposition 2.6 to get

$$ \lim_{m \to 0^+} \frac{H_f(m)}{m} \le \limsup_{u \to 0^+} \frac{f(u, 0)}{u} \stackrel{(S)}{\le} \liminf_{m \to 0^+} \frac{f(u, 0)}{u} = f'_-(0^+, 0), $$

and if $N = 2$ taking a function $\rho$ as in (S) and applying Proposition 2.5 yields

$$ \lim_{m \to 0^+} \frac{H_f(m)}{m} \le \limsup_{u \to 0^+} \sup_{|\xi|=1} \frac{f(x_0, u, \rho(|u|)\xi)}{|u|} \stackrel{(S)}{\le} \liminf_{m \to 0^+} \frac{f(u, 0)}{u} = f'_-(0^+, 0). $$
Proof. — One can assume that there exists $u \in W_{\text{loc}}^{1,1}(\mathbb{R}, \mathbb{R}_+)$ with $0 < \int_{\mathbb{R}} u < +\infty$ and $\mathcal{E}(u) < +\infty$. In particular, up to changing the value of $u$ on a negligible set, $u$ is continuous on $\mathbb{R}$. Let $\varepsilon \in (0, \sup_{\mathbb{R}} u)$, set $A_\varepsilon := \{x : u(x) = \varepsilon\}$ which is non-empty by the intermediate value theorem and integrability of $u$, and define

$$ 
\begin{aligned}
a_\varepsilon &= \begin{cases} \inf A_\varepsilon & \text{if } \inf A_\varepsilon > -\infty, \\ \text{any point in } (-\infty, -\varepsilon^{-1}) \cap A_\varepsilon & \text{otherwise,} \end{cases} \\ 
b_\varepsilon &= \begin{cases} \sup A_\varepsilon & \text{if } \sup A_\varepsilon < +\infty, \\ \text{any point in } (\varepsilon^{-1}, +\infty) \cap A_\varepsilon & \text{otherwise.} \end{cases}
\end{aligned}
$$

By continuity and integrability of $u$, $u(a_\varepsilon) = u(b_\varepsilon) = \varepsilon$ and

$$ \sup_{x \in \mathbb{R} \setminus [a_\varepsilon, b_\varepsilon]} u(x) \le \varepsilon \lor \sup_{|x| > \varepsilon^{-1}} u(x) \mathrel{\mathop{\longrightarrow}}_{\varepsilon \to 0} 0. $$

Moreover $a_\varepsilon, b_\varepsilon$ converge to points $-\infty \le a \le b \le +\infty$, hence $u = 0$ on $\mathbb{R} \setminus (a, b)$ and by dominated convergence, since $\nabla u = 0$ a.e. on $\{u = 0\}$,

$$ +\infty > \lim_{\varepsilon \to 0^+} \int_{\mathbb{R} \setminus [a_\varepsilon, b_\varepsilon]} u + f(u, \nabla u) = f(0, 0) \mathcal{L}(\mathbb{R} \setminus (a, b)). $$

Notice that this limit is necessary zero. Let $m > 0$. If $\varepsilon$ is small enough, then $\int_{\mathbb{R} \setminus [a_\varepsilon, b_\varepsilon]} u < m$ so that we can take $R_\varepsilon > 0$ such that $\varepsilon R_\varepsilon = m - \int_{\mathbb{R} \setminus [a_\varepsilon, b_\varepsilon]} u$. We then define

$$ u_\varepsilon(x) = \begin{cases} u(x) & \text{if } x \le a_\varepsilon, \\ \varepsilon & \text{if } a_\varepsilon < x < a_\varepsilon + R_\varepsilon, \\ u(b_\varepsilon + x - (a_\varepsilon + R_\varepsilon)) & \text{if } x \ge a_\varepsilon + R_\varepsilon, \end{cases} $$

so that $\int_{\mathbb{R}} v_\varepsilon = m$. Moreover,

$$ \mathcal{E}(v_\varepsilon) = \mathcal{E}(u, \mathbb{R} \setminus [a_\varepsilon, b_\varepsilon]) + R_\varepsilon f(\varepsilon, 0). $$

Hence, as $R_\varepsilon = (m + o(1))/\varepsilon$ as $\varepsilon \to 0$,

$$ H_f(m) \le \limsup_{\varepsilon \to 0^+} \mathcal{E}(v_\varepsilon) = m \limsup_{\varepsilon \to 0^+} \frac{f(\varepsilon, 0)}{\varepsilon}. \quad \square $$

From Proposition 2.5 and Proposition 2.6 we obtain the following corollary.

COROLLARY 2.7. — Let $f: \mathbb{R} \times \mathbb{R}^N \to [0, +\infty]$ satisfy $(H_1)$, $(H_3)$ and (S). Assume in addition that $H_f < +\infty$ if $N = 1$. Then $(H_6)$ holds.

Proof. — If $N = 1$ we apply Proposition 2.6 to get

$$ \lim_{m \to 0^+} \frac{H_f(m)}{m} \le \limsup_{u \to 0^+} \frac{f(u, 0)}{u} \stackrel{(S)}{\le} \liminf_{m \to 0^+} \frac{f(u, 0)}{u} = f'_-(0^+, 0), $$

and if $N = 2$ taking a function $\rho$ as in (S) and applying Proposition 2.5 yields

$$ \lim_{m \to 0^+} \frac{H_f(m)}{m} \le \limsup_{u \to 0^+} \sup_{|\xi|=1} \frac{f(x_0, u, \rho(|u|)\xi)}{|u|} \stackrel{(S)}{\le} \liminf_{m \to 0^+} \frac{f(u, 0)}{u} = f'_-(0^+, 0). $$
The analogous inequality when $m \to 0^-$ is obtained by considering the symmetric Lagrangian $(u, \xi) \mapsto f(-u, -\xi)$. $\square$

## 2.4. STRICT CONCAVITY OF THE MINIMAL COST FUNCTION IN DIMENSION $N \ge 2$

We show that in dimension $N \ge 2$, the minimal cost function must be strictly concave away from the possible initial interval where it is linear:

**PROPOSITION 2.8.** — Assume that $N \ge 2$ and that $f: \mathbb{R} \times \mathbb{R}^N \ni (u, \xi) \mapsto f(u, \xi) \in [0, +\infty]$ satisfies (H$_1$), (H$_2$), (H$_3$), (H$_5$) and (H$_6$). Let

$$m_* = \sup\{m \ge 0 : H_f \text{ is linear on } [0, m]\},$$

where $H_f$ is defined in (1.3). Then, $H_f$ is strictly concave on $(m_*, +\infty)$. A similar statement holds on $\mathbb{R}_-$.

A similar result does not hold in dimension 1 since any continuous concave function $H: \mathbb{R}_+ \to \mathbb{R}_+$ with $H(0) = 0$ can be written as $H = H_f$ with $f$ satisfying all our assumptions (H$_1$)-(H$_8$) (see Section 5.2).

We denote by $M_m^f$ the set of non-negative minimizers of mass $m \in \mathbb{R}_+$:

$$(2.13) \qquad M_m^f := \{u \in W_{\text{loc}}^{1,1}(\mathbb{R}^N, \mathbb{R}_+) : \mathcal{E}_f(u) = H_f(m) \text{ and } \int_{\mathbb{R}^N} u = m\}.$$

The proof of Proposition 2.8 is based on the following observation:

**LEMMA 2.9.** — Let $f: \mathbb{R}_+ \times \mathbb{R}^N \to [0, +\infty]$ be Borel measurable and let $u_i \in M_{m_i}^f$ with $m_i \in \mathbb{R}_+$ for $i = 1, 2$. Let also $u_* := \min\{u_1, u_2\}$, $u^* := \max\{u_1, u_2\}$, $m_* := \int_{\mathbb{R}^N} u_*$ and $m^* := \int_{\mathbb{R}^N} u^*$. If $H_f$ is affine on $[m_*, m^*]$ then $u_* \in M_{m_*}^f$ and $u^* \in M_{m^*}^f$.

*Proof of Lemma 2.9.* — We use the same observations as in the proof of Theorem 1.1. In particular, we have $m_* + m^* = m_1 + m_2$; since $H_f$ is affine on $[m_*, m^*]$, it yields

$$H_f(m_*) + H_f(m^*) = H_f(m_1) + H_f(m_2).$$

But we have also

$$H_f(m_*) + H_f(m^*) \le \mathcal{E}_f(u_*) + \mathcal{E}_f(u^*) = \mathcal{E}_f(u_1) + \mathcal{E}_f(u_2) = H_f(m_1) + H_f(m_2),$$

so that the inequalities we used, i.e., $H_f(m_*) \le \mathcal{E}_f(u_*)$ and $H_f(m^*) \le \mathcal{E}_f(u^*)$, are actually equalities. $\square$

We also use an elementary Sobolev type inequality:

**LEMMA 2.10.** — Let $N \ge 2$, $p \in (1, +\infty)$ and $\omega \subset \mathbb{R}^{N-1}$ be a bounded open set. For every $u \in W_{\text{loc}}^{1,p}(\mathbb{R} \times \omega)$,

$$\int_{\omega} \|u(\cdot, x')\|_{L^{\infty}(\mathbb{R})} dx' \le \|u\|_{L^1(\mathbb{R} \times \omega)} + |\omega|^{(p-1)/p} \left\| \frac{\partial u}{\partial x_1} \right\|_{L^p(\mathbb{R} \times \omega)}.$$
The analogous inequality when $m \to 0^-$ is obtained by considering the symmetric Lagrangian $(u, \xi) \mapsto f(-u, -\xi)$. $\square$

## 2.4. STRICT CONCAVITY OF THE MINIMAL COST FUNCTION IN DIMENSION $N \ge 2$

We show that in dimension $N \ge 2$, the minimal cost function must be strictly concave away from the possible initial interval where it is linear:

**PROPOSITION 2.8.** — Assume that $N \ge 2$ and that $f: \mathbb{R} \times \mathbb{R}^N \ni (u, \xi) \mapsto f(u, \xi) \in [0, +\infty]$ satisfies $(H_1), (H_2), (H_3), (H_5)$ and $(H_6)$. Let

$$m_* = \sup\{m \ge 0 : H_f \text{ is linear on } [0, m]\},$$

where $H_f$ is defined in (1.3). Then, $H_f$ is strictly concave on $(m_*, +\infty)$. A similar statement holds on $\mathbb{R}_-$.

A similar result does not hold in dimension 1 since any continuous concave function $H: \mathbb{R}_+ \to \mathbb{R}_+$ with $H(0) = 0$ can be written as $H = H_f$ with $f$ satisfying all our assumptions $(H_1)-(H_8)$ (see Section 5.2).

We denote by $\mathcal{M}_m^f$ the set of non-negative minimizers of mass $m \in \mathbb{R}_+$:

$$(2.13) \qquad \mathcal{M}_m^f := \{u \in W_{\text{loc}}^{1,1}(\mathbb{R}^N, \mathbb{R}_+) : \mathcal{E}_f(u) = H_f(m) \text{ and } \int_{\mathbb{R}^N} u = m\}.$$

The proof of Proposition 2.8 is based on the following observation:

**LEMMA 2.9.** — Let $f: \mathbb{R}_+ \times \mathbb{R}^N \to [0, +\infty]$ be Borel measurable and let $u_i \in \mathcal{M}_{m_i}^f$ with $m_i \in \mathbb{R}_+$ for $i = 1, 2$. Let also $u_* := \min\{u_1, u_2\}$, $u^* := \max\{u_1, u_2\}$, $m_* := \int_{\mathbb{R}^N} u_*$ and $m^* := \int_{\mathbb{R}^N} u^*$. If $H_f$ is affine on $[m_*, m^*]$ then $u_* \in \mathcal{M}_{m_*}^f$ and $u^* \in \mathcal{M}_{m^*}^f$.

*Proof of Lemma 2.9.* — We use the same observations as in the proof of Theorem 1.1. In particular, we have $m_* + m^* = m_1 + m_2$; since $H_f$ is affine on $[m_*, m^*]$, it yields

$$H_f(m_*) + H_f(m^*) = H_f(m_1) + H_f(m_2).$$

But we have also

$$H_f(m_*) + H_f(m^*) \le \mathcal{E}_f(u_*) + \mathcal{E}_f(u^*) = \mathcal{E}_f(u_1) + \mathcal{E}_f(u_2) = H_f(m_1) + H_f(m_2),$$

so that the inequalities we used, i.e., $H_f(m_*) \le \mathcal{E}_f(u_*)$ and $H_f(m^*) \le \mathcal{E}_f(u^*)$, are actually equalities. $\square$

We also use an elementary Sobolev type inequality:

**LEMMA 2.10.** — Let $N \ge 2$, $p \in (1, +\infty)$ and $\omega \subset \mathbb{R}^{N-1}$ be a bounded open set. For every $u \in W_{\text{loc}}^{1,p}(\mathbb{R} \times \omega)$,

$$\int_{\omega} \|u(\cdot, x')\|_{L^{\infty}(\mathbb{R})} dx' \le \|u\|_{L^1(\mathbb{R} \times \omega)} + |\omega|^{(p-1)/p} \left\| \frac{\partial u}{\partial x_1} \right\|_{L^p(\mathbb{R} \times \omega)}.$$
*Proof of Lemma 2.10.* — We prove the lemma when $u \in \mathcal{C}^1(\mathbb{R} \times \omega)$; the general case follows by approximation. For every $x_1, y_1 \in \mathbb{R}$, $x' \in \omega$, we have

$$u(x_1, x') = u(y_1, x') + \int_{y_1}^{x_1} \frac{\partial u}{\partial x_1}(t, x') dt.$$

By averaging in the variable $y_1$, we deduce

$$|u(x_1, x')| \le \int_{x_1-1/2}^{x_1+1/2} |u(y_1, x')| dy_1 + \int_{x_1-1/2}^{x_1+1/2} \left| \frac{\partial u}{\partial x_1}(t, x') \right| dt.$$

The result follows from Hölder inequality after integrating over $\omega$. $\square$

<figure><img src="image_3.png" /></figure>

*Proof of Proposition 2.8.* — Assume by contradiction that the concave function $H_f$ is not strictly concave on $(m_*, +\infty)$ which means that there exists $m \in (m_*, +\infty)$ and $\eta > 0$ such that $H_f$ is affine on $[m - \eta, m + \eta]$. (Note that $\eta \le m - m_*$ by definition of $m_*$.) Moreover, we will see in Proposition 3.12 that $M_m^f$ is not empty because $H_f$ is not linear on $[0, m]$. We let $u \in M_m^f$.

As before, we shall use the notations $\wedge$ and $\vee$ for the minimum and maximum; we also let $(e_1, \dots, e_N)$ be the canonical basis of $\mathbb{R}^N$. Knowing that $\tau \mapsto u(\cdot + \tau)$ is continuous in $\mathbb{R}^N$ for every $u \in L^1(\mathbb{R}^N)$, that $u \mapsto u(\cdot + \tau)$ is isometric in $L^1(\mathbb{R}^N)$ for every $\tau \in \mathbb{R}^N$, that the map $(x, y) \mapsto x \wedge y$ is Lipschitz on $\mathbb{R}^N \times \mathbb{R}^N$, and since the set $M_{m+\eta/2}$ is compact in $L^1$ up to translations in view of Remark 3.13, we deduce that there exists $\delta_0 > 0$ such that

$$(2.14) \qquad \|u \wedge u(\cdot + \delta_2)\|_{L^1(\mathbb{R}^N)} > m \quad \text{for all } \delta \in (0, \delta_0) \text{ and } u \in \mathcal{M}_{m+\eta/2}^f.$$

We now construct by induction a sequence $(t_n)_{n \in \mathbb{N}}$ in $\mathbb{R}_+$ and a sequence $(u_n)_{n \in \mathbb{N}}$ in $M_m^f$ such that

$$(2.15) \qquad t_{n+1} \ge t_n + \delta_0 \quad \text{and} \quad u_n(x) \le U(x) \wedge U(x + t_n e_2) \quad \forall x \in \mathbb{R}^N,$$

where we have set

$$U(x) := \ess \sup_{t \in \mathbb{R}} u(x + te_1).$$

To this aim, we first set $u_0 := u$ and $t_0 = 0$. Then, if we assume that $t_n$ and $u_n$ are constructed as before, we first pick an $\delta_n^1 \in \mathbb{R}_+$ such that

$$v_n := u_n \vee u_n(\cdot + \delta_n^1 e_1) \quad \text{satisfies} \quad \int_{\mathbb{R}^N} v_n = m + \frac{\eta}{2},$$

which is possible since $\eta \le m$, as we argued in the proof of Theorem 1.1. Similarly, we pick a $\delta_n^2 \in \mathbb{R}_+$ such that $u_{n+1} := v_n \wedge v_n(\cdot + \delta_n^2 e_2)$ satisfies $\int_{\mathbb{R}^N} u_{n+1} = m$, and we set $t_{n+1} = t_n + \delta_n^2$. By Lemma 2.9, $v_n \in \mathcal{M}_{m+\eta/2}^f$ and $u_n \in \mathcal{M}_m^f$. By (2.14), we have $\delta_n^2 \ge \delta_0$, thus insuring the first condition in (2.15). For the second condition, we observe that for all $x = (x_1, x') \in \mathbb{R}^N$,

$$u_{n+1}(x) \le (\sup_{t \in \mathbb{R}} u_n(x + te_1)) \wedge (\sup_{t \in \mathbb{R}} u_n(x + te_1 + \delta_n^2 e_2)) \le U(x) \wedge U(x + t_{n+1}e_2),$$

where in the last inequality we have used the induction hypothesis (2.15).

We now show that the sequence $(u_n \mathcal{L}^N)_{n \in \mathbb{N}}$ is vanishing which will contradict the compactness of $M_m^f$ in $L^1$ up to translations.
Proof of Lemma 2.10. — We prove the lemma when $u \in \mathcal{C}^1(\mathbb{R} \times \omega)$; the general case follows by approximation. For every $x_1, y_1 \in \mathbb{R}$, $x' \in \omega$, we have

$$u(x_1, x') = u(y_1, x') + \int_{y_1}^{x_1} \frac{\partial u}{\partial x_1}(t, x') dt.$$

By averaging in the variable $y_1$, we deduce

$$|u(x_1, x')| \le \int_{x_1-1/2}^{x_1+1/2} |u(y_1, x')| dy_1 + \int_{x_1-1/2}^{x_1+1/2} \left| \frac{\partial u}{\partial x_1}(t, x') \right| dt.$$

The result follows from Hölder inequality after integrating over $\omega$. $\square$

<figure><img src="image_5.png" /></figure>

Proof of Proposition 2.8. — Assume by contradiction that the concave function $H_f$ is not strictly concave on $(m_*, +\infty)$ which means that there exists $m \in (m_*, +\infty)$ and $\eta > 0$ such that $H_f$ is affine on $[m - \eta, m + \eta]$. (Note that $\eta \le m - m_*$ by definition of $m_*$.) Moreover, we will see in Proposition 3.12 that $M_m^f$ is not empty because $H_f$ is not linear on $[0, m]$. We let $u \in M_m^f$.

As before, we shall use the notations $\wedge$ and $\vee$ for the minimum and maximum; we also let $(e_1, \dots, e_N)$ be the canonical basis of $\mathbb{R}^N$. Knowing that $\tau \mapsto u(\cdot + \tau)$ is continuous in $\mathbb{R}^N$ for every $u \in L^1(\mathbb{R}^N)$, that $u \mapsto u(\cdot + \tau)$ is isometric in $L^1(\mathbb{R}^N)$ for every $\tau \in \mathbb{R}^N$, that the map $(x, y) \mapsto x \wedge y$ is Lipschitz on $\mathbb{R}^N \times \mathbb{R}^N$, and since the set $M_{m+\eta/2}$ is compact in $L^1$ up to translations in view of Remark 3.13, we deduce that there exists $\delta_0 > 0$ such that

$$(2.14) \qquad \|u \wedge u(\cdot + \delta_2)\|_{L^1(\mathbb{R}^N)} > m \quad \text{for all } \delta \in (0, \delta_0) \text{ and } u \in \mathcal{M}_{m+\eta/2}^f.$$

We now construct by induction a sequence $(t_n)_{n \in \mathbb{N}}$ in $\mathbb{R}_+$ and a sequence $(u_n)_{n \in \mathbb{N}}$ in $M_m^f$ such that

$$(2.15) \qquad t_{n+1} \ge t_n + \delta_0 \quad \text{and} \quad u_n(x) \le U(x) \wedge U(x + t_n e_2) \quad \forall x \in \mathbb{R}^N,$$

where we have set

$$U(x) := \ess \sup_{t \in \mathbb{R}} u(x + te_1).$$

To this aim, we first set $u_0 := u$ and $t_0 = 0$. Then, if we assume that $t_n$ and $u_n$ are constructed as before, we first pick an $\delta_n^1 \in \mathbb{R}_+$ such that

$$v_n := u_n \vee u_n(\cdot + \delta_n^1 e_1) \quad \text{satisfies} \quad \int_{\mathbb{R}^N} v_n = m + \frac{\eta}{2},$$

which is possible since $\eta \le m$, as we argued in the proof of Theorem 1.1. Similarly, we pick a $\delta_n^2 \in \mathbb{R}_+$ such that $u_{n+1} := v_n \wedge v_n(\cdot + \delta_n^2 e_2)$ satisfies $\int_{\mathbb{R}^N} u_{n+1} = m$, and we set $t_{n+1} = t_n + \delta_n^2$. By Lemma 2.9, $v_n \in \mathcal{M}_{m+\eta/2}^f$ and $u_n \in \mathcal{M}_m^f$. By (2.14), we have $\delta_n^2 \ge \delta_0$, thus insuring the first condition in (2.15). For the second condition, we observe that for all $x = (x_1, x') \in \mathbb{R}^N$,

$$u_{n+1}(x) \le (\sup_{t \in \mathbb{R}} u_n(x + te_1)) \wedge (\sup_{t \in \mathbb{R}} u_n(x + te_1 + \delta_n^2 e_2)) \le U(x) \wedge U(x + t_{n+1}e_2),$$

where in the last inequality we have used the induction hypothesis (2.15).

We now show that the sequence $(u_n \mathcal{L}^N)_{n \in \mathbb{N}}$ is vanishing which will contradict the compactness of $M_m^f$ in $L^1$ up to translations.
For this, we let $(x_k)_{k \in \mathbb{N}}$ be a sequence in $\mathbb{R}^N$ and $(u_{n_k})_{k \in \mathbb{N}}$ be a subsequence of $(u_n)_{n \in \mathbb{N}}$ such that

$$ \limsup_{n \to \infty} \sup_{x \in \mathbb{R}^N} \int_{x+[0,1)^N} u_n = \lim_{k \to \infty} \int_{x_k+[0,1)^N} u_{n_k} $$

By $(H_5)$, we have $\partial u / \partial x_1 \in L^p(\mathbb{R}^N)$. Using this fact, estimate (2.15), and Lemma 2.10 with $\omega$ a unit cube in $\mathbb{R}^{N-1}$, we obtain

$$ \lim_{k \to \infty} \int_{x_k+[0,1)^N} u_{n_k} \le \liminf_{k \to \infty} \int_{x_k+[0,1)^N} U \wedge \int_{x_k+t_{n_k}e_2+[0,1)^N} U \\ \le \liminf_{k \to \infty} \left( \|u\|_{L^1(\{0 \le (x-x_k) \cdot e_2 \le 1\})} + \left\| \frac{\partial u}{\partial x_1} \right\|_{L^p(\{0 \le (x-x_k) \cdot e_2 \le 1\})} \right) \\ \wedge \liminf_{k \to \infty} \left( \|u\|_{L^1(\{t_{n_k} \le (x-x_k) \cdot e_2 \le t_{n_k}+1\})} + \left\| \frac{\partial u}{\partial x_1} \right\|_{L^p(\{t_{n_k} \le (x-x_k) \cdot e_2 \le t_{n_k}+1\})} \right), $$

and the conclusion follows since the sequences $(x_k \cdot e_2)_{k \in \mathbb{N}}$ and $(t_{n_k} + x_k \cdot e_2)_{k \in \mathbb{N}}$ cannot be both bounded as $\lim_{k \to \infty} t_{n_k} = \infty$. $\square$

### 3. LOWER BOUND FOR THE ENERGY AND EXISTENCE OF OPTIMAL PROFILES

Our main tool to localize the energy and obtain a lower bound relies on a profile decomposition for bounded sequences of positive measures, which is reminiscent of the concentration-compactness principle of P.-L. Lions. This differs from classical strategies to localize the energy which are based on suitable cut-offs. Naturally, this concentration-compactness result also provides a criterion for the existence of optimal profiles in (1.2). Nothing can be said beyond existence of a minimizer at this level of generality. Further properties such as uniqueness and radial symmetry would require conditions on the Lagrangian $f$ and not only on the cost function $H_f$. We deal with these questions in a particular case in Section 5.3.

#### 3.1. PROFILE DECOMPOSITION BY CONCENTRATION-COMPACTNESS.
We prove a profile decomposition theorem for bounded sequences of positive measures over $\mathbb{R}^N$, which is essentially equivalent to [Mar14, Th. 1.5] in the Euclidean case. We have added an extra information on mass conservation that will be useful, and provide a self-contained simple proof. We start with a definition.

**Definition 3.1.** — A sequence of positive measures $(\mu_n)_{n \in \mathbb{N}} \in \mathcal{M}_+(\mathbb{R}^N)$ is vanishing if

$$ \sup_{x \in \mathbb{R}^N} \mu_n(B_1(x)) \xrightarrow{n \to \infty} 0. $$

Any bounded sequence of positive measures over $\mathbb{R}^N$ may be decomposed (up to subsequence) into a countable collection of narrowly converging “bubbles” and a vanishing part, accounting for the total mass of the sequence, as stated in the following theorem.
For this, we let $(x_k)_{k \in \mathbb{N}}$ be a sequence in $\mathbb{R}^N$ and $(u_{n_k})_{k \in \mathbb{N}}$ be a subsequence of $(u_n)_{n \in \mathbb{N}}$ such that

$$ \limsup_{n \to \infty} \sup_{x \in \mathbb{R}^N} \int_{x+[0,1)^N} u_n = \lim_{k \to \infty} \int_{x_k+[0,1)^N} u_{n_k} $$

By $(H_5)$, we have $\partial u / \partial x_1 \in L^p(\mathbb{R}^N)$. Using this fact, estimate (2.15), and Lemma 2.10 with $\omega$ a unit cube in $\mathbb{R}^{N-1}$, we obtain

$$ \lim_{k \to \infty} \int_{x_k+[0,1)^N} u_{n_k} \le \liminf_{k \to \infty} \int_{x_k+[0,1)^N} U \wedge \int_{x_k+t_{n_k}e_2+[0,1)^N} U \\ \le \liminf_{k \to \infty} \left( \|u\|_{L^1(\{0 \le (x-x_k) \cdot e_2 \le 1\})} + \left\| \frac{\partial u}{\partial x_1} \right\|_{L^p(\{0 \le (x-x_k) \cdot e_2 \le 1\})} \right) \\ \wedge \liminf_{k \to \infty} \left( \|u\|_{L^1(\{t_{n_k} \le (x-x_k) \cdot e_2 \le t_{n_k}+1\})} + \left\| \frac{\partial u}{\partial x_1} \right\|_{L^p(\{t_{n_k} \le (x-x_k) \cdot e_2 \le t_{n_k}+1\})} \right), $$

and the conclusion follows since the sequences $(x_k \cdot e_2)_{k \in \mathbb{N}}$ and $(t_{n_k} + x_k \cdot e_2)_{k \in \mathbb{N}}$ cannot be both bounded as $\lim_{k \to \infty} t_{n_k} = \infty$. $\square$

### 3. LOWER BOUND FOR THE ENERGY AND EXISTENCE OF OPTIMAL PROFILES

Our main tool to localize the energy and obtain a lower bound relies on a profile decomposition for bounded sequences of positive measures, which is reminiscent of the concentration-compactness principle of P.-L. Lions. This differs from classical strategies to localize the energy which are based on suitable cut-offs. Naturally, this concentration-compactness result also provides a criterion for the existence of optimal profiles in (1.2). Nothing can be said beyond existence of a minimizer at this level of generality. Further properties such as uniqueness and radial symmetry would require conditions on the Lagrangian $f$ and not only on the cost function $H_f$. We deal with these questions in a particular case in Section 5.3.

#### 3.1. PROFILE DECOMPOSITION BY CONCENTRATION-COMPACTNESS.
We prove a profile decomposition theorem for bounded sequences of positive measures over $\mathbb{R}^N$, which is essentially equivalent to [Mar14, Th. 1.5] in the Euclidean case. We have added an extra information on mass conservation that will be useful, and provide a self-contained simple proof. We start with a definition.

**Definition 3.1.** — A sequence of positive measures $(\mu_n)_{n \in \mathbb{N}} \in \mathcal{M}_+(\mathbb{R}^N)$ is vanishing if

$$ \sup_{x \in \mathbb{R}^N} \mu_n(B_1(x)) \xrightarrow{n \to \infty} 0. $$

Any bounded sequence of positive measures over $\mathbb{R}^N$ may be decomposed (up to subsequence) into a countable collection of narrowly converging “bubbles” and a vanishing part, accounting for the total mass of the sequence, as stated in the following theorem.
**THEOREM 3.2.** — For every bounded sequence $(\mu_n)_{n \in \mathbb{N}}$ of positive Borel measures on $\mathbb{R}^N$, there exists a subsequence $(\mu_n)_{n \in \sigma(\mathbb{N})}$, $\sigma \in \Sigma$, a non-decreasing sequence of integers $(k_n)_{n \in \sigma(\mathbb{N})}$ converging to some $k \in \mathbb{N} \cup \{+\infty\}$, a sequence of non-trivial positive Borel measures $(\mu^i)_{0 \le i < k}$, and for every $n \in \sigma(\mathbb{N})$, a collection of balls $(B_n^i)_{0 \le i < k_n}$ centered at points of supp $\mu_n$ such that, writing for all $n \in \sigma(\mathbb{N})$,

$$ (3.1) \qquad \mu_n = \mu_n^b + \mu_n^v, \quad \text{where } \mu_n^b = \sum_{0 \le i < k_n} \mu_n \llcorner B_n^i, $$

(A) bubbles emerge: $(c_{B_n^i} \mu_n)_{n \in \sigma(\mathbb{N})} \xrightarrow{n \to \infty} \mu^i$ for every $i < k$, (5)

(B) bubbles split: $\min_{0 \le i < j < k_n} \text{dist}(B_n^i, B_n^j) \xrightarrow{n \to \infty} +\infty$,

(C) bubbles diverge: $\min_{0 \le i < k_n} \text{diam}(B_n^i) \xrightarrow{n \to \infty} +\infty$,

(D) the bubbling mass is conserved: $\|\mu_n^b\| \xrightarrow{\ell \to \infty} \sum_{0 \le i < k} \|\mu^i\|$,

(E) the remaining part is vanishing: $\sup_{x \in \mathbb{R}^N} \mu_n^v(B_1(x)) \xrightarrow{n \to \infty} 0$.

Before proving Theorem 3.2, we introduce the “bubbling” function of a sequence of finite signed measures $(\mu_n)_{n \in \mathbb{N}}$:

$$ (3.2) \quad m((\mu_n)_{n \in \mathbb{N}}) := \sup \left\{ \|\mu\| : (\tau_{-x_{\sigma(\ell)}} \mu_{\sigma(\ell)})_{\ell \in \mathbb{N}} \xrightarrow{\mathscr{C}'_0} \mu, \sigma \in \Sigma, x_{\sigma(\ell)} \in \mathbb{R}^N (\forall \ell) \right\}. $$

Although we will use this function on signed measures, we will start from a sequence of positive measures and use the following characterization of vanishing sequences, which holds only in the case of positive measures:

**LEMMA 3.3.** — A sequence $(\mu_n)_{n \in \mathbb{N}}$ of finite positive measures over $\mathbb{R}^N$ is vanishing if and only if $m((\mu_n)_{n \in \mathbb{N}}) = 0$.

*Proof.* — Assume that $(\mu_n)_{n \in \mathbb{N}}$ is vanishing and that $(\tau_{-x_{\sigma(\ell)}} \mu_{\sigma(\ell)})_{\ell \in \mathbb{N}} \xrightarrow{\mathscr{C}'_0} \mu$ for some $\sigma \in \Sigma$ and some sequence of points $(x_{\sigma(\ell)})_{\ell \in \mathbb{N}}$. Then, for every $x \in \mathbb{R}^N$,

$$ \mu(B_1(x)) \le \liminf_{\ell \to \infty} \tau_{-x_{\sigma(\ell)}} \mu_{\sigma(\ell)}(B_1(x)) = \liminf_{\ell \to \infty} \mu_{\sigma(\ell)}(B_1(x + x_{\sigma(\ell)})) = 0, $$

i.e., $\mu = 0$ and thus $m((\mu_\ell)_{\ell \in \mathbb{N}}) = 0$.

Conversely, if $(\mu_n)_{n \in \mathbb{N}}$ is not vanishing, then there exists $\varepsilon > 0$, $\sigma \in \Sigma$ and a sequence of points $(x_n)_{n \in \sigma(\mathbb{N})}$ in $\mathbb{R}^N$ such that $\mu_n(B_1(x_n)) \ge \varepsilon$ for every $n \in \sigma(\mathbb{N})$.

Up to further extraction, one can assume that $(\tau_{-x_{\sigma(\ell)}} \mu_{\sigma(\ell)})_{\ell \in \mathbb{N}} \xrightarrow{\mathscr{C}'_0} \mu \in \mathcal{M}(\mathbb{R}^N)$. We have

$$ \mu(\bar{B}_1(0)) \ge \limsup_{\ell \to \infty} \tau_{-x_{\sigma(\ell)}} \mu_{\sigma(\ell)}(\bar{B}_1(0)) = \limsup_{\ell \to \infty} \mu_{\sigma(\ell)}(\bar{B}_1(x_{\sigma(\ell)})) \ge \varepsilon > 0, $$

which entails $m((\mu_\ell)_{\ell \in \mathbb{N}}) \ge \varepsilon > 0$. $\square$

<figure><img src="image_4.png" /></figure>

*Proof of Theorem 3.2.* — If $(\mu_n)_{n \in \mathbb{N}}$ is vanishing, then we take $\sigma = \text{Id}$ and $k = 0$, so that $\mu_{\sigma(\ell)} = \mu_\ell = \mu_\ell^v$, (A) to (D) are empty statements and (E) is satisfied since $(\mu_n)_{n \in \mathbb{N}}$ is vanishing. Assume on the contrary that $(\mu_n)_{n \in \mathbb{N}}$ is not vanishing. We shall construct the bubbles by induction and prove their properties in several steps.

<footnote>(5)Recall that $c_B \mu = (x \mapsto x - y)_\sharp(\mu \llcorner B)$ if $B = B_r(y)$ and $\mu \in \mathcal{M}(\mathbb{R}^N)$.</footnote>
**THEOREM 3.2.** — For every bounded sequence $(\mu_n)_{n \in \mathbb{N}}$ of positive Borel measures on $\mathbb{R}^N$, there exists a subsequence $(\mu_n)_{n \in \sigma(\mathbb{N})}$, $\sigma \in \Sigma$, a non-decreasing sequence of integers $(k_n)_{n \in \sigma(\mathbb{N})}$ converging to some $k \in \mathbb{N} \cup \{+\infty\}$, a sequence of non-trivial positive Borel measures $(\mu^i)_{0 \le i < k}$, and for every $n \in \sigma(\mathbb{N})$, a collection of balls $(B_n^i)_{0 \le i < k_n}$ centered at points of supp $\mu_n$ such that, writing for all $n \in \sigma(\mathbb{N})$,

$$ (3.1) \qquad \mu_n = \mu_n^b + \mu_n^v, \quad \text{where } \mu_n^b = \sum_{0 \le i < k_n} \mu_n \llcorner B_n^i, $$

(A) bubbles emerge: $(c_{B_n^i} \mu_n)_{n \in \sigma(\mathbb{N})} \xrightarrow{n \to \infty} \mu^i$ for every $i < k$, (5)

(B) bubbles split: $\min_{0 \le i < j < k_n} \text{dist}(B_n^i, B_n^j) \xrightarrow{n \to \infty} +\infty$,

(C) bubbles diverge: $\min_{0 \le i < k_n} \text{diam}(B_n^i) \xrightarrow{n \to \infty} +\infty$,

(D) the bubbling mass is conserved: $\|\mu_n^b\| \xrightarrow{\ell \to \infty} \sum_{0 \le i < k} \|\mu^i\|$,

(E) the remaining part is vanishing: $\sup_{x \in \mathbb{R}^N} \mu_n^v(B_1(x)) \xrightarrow{n \to \infty} 0$.

Before proving Theorem 3.2, we introduce the “bubbling” function of a sequence of finite signed measures $(\mu_n)_{n \in \mathbb{N}}$:

$$ (3.2) \quad m((\mu_n)_{n \in \mathbb{N}}) := \sup \left\{ \|\mu\| : (\tau_{-x_{\sigma(\ell)}} \mu_{\sigma(\ell)})_{\ell \in \mathbb{N}} \xrightarrow{\mathscr{C}'_0} \mu, \sigma \in \Sigma, x_{\sigma(\ell)} \in \mathbb{R}^N (\forall \ell) \right\}. $$

Although we will use this function on signed measures, we will start from a sequence of positive measures and use the following characterization of vanishing sequences, which holds only in the case of positive measures:

**LEMMA 3.3.** — A sequence $(\mu_n)_{n \in \mathbb{N}}$ of finite positive measures over $\mathbb{R}^N$ is vanishing if and only if $m((\mu_n)_{n \in \mathbb{N}}) = 0$.

*Proof.* — Assume that $(\mu_n)_{n \in \mathbb{N}}$ is vanishing and that $(\tau_{-x_{\sigma(\ell)}} \mu_{\sigma(\ell)})_{\ell \in \mathbb{N}} \xrightarrow{\mathscr{C}'_0} \mu$ for some $\sigma \in \Sigma$ and some sequence of points $(x_{\sigma(\ell)})_{\ell \in \mathbb{N}}$. Then, for every $x \in \mathbb{R}^N$,

$$ \mu(B_1(x)) \le \liminf_{\ell \to \infty} \tau_{-x_{\sigma(\ell)}} \mu_{\sigma(\ell)}(B_1(x)) = \liminf_{\ell \to \infty} \mu_{\sigma(\ell)}(B_1(x + x_{\sigma(\ell)})) = 0, $$

i.e., $\mu = 0$ and thus $m((\mu_\ell)_{\ell \in \mathbb{N}}) = 0$.

Conversely, if $(\mu_n)_{n \in \mathbb{N}}$ is not vanishing, then there exists $\varepsilon > 0$, $\sigma \in \Sigma$ and a sequence of points $(x_n)_{n \in \sigma(\mathbb{N})}$ in $\mathbb{R}^N$ such that $\mu_n(B_1(x_n)) \ge \varepsilon$ for every $n \in \sigma(\mathbb{N})$.

Up to further extraction, one can assume that $(\tau_{-x_{\sigma(\ell)}} \mu_{\sigma(\ell)})_{\ell \in \mathbb{N}} \xrightarrow{\mathscr{C}'_0} \mu \in \mathcal{M}(\mathbb{R}^N)$. We have

$$ \mu(\bar{B}_1(0)) \ge \limsup_{\ell \to \infty} \tau_{-x_{\sigma(\ell)}} \mu_{\sigma(\ell)}(\bar{B}_1(0)) = \limsup_{\ell \to \infty} \mu_{\sigma(\ell)}(\bar{B}_1(x_{\sigma(\ell)})) \ge \varepsilon > 0, $$

which entails $m((\mu_\ell)_{\ell \in \mathbb{N}}) \ge \varepsilon > 0$. $\square$

<figure><img src="image_6.png" /></figure>

*Proof of Theorem 3.2.* — If $(\mu_n)_{n \in \mathbb{N}}$ is vanishing, then we take $\sigma = \text{Id}$ and $k = 0$, so that $\mu_{\sigma(\ell)} = \mu_\ell = \mu_\ell^v$, (A) to (D) are empty statements and (E) is satisfied since $(\mu_n)_{n \in \mathbb{N}}$ is vanishing. Assume on the contrary that $(\mu_n)_{n \in \mathbb{N}}$ is not vanishing. We shall construct the bubbles by induction and prove their properties in several steps.

<footnote>(5)Recall that $c_B \mu = (x \mapsto x - y)_\sharp(\mu \llcorner B)$ if $B = B_r(y)$ and $\mu \in \mathcal{M}(\mathbb{R}^N)$.</footnote>
*Step 1: construction of bubbles centers.* — At first step (step 0), since $m((\mu_n)_{n \in \mathbb{N}}) > 0$, there exists $\sigma_0 \in \Sigma$ and a sequence of points $(x_n^0)_{n \in \sigma_0(\mathbb{N})}$, such that

$$ (3.3) \quad (\tau_{-x_n^0} \mu_n)_{n \in \sigma_0(\mathbb{N})} \xrightarrow{\mathscr{C}'_{0,\sigma}} \mu^0 \in \mathcal{M}(\mathbb{R}^N) \quad \text{with} \quad \|\mu^0\| \ge \frac{1}{2} m((\mu_n)_{n \in \mathbb{N}}). $$

We then set $\mu_n^0 := \mu_n - \tau_{x_n^0} \mu^0$ and we continue by induction, starting from the sequence $(\mu_n^0)_{n \in \sigma_0(\mathbb{N})}$. More precisely, assume that for a fixed step $k-1 \in \mathbb{N}$, for every $i \in \mathbb{N}$ such that $0 \le i \le k-1$, we have built $\mu^i \in \mathcal{M}(\mathbb{R}^N)$, $\sigma_i \in \Sigma$, points $(x_n^i)_{n \in \sigma_i(\mathbb{N})}$ and sequences $(\mu_n^i)_{n \in \sigma_i(\mathbb{N})} \in \mathcal{M}(\mathbb{R}^N)$ such that for every $i$,

$$ (3.4) \qquad \sigma_i \le \sigma_{i-1}, $$

$$ (3.5) \qquad \mu_n^i = \mu_n - \sum_{0 \le j \le i} \tau_{x_n^j} \mu^j, \quad (\forall n \in \sigma_i(\mathbb{N})), $$

$$ (3.6) \qquad (\tau_{-x_n^i} \mu_n^{i-1})_{n \in \sigma_i(\mathbb{N})} \xrightarrow{\mathscr{C}'_{0,\sigma}} \mu^i, $$

$$ (3.7) \qquad \|\mu^i\| \ge \frac{1}{2} m((\mu_n^i)_{n \in \sigma_i(\mathbb{N})}) > 0, $$

where $\sigma_{-1} := \text{Id}, (\mu_n^{-1}) := (\mu_n)$. If $m((\mu_n^{k-1})_{n \in \sigma_{k-1}(\mathbb{N})}) = 0$, we stop; otherwise, we proceed to the next step $k$ to build $\sigma_k, \mu^k, (x_n^k)_{n \in \sigma_k(\mathbb{N})}, (\mu_n^k)$ as we did at step $k=0$, starting with $(\mu_n^{k-1})_{n \in \sigma_{k-1}(\mathbb{N})}$. Either the induction stops at some step $k-1 \in \mathbb{N}$ for which $m((\mu_n^{k-1})_{n \in \sigma_{k-1}(\mathbb{N})}) = 0$ or the previous objects are defined for every $i \in \mathbb{N}$, in which case we let $k := +\infty$.

*Step 2: splitting of bubbles centers.* — We prove that

$$ (3.8) \qquad \lim_{\sigma_i(\mathbb{N}) \ni n \to \infty} \text{dist}(x_n^i, x_n^j) = +\infty \quad \text{for every } i, j \in \mathbb{N} \text{ with } 0 \le j < i < k. $$

Indeed, assume by contradiction that there is a first index $i < k$ such that for some $j_0 < i$, $(\text{dist}(x_n^i, x_n^{j_0}))_{n \in \sigma_i(\mathbb{N})}$ is not divergent. In particular, there exists $\sigma \le \sigma_i$ such that $(x_n^i - x_n^{j_0})_{n \in \sigma(\mathbb{N})} \to x \in \mathbb{R}^N$. Moreover, $(\text{dist}(x_n^i, x_n^{j_0}))_{n \in \sigma_i(\mathbb{N})} \to \infty$, for every $j < i$, $j \ne j_0$ by minimality of $i$ and the triangle inequality $\text{dist}(x_n^j, x_n^{j_0}) \le \text{dist}(x_n^j, x_n^i) + \text{dist}(x_n^i, x_n^{j_0})$. Notice by (3.5) that for every $n \in \sigma(\mathbb{N})$,

$$ \mu_n^{i-1} = \mu_n^{j_0-1} - \tau_{x_n^{j_0}} \mu^{j_0} - \sum_{j_0 < j < i} \tau_{x_n^j} \mu^j, $$

hence taking the translation $\tau_{-x_n^i}$,

$$ \tau_{-x_n^i} \mu_n^{i-1} = \tau_{x_n^{j_0}-x_n^i} (\tau_{-x_n^{j_0}} \mu_n^{j_0-1} - \mu^{j_0}) - \sum_{j_0 < j < i} \tau_{x_n^j-x_n^i} \mu^j, $$

and passing to the weak limit, knowing that $x_n^{j_0} - x_n^i \to -x$ and $\text{dist}(x_n^j, x_n^i) \to +\infty$ for $j_0 < j < i$,

$$ \mu^i = \tau_{-x}(\mu^{j_0} - \mu^{j_0}) - \sum_{j_0 < j < i} 0 = 0. $$

This contradicts the fact that $(\tau_{-x_n^i} \mu_n^{i-1})_{n \in \sigma(\mathbb{N})} \xrightarrow{\mathscr{C}'_{0,\sigma}} \mu^i \ne 0$ and proves (3.8).
**Step 1: construction of bubbles centers.** — At first step (step 0), since $m((\mu_n)_{n \in \mathbb{N}}) > 0$, there exists $\sigma_0 \in \Sigma$ and a sequence of points $(x_n^0)_{n \in \sigma_0(\mathbb{N})}$, such that

$$ (3.3) \quad (\tau_{-x_n^0} \mu_n)_{n \in \sigma_0(\mathbb{N})} \xrightarrow{\mathscr{C}'_{0,\sigma}} \mu^0 \in \mathcal{M}(\mathbb{R}^N) \quad \text{with} \quad \|\mu^0\| \ge \frac{1}{2} m((\mu_n)_{n \in \mathbb{N}}). $$

We then set $\mu_n^0 := \mu_n - \tau_{x_n^0} \mu^0$ and we continue by induction, starting from the sequence $(\mu_n^0)_{n \in \sigma_0(\mathbb{N})}$. More precisely, assume that for a fixed step $k-1 \in \mathbb{N}$, for every $i \in \mathbb{N}$ such that $0 \le i \le k-1$, we have built $\mu^i \in \mathcal{M}(\mathbb{R}^N)$, $\sigma_i \in \Sigma$, points $(x_n^i)_{n \in \sigma_i(\mathbb{N})}$ and sequences $(\mu_n^i)_{n \in \sigma_i(\mathbb{N})} \in \mathcal{M}(\mathbb{R}^N)$ such that for every $i$,

$$ (3.4) \qquad \sigma_i \le \sigma_{i-1}, $$

$$ (3.5) \qquad \mu_n^i = \mu_n - \sum_{0 \le j \le i} \tau_{x_n^j} \mu^j, \quad (\forall n \in \sigma_i(\mathbb{N})), $$

$$ (3.6) \qquad (\tau_{-x_n^i} \mu_n^{i-1})_{n \in \sigma_i(\mathbb{N})} \xrightarrow{\mathscr{C}'_{0,\sigma}} \mu^i, $$

$$ (3.7) \qquad \|\mu^i\| \ge \frac{1}{2} m((\mu_n^i)_{n \in \sigma_i(\mathbb{N})}) > 0, $$

where $\sigma_{-1} := \text{Id}, (\mu_n^{-1}) := (\mu_n)$. If $m((\mu_n^{k-1})_{n \in \sigma_{k-1}(\mathbb{N})}) = 0$, we stop; otherwise, we proceed to the next step $k$ to build $\sigma_k, \mu^k, (x_n^k)_{n \in \sigma_k(\mathbb{N})}, (\mu_n^k)$ as we did at step $k=0$, starting with $(\mu_n^{k-1})_{n \in \sigma_{k-1}(\mathbb{N})}$. Either the induction stops at some step $k-1 \in \mathbb{N}$ for which $m((\mu_n^{k-1})_{n \in \sigma_{k-1}(\mathbb{N})}) = 0$ or the previous objects are defined for every $i \in \mathbb{N}$, in which case we let $k := +\infty$.

**Step 2: splitting of bubbles centers.** — We prove that

$$ (3.8) \qquad \lim_{\sigma_i(\mathbb{N}) \ni n \to \infty} \text{dist}(x_n^i, x_n^{j_0}) = +\infty \quad \text{for every } i, j \in \mathbb{N} \text{ with } 0 \le j < i < k. $$

Indeed, assume by contradiction that there is a first index $i < k$ such that for some $j_0 < i$, $(\text{dist}(x_n^i, x_n^{j_0}))_{n \in \sigma_i(\mathbb{N})}$ is not divergent. In particular, there exists $\sigma \le \sigma_i$ such that $(x_n^i - x_n^{j_0})_{n \in \sigma(\mathbb{N})} \to x \in \mathbb{R}^N$. Moreover, $(\text{dist}(x_n^i, x_n^{j_0}))_{n \in \sigma_i(\mathbb{N})} \to \infty$, for every $j < i$, $j \ne j_0$ by minimality of $i$ and the triangle inequality $\text{dist}(x_n^j, x_n^{j_0}) \le \text{dist}(x_n^j, x_n^i) + \text{dist}(x_n^i, x_n^{j_0})$. Notice by (3.5) that for every $n \in \sigma(\mathbb{N})$,

$$ \mu_n^{i-1} = \mu_n^{j_0-1} - \tau_{x_n^{j_0}} \mu^{j_0} - \sum_{j_0 < j < i} \tau_{x_n^j} \mu^j, $$

hence taking the translation $\tau_{-x_n^i}$,

$$ \tau_{-x_n^i} \mu_n^{i-1} = \tau_{x_n^{j_0}-x_n^i} (\tau_{-x_n^{j_0}} \mu_n^{j_0-1} - \mu^{j_0}) - \sum_{j_0 < j < i} \tau_{x_n^j-x_n^i} \mu^j, $$

and passing to the weak limit, knowing that $x_n^{j_0} - x_n^i \to -x$ and $\text{dist}(x_n^j, x_n^i) \to +\infty$ for $j_0 < j < i$,

$$ \mu^i = \tau_{-x}(\mu^{j_0} - \mu^{j_0}) - \sum_{j_0 < j < i} 0 = 0. $$

This contradicts the fact that $(\tau_{-x_n^i} \mu_n^{i-1})_{n \in \sigma(\mathbb{N})} \xrightarrow{\mathscr{C}'_{0,\sigma}} \mu^i \ne 0$ and proves (3.8).
**Step 3: weak convergence of bubbles.** — From (3.5) we get

$$ (3.9) \qquad \tau_{-x_n^i} \mu_n^{i-1} = \tau_{-x_n^i} \mu_n - \sum_{0 \le j < i} \tau_{-x_n^i + x_j^i} \mu^j, $$

and by (3.8), the sum converges weakly to 0, and so

$$ (3.10) \qquad (\tau_{-x_n^i} \mu_n)_{n \in \sigma_i(\mathbb{N})} \xrightarrow{\mathscr{C}'_0} \mu^i \quad \text{for every } i \in \mathbb{N} \text{ with } i < k. $$

**Step 4: construction of the bubbles with mass conservation.** — We now construct the extraction $\sigma \in \Sigma$ that we need by induction: we set $\sigma(0) = 0$ and, assuming that $\sigma(0) < \dots < \sigma(\ell - 1)$, with $\ell \in \mathbb{N}^*$, have been constructed, we set $\sigma(\ell) := n$ with $n \in \sigma_{\ell \wedge k-1}(\mathbb{N})$ large enough so that $n > \sigma(\ell - 1)$ and for every $i < \ell \wedge k$,

$$ (3.11) \qquad \mu_n(B_\ell(x_n^i)) \le \|\mu^i\| + 2^{-\ell}, $$

and

$$ (3.12) \qquad \min_{0 \le j < i} \mathrm{dist}(x_n^i, x_n^j) \ge 4\ell. $$

Such an $n$ exists by (3.8) and (3.10), noticing that $\mu_n(B_\ell(x_n^i)) = (\tau_{-x_n^i} \mu_n)(B_\ell)$. Then for each $n = \sigma(\ell)$, $\ell \in \mathbb{N}$, we set $k_n = \ell \wedge k$, and for each $i \in \{0, \dots, k_n - 1\}$,

$$ B_n^i := B_\ell(x_n^i). $$

Finally, for every $n \in \sigma(\mathbb{N})$, we decompose $\mu_n$ as expected:

$$ \mu_n = \mu_n^b + \mu_n^v, \quad \text{where } \mu_n^b = \sum_{0 \le i < k_n} \mu_n \llcorner B_n^i. $$

Let us check the four first items (A)–(D). Notice that (C) is fulfilled because $\mathrm{diam}(B_{\sigma(\ell)}^i) = \ell \to +\infty$ as $\ell \to \infty$, and (B) because of (3.12). Since for every $i < k$, $\lim_{\sigma(\mathbb{N}) \ni n \to \infty} \mathrm{diam}(B_n^i) = +\infty$ and $c_{B_n^i} \mu_n = (\tau_{-x_n^i}(\mu_n \llcorner B_n^i))$ for every $n \in \sigma_i(\mathbb{N})$, $(c_{B_n^i} \mu_n)_{n \in \sigma(\mathbb{N})}$ converges weakly to $\mu^i$ by (3.10), and together with (3.11) it implies that

$$ (c_{B_n^i} \mu_n)_{n \in \sigma(\mathbb{N})} \xrightarrow{\mathscr{C}'_b} \mu^i, $$

i.e., (A) is satisfied. Moreover, by (3.11) again,

$$ \limsup_{\ell \to \infty} \sum_{0 \le i < k_{\sigma(\ell)}} \mu_{\sigma(\ell)}(B_{\sigma(\ell)}^i) \le \sum_{0 \le i < k} \|\mu^i\| + \limsup_{\ell \to \infty} (\ell \wedge k) 2^{-\ell} = \sum_{0 \le i < k} \|\mu^i\|, $$

and since $k_n \to k$, by Fatou's lemma we have,

$$ \sum_{0 \le i < k} \|\mu^i\| \le \liminf_{\ell \to \infty} \sum_{0 \le i < k_{\sigma(\ell)}} \mu_{\sigma(\ell)}(B_{\sigma(\ell)}^i), $$

$$ \text{which proves (D) because } \sum_{0 \le i < k_{\sigma(\ell)}} \mu_{\sigma(\ell)}(B_{\sigma(\ell)}^i) = \|\mu_{\sigma(\ell)}^b\|. $$

**Step 5: vanishing of the remaining part, proof of (E).** — By Lemma 3.3, it suffices to prove that $m((\mu_n^v)_{n \in \sigma(\mathbb{N})}) = 0$. We claim that:

$$ (3.13) \qquad m((\mu_n^v)_{n \in \sigma(\mathbb{N})}) \le m((\mu_n^i)_{n \in \sigma_i(\mathbb{N})}), \quad \text{for every } i \in \mathbb{N} \text{ with } i < k, $$

which concludes since $m((\mu_n^k)_{n \in \sigma_{k-1}(\mathbb{N})}) = 0$ if $k < \infty$, and $m((\mu_n^i)_{n \in \sigma_i(\mathbb{N})}) \to 0$ as $i \to \infty$ if $k = \infty$. Indeed, if $k = \infty$, we have by (3.7) and (D),

$$ \frac{1}{2} \sum_{i \in \mathbb{N}} m((\mu_n^i)_{n \in \sigma_i(\mathbb{N})}) \le \sum_{i \in \mathbb{N}} \|\mu^i\| = \lim_{\ell \to \infty} \|\mu_{\sigma(\ell)}^b\| \le \liminf_{\ell \to \infty} \|\mu_{\sigma(\ell)}\| < \infty. $$
**Step 3: weak convergence of bubbles.** — From (3.5) we get

$$ (3.9) \qquad \tau_{-x_n^i} \mu_n^{i-1} = \tau_{-x_n^i} \mu_n - \sum_{0 \le j < i} \tau_{-x_n^i + x_j^i} \mu^j, $$

and by (3.8), the sum converges weakly to 0, and so

$$ (3.10) \qquad (\tau_{-x_n^i} \mu_n)_{n \in \sigma_i(\mathbb{N})} \xrightarrow{\mathscr{C}'_0} \mu^i \quad \text{for every } i \in \mathbb{N} \text{ with } i < k. $$

**Step 4: construction of the bubbles with mass conservation.** — We now construct the extraction $\sigma \in \Sigma$ that we need by induction: we set $\sigma(0) = 0$ and, assuming that $\sigma(0) < \dots < \sigma(\ell - 1)$, with $\ell \in \mathbb{N}^*$, have been constructed, we set $\sigma(\ell) := n$ with $n \in \sigma_{\ell \wedge k-1}(\mathbb{N})$ large enough so that $n > \sigma(\ell - 1)$ and for every $i < \ell \wedge k$,

$$ (3.11) \qquad \mu_n(B_\ell(x_n^i)) \le \|\mu^i\| + 2^{-\ell}, $$

and

$$ (3.12) \qquad \min_{0 \le j < i} \mathrm{dist}(x_n^i, x_n^j) \ge 4\ell. $$

Such an $n$ exists by (3.8) and (3.10), noticing that $\mu_n(B_\ell(x_n^i)) = (\tau_{-x_n^i} \mu_n)(B_\ell)$. Then for each $n = \sigma(\ell)$, $\ell \in \mathbb{N}$, we set $k_n = \ell \wedge k$, and for each $i \in \{0, \dots, k_n - 1\}$,

$$ B_n^i := B_\ell(x_n^i). $$

Finally, for every $n \in \sigma(\mathbb{N})$, we decompose $\mu_n$ as expected:

$$ \mu_n = \mu_n^b + \mu_n^v, \quad \text{where } \mu_n^b = \sum_{0 \le i < k_n} \mu_n \llcorner B_n^i. $$

Let us check the four first items (A)–(D). Notice that (C) is fulfilled because $\mathrm{diam}(B_{\sigma(\ell)}^i) = \ell \to +\infty$ as $\ell \to \infty$, and (B) because of (3.12). Since for every $i < k$, $\lim_{\sigma(\mathbb{N}) \ni n \to \infty} \mathrm{diam}(B_n^i) = +\infty$ and $c_{B_n^i} \mu_n = (\tau_{-x_n^i}(\mu_n \llcorner B_n^i))$ for every $n \in \sigma_i(\mathbb{N})$, $(c_{B_n^i} \mu_n)_{n \in \sigma(\mathbb{N})}$ converges weakly to $\mu^i$ by (3.10), and together with (3.11) it implies that

$$ (c_{B_n^i} \mu_n)_{n \in \sigma(\mathbb{N})} \xrightarrow{\mathscr{C}'_b} \mu^i, $$

i.e., (A) is satisfied. Moreover, by (3.11) again,

$$ \limsup_{\ell \to \infty} \sum_{0 \le i < k_{\sigma(\ell)}} \mu_{\sigma(\ell)}(B_{\sigma(\ell)}^i) \le \sum_{0 \le i < k} \|\mu^i\| + \limsup_{\ell \to \infty} (\ell \wedge k) 2^{-\ell} = \sum_{0 \le i < k} \|\mu^i\|, $$

and since $k_n \to k$, by Fatou's lemma we have,

$$ \sum_{0 \le i < k} \|\mu^i\| \le \liminf_{\ell \to \infty} \sum_{0 \le i < k_{\sigma(\ell)}} \mu_{\sigma(\ell)}(B_{\sigma(\ell)}^i), $$

$$ \text{which proves (D) because } \sum_{0 \le i < k_{\sigma(\ell)}} \mu_{\sigma(\ell)}(B_{\sigma(\ell)}^i) = \|\mu_{\sigma(\ell)}^b\|. $$

**Step 5: vanishing of the remaining part, proof of (E).** — By Lemma 3.3, it suffices to prove that $m((\mu_n^v)_{n \in \sigma(\mathbb{N})}) = 0$. We claim that:

$$ (3.13) \qquad m((\mu_n^v)_{n \in \sigma(\mathbb{N})}) \le m((\mu_n^i)_{n \in \sigma_i(\mathbb{N})}), \quad \text{for every } i \in \mathbb{N} \text{ with } i < k, $$

which concludes since $m((\mu_n^k)_{n \in \sigma_{k-1}(\mathbb{N})}) = 0$ if $k < \infty$, and $m((\mu_n^i)_{n \in \sigma_i(\mathbb{N})}) \to 0$ as $i \to \infty$ if $k = \infty$. Indeed, if $k = \infty$, we have by (3.7) and (D),

$$ \frac{1}{2} \sum_{i \in \mathbb{N}} m((\mu_n^i)_{n \in \sigma_i(\mathbb{N})}) \le \sum_{i \in \mathbb{N}} \|\mu^i\| = \lim_{\ell \to \infty} \|\mu_{\sigma(\ell)}^b\| \le \liminf_{\ell \to \infty} \|\mu_{\sigma(\ell)}\| < \infty. $$
Let us show (3.13). Let $\bar{\sigma} \preceq \sigma$ and $(x_n)_{n \in \bar{\sigma}(\mathbb{N})}$ be a sequence of points such that

$$ (\tau_{-x_n} \mu_n^v)_{n \in \bar{\sigma}(\mathbb{N})} \xrightarrow{\mathscr{C}'_0} \mu \in \mathcal{M}(\mathbb{R}^N). $$

We need to prove that $\|\mu\| \le m((\mu_n^i)_{n \in \sigma_i(\mathbb{N})})$ for every $i < k$. Assume without loss of generality that $\|\mu\| > 0$. Then for every $i < k$,

$$ (3.14) \qquad (\mathrm{dist}(x_n, x_n^i))_{n \in \bar{\sigma}(\mathbb{N})} \to \infty. $$

Otherwise, up to subsequence, $(\mathrm{dist}(x_n, x_n^i))_n$ would be bounded by some constant $M$, and for every $r > 0$,

$$ (\tau_{-x_n} \mu_n^v)(B_r) \le \mu_n^v(B_{r+M}(x_n^i)) \xrightarrow{n \to \infty} 0, $$

because $\mu_n^v$ is supported on $\mathbb{R}^N \setminus \bigcup_{0 \le i < k_n} B_n^i$ and $B_{r+M}(x_n^i) \subseteq B_n^i$ for $n$ large enough by (C). Hence $\mu$ would be 0, a contradiction. Up to further extraction, one can assume that $(\tau_{-x_n} \mu_n)_{n \in \bar{\sigma}(\mathbb{N})}$ converges weakly to a measure $\bar{\mu} \in \mathcal{M}(\mathbb{R}^N)$. Since $\mu_n^v \le \mu_n$, we have $\mu \le \bar{\mu}$. Moreover by (3.5), for every $i < k$ and $n \in \bar{\sigma}(\mathbb{N})$ large enough,

$$ \tau_{-x_n} \mu_n^i = \tau_{-x_n} \mu_n - \sum_{0 \le j \le i} \tau_{x_n^j - x_n} \mu^j, $$

and because of (3.14) the sum converges weakly to 0, so that $\tau_{-x_n} \mu_n^i \xrightarrow{\mathscr{C}'_0} \bar{\mu}$, and consequently,

$$ \|\mu\| \le \|\bar{\mu}\| \le m((\mu_n^i)_{n \in \sigma_i(\mathbb{N})}), $$

which is what had to be proved.

**Step 6: re-centering of the bubbles at points of supp $\mu_n$.** — By (3.10), $(\tau_{-x_n^i} \mu_n)_{n \in \sigma(\mathbb{N})}$ converges weakly to the non-trivial measure $\mu_i$ for every $i < k$, thus

$$ (3.15) \qquad R_i/2 := \limsup_{\sigma(\mathbb{N}) \ni n \to +\infty} \mathrm{dist}(\mathrm{supp} \, \mu_n, x_n^i) < +\infty. $$

Therefore, for every $n$ large enough, there is a point $\tilde{x}_n^i$ such that $|x_n^i - \tilde{x}_n^i| < R_i$ and $\tilde{x}_n^i \in \mathrm{supp} \, \mu_n$. After a further extraction, one may assume that for every $i$, $|x_n^i - \tilde{x}_n^i| < R_i < r_i^n$ where $\mathrm{diam} \, B_n^i = 2r_n^i$ for every $n$, and $(x_n^i - \tilde{x}_n^i)_{n \in \sigma(\mathbb{N})}$ converges to some $p_i \in \mathbb{R}^N$. Finally, we set $\tilde{r}_i^n := r_i^n - R_i$ and $\tilde{B}_n^i := B(\tilde{x}_n^i, \tilde{r}_i^n) \subseteq B_n^i$. After replacing the balls $B_n^i$ by $\tilde{B}_n^i$, (B) and (C) are satisfied by definition. Notice that $(\tau_{-\tilde{x}_n^i} \mu_n)_{n \in \sigma(\mathbb{N})}$ converges weakly to $\tilde{\mu}^i := \tau_{p_i} \mu^i$ with $\|\tilde{\mu}^i\| = \|\mu^i\|$, and

$$ \limsup_n \|c_{B_n^i} \mu_n\| = \limsup_n \mu_n(\tilde{B}_n^i) \le \limsup_n \mu_n(B_n^i) = \|\mu^i\| $$

hence (A) holds. Besides, using Fatou's lemma,

$$ \limsup_n \sum_{i<k_n} \mu_n(\tilde{B}_n^i) \le \limsup_n \sum_{i<k_n} \mu_n(B_n^i) \\ = \sum_{i<k} \|\mu^i\| \le \sum_{i<k} \liminf_n \mu_n(\tilde{B}_n^i) \le \liminf_n \sum_{i<k_n} \mu_n(\tilde{B}_n^i) $$
Let us show (3.13). Let $\bar{\sigma} \preceq \sigma$ and $(x_n)_{n \in \bar{\sigma}(\mathbb{N})}$ be a sequence of points such that

$$ (\tau_{-x_n} \mu_n^v)_{n \in \bar{\sigma}(\mathbb{N})} \xrightarrow{\mathscr{C}'_0} \mu \in \mathcal{M}(\mathbb{R}^N). $$

We need to prove that $\|\mu\| \le m((\mu_n^i)_{n \in \sigma_i(\mathbb{N})})$ for every $i < k$. Assume without loss of generality that $\|\mu\| > 0$. Then for every $i < k$,

$$ (3.14) \qquad (\mathrm{dist}(x_n, x_n^i))_{n \in \bar{\sigma}(\mathbb{N})} \to \infty. $$

Otherwise, up to subsequence, $(\mathrm{dist}(x_n, x_n^i))_n$ would be bounded by some constant $M$, and for every $r > 0$,

$$ (\tau_{-x_n} \mu_n^v)(B_r) \le \mu_n^v(B_{r+M}(x_n^i)) \xrightarrow{n \to \infty} 0, $$

because $\mu_n^v$ is supported on $\mathbb{R}^N \setminus \bigcup_{0 \le i < k_n} B_n^i$ and $B_{r+M}(x_n^i) \subseteq B_n^i$ for $n$ large enough by (C). Hence $\mu$ would be 0, a contradiction. Up to further extraction, one can assume that $(\tau_{-x_n} \mu_n)_{n \in \bar{\sigma}(\mathbb{N})}$ converges weakly to a measure $\bar{\mu} \in \mathcal{M}(\mathbb{R}^N)$. Since $\mu_n^v \le \mu_n$, we have $\mu \le \bar{\mu}$. Moreover by (3.5), for every $i < k$ and $n \in \bar{\sigma}(\mathbb{N})$ large enough,

$$ \tau_{-x_n} \mu_n^i = \tau_{-x_n} \mu_n - \sum_{0 \le j \le i} \tau_{x_n^j - x_n} \mu^j, $$

and because of (3.14) the sum converges weakly to 0, so that $\tau_{-x_n} \mu_n^i \xrightarrow{\mathscr{C}'_0} \bar{\mu}$, and consequently,

$$ \|\mu\| \le \|\bar{\mu}\| \le m((\mu_n^i)_{n \in \sigma_i(\mathbb{N})}), $$

which is what had to be proved.

**Step 6: re-centering of the bubbles at points of supp $\mu_n$.** — By (3.10), $(\tau_{-x_n^i} \mu_n)_{n \in \sigma(\mathbb{N})}$ converges weakly to the non-trivial measure $\mu_i$ for every $i < k$, thus

$$ (3.15) \qquad R_i/2 := \limsup_{\sigma(\mathbb{N}) \ni n \to +\infty} \mathrm{dist}(\mathrm{supp} \, \mu_n, x_n^i) < +\infty. $$

Therefore, for every $n$ large enough, there is a point $\tilde{x}_n^i$ such that $|x_n^i - \tilde{x}_n^i| < R_i$ and $\tilde{x}_n^i \in \mathrm{supp} \, \mu_n$. After a further extraction, one may assume that for every $i$, $|x_n^i - \tilde{x}_n^i| < R_i < r_i^n$ where $\mathrm{diam} \, B_n^i = 2r_n^i$ for every $n$, and $(x_n^i - \tilde{x}_n^i)_{n \in \sigma(\mathbb{N})}$ converges to some $p_i \in \mathbb{R}^N$. Finally, we set $\tilde{r}_i^n := r_i^n - R_i$ and $\tilde{B}_n^i := B(\tilde{x}_n^i, \tilde{r}_i^n) \subseteq B_n^i$. After replacing the balls $B_n^i$ by $\tilde{B}_n^i$, (B) and (C) are satisfied by definition. Notice that $(\tau_{-\tilde{x}_n^i} \mu_n)_{n \in \sigma(\mathbb{N})}$ converges weakly to $\tilde{\mu}^i := \tau_{p_i} \mu^i$ with $\|\tilde{\mu}^i\| = \|\mu^i\|$, and

$$ \limsup_n \|c_{B_n^i} \mu_n\| = \limsup_n \mu_n(\tilde{B}_n^i) \le \limsup_n \mu_n(B_n^i) = \|\mu^i\| $$

hence (A) holds. Besides, using Fatou's lemma,

$$ \limsup_n \sum_{i<k_n} \mu_n(\tilde{B}_n^i) \le \limsup_n \sum_{i<k_n} \mu_n(B_n^i) \\ = \sum_{i<k} \|\mu^i\| \le \sum_{i<k} \liminf_n \mu_n(\tilde{B}_n^i) \le \liminf_n \sum_{i<k_n} \mu_n(\tilde{B}_n^i) $$
so that $\lim_{n} \sum_{i<k_n} \mu_n(\tilde{B}_n^i) = \sum_i \|\mu_i\|$ and (D) is satisfied. In particular,

$$ \lim_{n} \sum_{i<k_n} \mu_n(B_n^i \setminus \tilde{B}_n^i) = \lim_{n} \sum_{i<k_n} \mu_n(B_n^i) - \lim_{n} \sum_{i<k_n} \mu_n(\tilde{B}_n^i) = 0, $$

and (E) holds as well. □

<figure><img src="image_5.png" /></figure>

**REMARK 3.4.** — If the sequence of families of balls $(B_n^i)_{0 \le i < k_n}$ satisfies the conclusion of the theorem, i.e., (A)–(E), then it is also the case for any family of balls $(\tilde{B}_n^i)_{0 \le i < k_n}$ with the same centers as those of $B_n^i$ and with smaller but still divergent radii (i.e., satisfying (C)). It can be easily seen following the arguments at Step 6 of the proof.

**3.2. LOWER BOUND BY CONCENTRATION-COMPACTNESS.** — We will first establish a lower bound for the minimal energy along vanishing sequences defined on varying subsets of $\mathbb{R}^N$. We say that a sequence of Borel functions $(u_n)_{n \in \mathbb{N}}$, each defined on some open set $\Omega_n \subseteq \mathbb{R}^N$, is vanishing if the sequence of measures $\langle |u_n|, \mathcal{L}^N \llcorner \Omega_n \rangle_{n \in \mathbb{N}}$ is vanishing in the sense of Definition 3.1, namely if $\|u_n\|_{L_{\text{uloc}}^1(\Omega_n)} \to 0$ as $n \to \infty$, where $L_{\text{uloc}}^1(\Omega)$ is the set of uniformly locally integrable functions on the open set $\Omega$, i.e., Borel functions $u$ on $\Omega$ such that

$$ (3.16) \qquad \|u\|_{L_{\text{uloc}}^1(\Omega)} := \sup_{x \in \mathbb{R}^N} \int_{\Omega \cap (x+[0,1)^N)} |u| < +\infty. $$

It will be convenient to first extend our Sobolev functions to a neighbourhood $\Omega_\delta$ of $\Omega$ where for every $\delta > 0$ and every set $X \subseteq \mathbb{R}^N$, we have set

$$ X_\delta := \{x \in \mathbb{R}^N : \text{dist}(x, X) < \delta\}. $$

We will need to consider sufficiently regular domains for which we have an extension operator $W^{1,p} \cap L_{\text{uloc}}^1(\Omega) \to W^{1,p} \cap L_{\text{uloc}}^1(\Omega_\delta)$. We will only apply it to domains with smooth boundary, in which case we can use a reflection technique. Since we want quantitative estimates, we will use the notion of *reach* of a set $X \subseteq \mathbb{R}^N$ (see [Fed59]). We say that $X$ has positive reach if there exists $\delta > 0$ such that every $x \in X_\delta$ has a unique nearest point $\pi(x)$ on $X$. The greatest $\delta$ for which this holds is denoted by $\text{reach}(X)$ and the map $x \in X_{\text{reach}(X)} \mapsto \pi(x) \in X$ is called the nearest point retraction.

**EXAMPLE 3.5.** — Assume that $\Omega$ is a perforated domain $B^0 \setminus \bigcup_{i=1}^k B^i$ where the $B^i$ are disjoint closed balls included in some open ball $B^0$ (possibly $B^0 = \mathbb{R}^N$). Then,

$$ \text{reach}(\partial\Omega) = \inf\{\text{radius}(B^i) : i = 0, \dots, k\} \cup \{\text{dist}(\partial B^i, \partial B^j) : i \neq j\}. $$

By [Fed59, Th. 4.8], we have

(i) if $x, y \in X_\delta$ with $0 < \delta < \delta_0 := \text{reach}(X)$, then $|\pi(x) - \pi(y)| \le \frac{\delta_0}{\delta_0 - \delta} |x - y|$,

(ii) if $x \in X$ and $D_x$ is the intersection of $X_{\text{reach}(X)}$ with the straight line crossing $\partial\Omega$ orthogonally at $x$, then $\pi(y) = x$ for every $y \in D_x$.
so that $\lim_n \sum_{i<k_n} \mu_n(\tilde{B}_n^i) = \sum_i \|\mu_i\|$ and (D) is satisfied. In particular,

$$ \lim_n \sum_{i<k_n} \mu_n(B_n^i \setminus \tilde{B}_n^i) = \lim_n \sum_{i<k_n} \mu_n(B_n^i) - \lim_n \sum_{i<k_n} \mu_n(\tilde{B}_n^i) = 0, $$

and (E) holds as well. □

**REMARK 3.4.** — If the sequence of families of balls $(B_n^i)_{0 \le i < k_n}$ satisfies the conclusion of the theorem, i.e., (A)–(E), then it is also the case for any family of balls $(\tilde{B}_n^i)_{0 \le i < k_n}$ with the same centers as those of $B_n^i$ and with smaller but still divergent radii (i.e., satisfying (C)). It can be easily seen following the arguments at Step 6 of the proof.

**3.2. LOWER BOUND BY CONCENTRATION-COMPACTNESS.** — We will first establish a lower bound for the minimal energy along vanishing sequences defined on varying subsets of $\mathbb{R}^N$. We say that a sequence of Borel functions $(u_n)_{n \in \mathbb{N}}$, each defined on some open set $\Omega_n \subseteq \mathbb{R}^N$, is vanishing if the sequence of measures $(|u_n| \mathcal{L}^N \llcorner \Omega_n)_{n \in \mathbb{N}}$ is vanishing in the sense of Definition 3.1, namely if $\|u_n\|_{L_{\text{uloc}}^1(\Omega_n)} \to 0$ as $n \to \infty$, where $L_{\text{uloc}}^1(\Omega)$ is the set of uniformly locally integrable functions on the open set $\Omega$, i.e., Borel functions $u$ on $\Omega$ such that

$$ (3.16) \qquad \|u\|_{L_{\text{uloc}}^1(\Omega)} := \sup_{x \in \mathbb{R}^N} \int_{\Omega \cap (x+[0,1)^N)} |u| < +\infty. $$

It will be convenient to first extend our Sobolev functions to a neighbourhood $\Omega_\delta$ of $\Omega$ where for every $\delta > 0$ and every set $X \subseteq \mathbb{R}^N$, we have set

$$ X_\delta := \{x \in \mathbb{R}^N : \text{dist}(x, X) < \delta\}. $$

We will need to consider sufficiently regular domains for which we have an extension operator $W^{1,p} \cap L_{\text{uloc}}^1(\Omega) \to W^{1,p} \cap L_{\text{uloc}}^1(\Omega_\delta)$. We will only apply it to domains with smooth boundary, in which case we can use a reflection technique. Since we want quantitative estimates, we will use the notion of *reach* of a set $X \subseteq \mathbb{R}^N$ (see [Fed59]). We say that $X$ has positive reach if there exists $\delta > 0$ such that every $x \in X_\delta$ has a unique nearest point $\pi(x)$ on $X$. The greatest $\delta$ for which this holds is denoted by $\text{reach}(X)$ and the map $x \in X_{\text{reach}(X)} \mapsto \pi(x) \in X$ is called the nearest point retraction.

**EXAMPLE 3.5.** — Assume that $\Omega$ is a perforated domain $B^0 \setminus \bigcup_{i=1}^k B^i$ where the $B^i$ are disjoint closed balls included in some open ball $B^0$ (possibly $B^0 = \mathbb{R}^N$). Then,

$$ \text{reach}(\partial\Omega) = \inf\{\text{radius}(B^i) : i = 0, \dots, k\} \cup \{\text{dist}(\partial B^i, \partial B^j) : i \neq j\}. $$

By [Fed59, Th. 4.8], we have

(i) if $x, y \in X_\delta$ with $0 < \delta < \delta_0 := \text{reach}(X)$, then $|\pi(x) - \pi(y)| \le \frac{\delta_0}{\delta_0 - \delta} |x - y|$,

(ii) if $x \in X$ and $D_x$ is the intersection of $X_{\text{reach}(X)}$ with the straight line crossing $\partial\Omega$ orthogonally at $x$, then $\pi(y) = x$ for every $y \in D_x$.
**LEMMA 3.6 (Extension).** — Let $\Omega \subseteq \mathbb{R}^N$ be an open set such that its boundary $\partial\Omega$ is $\mathscr{C}^1$ with positive reach.⁽⁶⁾ Then, for every $\delta \in (0, \text{reach}(\partial\Omega))$, every $p \in [1, +\infty)$ and every $u \in L^1 \cap W^{1,p}(\Omega)$, there exists $\bar{u} \in L^1 \cap W^{1,p}(\Omega_\delta)$ such that $\bar{u} = u$ a.e. on $\Omega$, and

$$ \|\bar{u}\|_{L^1(\Omega_\delta)} \le A \|u\|_{L^1(\Omega)}, \quad \|\bar{u}\|_{L_{\text{uloc}}^1(\Omega_\delta)} \le A \|u\|_{L_{\text{uloc}}^1(\Omega)}, \quad \|\nabla \bar{u}\|_{L^p(\Omega_\delta)} \le A \|\nabla u\|_{L^p(\Omega)}, $$
with a constant $A < +\infty$ depending only on $N, \delta$ and reach($\partial\Omega$).

*Proof.* — Let $\sigma : (\partial\Omega)_\delta \to (\partial\Omega)_\delta$ be the reflection through $\partial\Omega$, defined by $\sigma(x) = 2\pi(x) - x$. By the properties (i) and (ii) of the nearest point retraction, we have that $\sigma = \sigma^{-1}$ (simply because $\pi(\sigma(x)) = \pi(x)$) and $\sigma$ is $L$-Lipschitz with a constant $L < +\infty$ depending on $\delta$ and reach($\partial\Omega$) only.

We define⁽⁷⁾ $\bar{u}$ by $\bar{u} = u$ on $\Omega$ and $\bar{u} = u \circ \sigma$ on $\Omega_\delta \setminus \Omega$. This map is well-defined since $\sigma(\Omega_\delta \setminus \Omega) \subseteq \Omega$. Indeed, if we had $x, \sigma(x) \in \Omega_\delta \setminus \Omega$, then the line segment $[x, \sigma(x)]$ would meet $\partial\Omega$ orthogonally at its center $\pi(x)$, and would remain out of $\Omega$, because otherwise there would exist a point $y$ belonging either to $\partial\Omega \cap (x, \pi(x))$ or $\partial\Omega \cap (\pi(x), \sigma(x))$ thus contradicting the definition of $\pi(x)$. Such a situation is not possible for a $\mathscr{C}^1$ boundary.

Moreover, by the change of variable formula and the chain rule, $\bar{u}$ satisfies the desired estimates since $\sigma$ is bi-Lipschitz with its Lipschitz constants controlled in terms of $\delta$ and reach($\partial\Omega$). □

We will need a localized version of the Gagliardo–Nirenberg–Sobolev inequality in a particular case:

**LEMMA 3.7.** — Let $\Omega \subseteq \mathbb{R}^N$ be an open set such that $\partial\Omega$ is $\mathscr{C}^1$ with positive reach, let $p \in [1, +\infty)$, let $r \ge p(1 + 1/N)$, and assume that $r \le pN/(N - p)$ when $p < N$. Then for every $u \in L^1 \cap W^{1,p}(\Omega)$,

$$ \|u\|_{L^r(\Omega)} \le C (\|\nabla u\|_{L^p(\Omega)} + \|u\|_{L^1(\Omega)})^\alpha \|u\|_{L_{\text{uloc}}^1(\Omega)}^{1-\alpha}, $$

where $\alpha \in (0, 1]$ is the unique parameter such that $1/r = \alpha(1/p - 1/N) + (1 - \alpha)$, and the constant $C < +\infty$ depends on $N, r, p$ and reach($\partial\Omega$).

*Proof of Lemma 3.7.* — We let $u \in L^1 \cap W^{1,p}(\Omega)$ and we extend $u$ to $\bar{u} \in L^1 \cap W^{1,p}(\Omega_\delta)$ as in Lemma 3.6, with $\delta := \text{reach}(\Omega)/2$. By the Gagliardo–Nirenberg–Sobolev inequality (see [Nir59]) on the hypercube $Q_\delta = [-\delta/\sqrt{N}, \delta/\sqrt{N})^N \subseteq \bar{B}_\delta$, we have for some $C$ depending on $N, \delta$,

$$ \|\bar{u}\|_{L^r(Q_\delta)} \le C \|\nabla \bar{u}\|_{L^p(Q_\delta)}^\alpha \|\bar{u}\|_{L^1(Q_\delta)}^{1-\alpha} + C \|\bar{u}\|_{L^1(Q_\delta)}. $$

We then cover $\Omega$ with the disjoint hypercubes $Q_\delta(c) = c + Q_\delta \subseteq \Omega_\delta$ centered at points $c$ on the grid $\mathscr{C} := \Omega \cap (2\delta/\sqrt{N})\mathbb{Z}^N$. Since $r \ge p(1 + 1/N)$, we can check that

$$ (3.17) \qquad r\alpha = \frac{r-1}{1+1/N-1/p} \ge p. $$

<footnote>⁽⁶⁾Thanks to [Fed59, Rem. 4.20], $\partial\Omega$ is actually of class $\mathscr{C}^{1,1}$.</footnote>

<footnote>⁽⁷⁾Note that $\bar{u}$ is not defined on $\partial\Omega$, but this set is negligible.</footnote>
**LEMMA 3.6 (Extension).** — Let $\Omega \subseteq \mathbb{R}^N$ be an open set such that its boundary $\partial\Omega$ is $\mathcal{C}^1$ with positive reach.⁽⁶⁾ Then, for every $\delta \in (0, \text{reach}(\partial\Omega))$, every $p \in [1, +\infty)$ and every $u \in L^1 \cap W^{1,p}(\Omega)$, there exists $\bar{u} \in L^1 \cap W^{1,p}(\Omega_\delta)$ such that $\bar{u} = u$ a.e. on $\Omega$, and

$$ \|\bar{u}\|_{L^1(\Omega_\delta)} \le A\|u\|_{L^1(\Omega)}, \quad \|\bar{u}\|_{L_{\text{uloc}}^1(\Omega_\delta)} \le A\|u\|_{L_{\text{uloc}}^1(\Omega)}, \quad \|\nabla \bar{u}\|_{L^p(\Omega_\delta)} \le A\|\nabla u\|_{L^p(\Omega)}, $$
with a constant $A < +\infty$ depending only on $N, \delta$ and reach($\partial\Omega$).

*Proof.* — Let $\sigma : (\partial\Omega)_\delta \to (\partial\Omega)_\delta$ be the reflection through $\partial\Omega$, defined by $\sigma(x) = 2\pi(x) - x$. By the properties (i) and (ii) of the nearest point retraction, we have that $\sigma = \sigma^{-1}$ (simply because $\pi(\sigma(x)) = \pi(x)$) and $\sigma$ is $L$-Lipschitz with a constant $L < +\infty$ depending on $\delta$ and reach($\partial\Omega$) only.

We define⁽⁷⁾ $\bar{u}$ by $\bar{u} = u$ on $\Omega$ and $\bar{u} = u \circ \sigma$ on $\Omega_\delta \setminus \Omega$. This map is well-defined since $\sigma(\Omega_\delta \setminus \Omega) \subseteq \Omega$. Indeed, if we had $x, \sigma(x) \in \Omega_\delta \setminus \Omega$, then the line segment $[x, \sigma(x)]$ would meet $\partial\Omega$ orthogonally at its center $\pi(x)$, and would remain out of $\Omega$, because otherwise there would exist a point $y$ belonging either to $\partial\Omega \cap (x, \pi(x))$ or $\partial\Omega \cap (\pi(x), \sigma(x))$ thus contradicting the definition of $\pi(x)$. Such a situation is not possible for a $\mathcal{C}^1$ boundary.

Moreover, by the change of variable formula and the chain rule, $\bar{u}$ satisfies the desired estimates since $\sigma$ is bi-Lipschitz with its Lipschitz constants controlled in terms of $\delta$ and reach($\partial\Omega$). □

We will need a localized version of the Gagliardo–Nirenberg–Sobolev inequality in a particular case:

**LEMMA 3.7.** — Let $\Omega \subseteq \mathbb{R}^N$ be an open set such that $\partial\Omega$ is $\mathcal{C}^1$ with positive reach, let $p \in [1, +\infty)$, let $r \ge p(1 + 1/N)$, and assume that $r \le pN/(N - p)$ when $p < N$. Then for every $u \in L^1 \cap W^{1,p}(\Omega)$,

$$ \|u\|_{L^r(\Omega)} \le C(\|\nabla u\|_{L^p(\Omega)} + \|u\|_{L^1(\Omega)})^\alpha \|u\|_{L_{\text{uloc}}^1(\Omega)}^{1-\alpha}, $$

where $\alpha \in (0, 1]$ is the unique parameter such that $1/r = \alpha(1/p - 1/N) + (1 - \alpha)$, and the constant $C < +\infty$ depends on $N, r, p$ and reach($\partial\Omega$).

*Proof of Lemma 3.7.* — We let $u \in L^1 \cap W^{1,p}(\Omega)$ and we extend $u$ to $\bar{u} \in L^1 \cap W^{1,p}(\Omega_\delta)$ as in Lemma 3.6, with $\delta := \text{reach}(\Omega)/2$. By the Gagliardo–Nirenberg–Sobolev inequality (see [Nir59]) on the hypercube $Q_\delta = [-\delta/\sqrt{N}, \delta/\sqrt{N})^N \subseteq \bar{B}_\delta$, we have for some $C$ depending on $N, \delta$,

$$ \|\bar{u}\|_{L^r(Q_\delta)} \le C \|\nabla \bar{u}\|_{L^p(Q_\delta)}^\alpha \|\bar{u}\|_{L^1(Q_\delta)}^{1-\alpha} + C \|\bar{u}\|_{L^1(Q_\delta)}. $$

We then cover $\Omega$ with the disjoint hypercubes $Q_\delta(c) = c + Q_\delta \subseteq \Omega_\delta$ centered at points $c$ on the grid $\mathcal{C} := \Omega \cap (2\delta/\sqrt{N})\mathbb{Z}^N$. Since $r \ge p(1 + 1/N)$, we can check that

$$ (3.17) \qquad r\alpha = \frac{r-1}{1+1/N-1/p} \ge p. $$

<footnote>(⁶)Thanks to [Fed59, Rem. 4.20], $\partial\Omega$ is actually of class $\mathcal{C}^{1,1}$.</footnote>

<footnote>(⁷)Note that $\bar{u}$ is not defined on $\partial\Omega$, but this set is negligible.</footnote>
By superadditivity of $s \mapsto s^{r\alpha/p}$ and of $s \mapsto s^{r\alpha}$, we obtain

$$
\begin{align*}
\|u\|_{L^r(\Omega)}^r &\le \sum_{c \in \mathcal{C}} \|\bar{u}\|_{L^r(Q_\delta(c))}^r \\
&\le C' \sum_{c \in \mathcal{C}} \|\nabla \bar{u}\|_{L^p(Q_\delta(c))}^{p \frac{r\alpha}{p}} \|\bar{u}\|_{L^1(Q_\delta(c))}^{r(1-\alpha)} + C' \|\bar{u}\|_{L^1(Q_\delta(c))}^r \\
&\le C' \|\nabla \bar{u}\|_{L^p(\Omega_\delta)}^{r\alpha} \|\bar{u}\|_{L_{\text{uloc}}^1(\Omega_\delta)}^{r(1-\alpha)} + C' \|\bar{u}\|_{L^1(\Omega_\delta)}^{r\alpha} \|\bar{u}\|_{L_{\text{uloc}}^1(\Omega_\delta)}^{r(1-\alpha)} \\
&\le C'' (\|\nabla u\|_{L^p(\Omega)} + \|u\|_{L^1(\Omega)}^{r\alpha}) \|u\|_{L_{\text{uloc}}^1(\Omega)}^{r(1-\alpha)}. \quad \square
\end{align*}
$$

**PROPOSITION 3.8.** — Assume that $f: \mathbb{R}^N \times \mathbb{R} \times \mathbb{R}^N \to [0, +\infty]$ satisfies (H$_1$) and (H$_5$) for some $p \in (1, +\infty)$. Consider a vanishing sequence $(u_n)_{n \in \mathbb{N}}$ in $W_{\text{loc}}^{1,1}(\Omega_n, \mathbb{R}_{\pm})$, where the $\Omega_n \subseteq \mathbb{R}^N$ are open sets with $\mathcal{C}^1$ boundary such that $\inf_{n \in \mathbb{N}} \text{reach}(\partial \Omega_n) > 0$, and a sequence $(\Phi_n)_{n \in \mathbb{N}}$ of Borel maps $\Phi_n: \Omega_n \to \mathbb{R}^N$ such that $\sup_{y \in \Omega_n} |\Phi_n(y) - x_0| \to 0$ as $n \to +\infty$ for some $x_0 \in \mathbb{R}^N$. If $\theta_n := \int_{\Omega_n} u_n \ne 0$ for every $n$ and $(\theta_n)_{n \in \mathbb{N}}$ is bounded, then:

$$ \liminf_{n \to +\infty} \frac{1}{|\theta_n|} \int_{\Omega_n} f(\Phi_n(y), u_n(y), \nabla u_n(y)) dy \ge f'_-(x_0, 0^{\pm}, 0), $$

where $f'_-(x_0, 0^{\pm}, 0)$ was defined in (2.10).

**Proof of Proposition 3.8.** — Suppose for example that $u_n \ge 0$ a.e. for every $n$. Without loss of generality, we may assume after extracting a subsequence that:

$$ (3.18) \qquad K := \sup_n \frac{1}{\theta_n} \int_{\Omega_n} f(\Phi_n(y), u_n(y), \nabla u_n(y)) dy + \theta_n < +\infty. $$

For all $n \in \mathbb{N}$, we consider the measure $\nu_n \in \mathcal{M}_+(\mathbb{R}^N \times \mathbb{R} \times \mathbb{R}^N)$ defined as the pushforward of the probability measure $\bar{\mu}_n = \frac{1}{\theta_n} u_n \mathcal{L}^N \llcorner \Omega_n$ by the map $(\Phi_n, u_n, \nabla u_n)$, that is:

$$ \nu_n := (\Phi_n, u_n, \nabla u_n)_\#(\bar{\mu}_n). $$

We are going to show in several steps that $\nu_n \xrightarrow{\mathcal{C}_b'} \delta_{(x_0,0,0)}$ and deduce the result. It suffices to show that the three projections $\nu_n^i := (\pi^i)_\# \nu_n$, $i \in \{1, 2, 3\}$ converge narrowly to $\delta_{x_0}$, $\delta_0$ and $\delta_0$ respectively. Indeed, this would imply that $(\nu_n)$ converges narrowly to a measure concentrated on $(x_0, 0, 0)$, hence to $\delta_{(x_0,0,0)}$ since the $\nu_n$ are probability measures. First of all, since $(\nu_n)$ has bounded mass and $(\theta_n)$ is bounded, we may take a subsequence (not relabeled) such that $\nu_n \xrightarrow{\mathcal{C}_0'} \nu$ and $\theta_n \to \theta$ as $n \to \infty$ for some $\nu \in \mathcal{M}_+(\mathbb{R}^N \times \mathbb{R} \times \mathbb{R}^N)$ and $\theta \ge 0$.

**Step 1:** $\nu_n^1 \xrightarrow{\mathcal{C}_b'} \delta_{x_0}$. — This is a direct consequence of the fact that $\nu_n^1$ is concentrated on $\Phi_n(\mathbb{R}^N)$ for every $n$ and $\text{dist}(\Phi_n(\mathbb{R}^N), x_0) \to 0$ as $n \to \infty$.

**Step 2:** $\nu_n^2 \xrightarrow{\mathcal{C}_b'} \delta_0$. — By (3.18) and our assumption (H$_5$), there is a constant $K_1 > 0$ with

$$ (3.19) \qquad \int_{\Omega_n} |\nabla u_n|^p \le K_1 \int_{\Omega_n} u_n, \quad n \in \mathbb{N}. $$
By superadditivity of $s \mapsto s^{r\alpha/p}$ and of $s \mapsto s^{r\alpha}$, we obtain

$$
\begin{align*}
\|u\|_{L^r(\Omega)}^r &\le \sum_{c \in \mathcal{C}} \|\bar{u}\|_{L^r(Q_\delta(c))}^r \\
&\le C' \sum_{c \in \mathcal{C}} \|\nabla \bar{u}\|_{L^p(Q_\delta(c))}^{p \frac{r\alpha}{p}} \|\bar{u}\|_{L^1(Q_\delta(c))}^{r(1-\alpha)} + C' \|\bar{u}\|_{L^1(Q_\delta(c))}^r \\
&\le C' \|\nabla \bar{u}\|_{L^p(\Omega_\delta)}^{r\alpha} \|\bar{u}\|_{L_{\text{uloc}}^1(\Omega_\delta)}^{r(1-\alpha)} + C' \|\bar{u}\|_{L^1(\Omega_\delta)}^{r\alpha} \|\bar{u}\|_{L_{\text{uloc}}^1(\Omega_\delta)}^{r(1-\alpha)} \\
&\le C'' (\|\nabla u\|_{L^p(\Omega)} + \|u\|_{L^1(\Omega)}^{r\alpha}) \|u\|_{L_{\text{uloc}}^1(\Omega)}^{r(1-\alpha)}. \quad \square
\end{align*}
$$

**PROPOSITION 3.8.** — Assume that $f: \mathbb{R}^N \times \mathbb{R} \times \mathbb{R}^N \to [0, +\infty]$ satisfies (H$_1$) and (H$_5$) for some $p \in (1, +\infty)$. Consider a vanishing sequence $(u_n)_{n \in \mathbb{N}}$ in $W_{\text{loc}}^{1,1}(\Omega_n, \mathbb{R}_{\pm})$, where the $\Omega_n \subseteq \mathbb{R}^N$ are open sets with $\mathcal{C}^1$ boundary such that $\inf_{n \in \mathbb{N}} \text{reach}(\partial \Omega_n) > 0$, and a sequence $(\Phi_n)_{n \in \mathbb{N}}$ of Borel maps $\Phi_n: \Omega_n \to \mathbb{R}^N$ such that $\sup_{y \in \Omega_n} |\Phi_n(y) - x_0| \to 0$ as $n \to +\infty$ for some $x_0 \in \mathbb{R}^N$. If $\theta_n := \int_{\Omega_n} u_n \ne 0$ for every $n$ and $(\theta_n)_{n \in \mathbb{N}}$ is bounded, then:

$$
\liminf_{n \to +\infty} \frac{1}{|\theta_n|} \int_{\Omega_n} f(\Phi_n(y), u_n(y), \nabla u_n(y)) \, dy \ge f'_-(x_0, 0^{\pm}, 0),
$$

where $f'_-(x_0, 0^{\pm}, 0)$ was defined in (2.10).

*Proof of Proposition 3.8.* — Suppose for example that $u_n \ge 0$ a.e. for every $n$. Without loss of generality, we may assume after extracting a subsequence that:

$$
(3.18) \quad K := \sup_n \frac{1}{\theta_n} \int_{\Omega_n} f(\Phi_n(y), u_n(y), \nabla u_n(y)) \, dy + \theta_n < +\infty.
$$

For all $n \in \mathbb{N}$, we consider the measure $\nu_n \in \mathcal{M}_+(\mathbb{R}^N \times \mathbb{R} \times \mathbb{R}^N)$ defined as the pushforward of the probability measure $\bar{\mu}_n = \frac{1}{\theta_n} u_n \mathcal{L}^N \llcorner \Omega_n$ by the map $(\Phi_n, u_n, \nabla u_n)$, that is:

$$
\nu_n := (\Phi_n, u_n, \nabla u_n)_\#(\bar{\mu}_n).
$$

We are going to show in several steps that $\nu_n \xrightarrow{\mathcal{C}_b'} \delta_{(x_0,0,0)}$ and deduce the result. It suffices to show that the three projections $\nu_n^i := (\pi^i)_\# \nu_n$, $i \in \{1,2,3\}$ converge narrowly to $\delta_{x_0}$, $\delta_0$ and $\delta_0$ respectively. Indeed, this would imply that $(\nu_n)$ converges narrowly to a measure concentrated on $(x_0,0,0)$, hence to $\delta_{(x_0,0,0)}$ since the $\nu_n$ are probability measures. First of all, since $(\nu_n)$ has bounded mass and $(\theta_n)$ is bounded, we may take a subsequence (not relabeled) such that $\nu_n \xrightarrow{\mathcal{C}_0'} \nu$ and $\theta_n \to \theta$ as $n \to \infty$ for some $\nu \in \mathcal{M}_+(\mathbb{R}^N \times \mathbb{R} \times \mathbb{R}^N)$ and $\theta \ge 0$.

*Step 1:* $\nu_n^1 \xrightarrow{\mathcal{C}_b'} \delta_{x_0}$. — This is a direct consequence of the fact that $\nu_n^1$ is concentrated on $\Phi_n(\mathbb{R}^N)$ for every $n$ and $\text{dist}(\Phi_n(\mathbb{R}^N), x_0) \to 0$ as $n \to \infty$.

*Step 2:* $\nu_n^2 \xrightarrow{\mathcal{C}_b'} \delta_0$. — By (3.18) and our assumption (H$_5$), there is a constant $K_1 > 0$ with

$$
(3.19) \qquad \int_{\Omega_n} |\nabla u_n|^p \le K_1 \int_{\Omega_n} u_n, \quad n \in \mathbb{N}.
$$
We deduce from Markov's inequality, and Lemma 3.7 applied with $r = p(1 + 1/N)$, corresponding to $\alpha = N/(N + 1)$, that

$$ 
\begin{aligned}
\nu_n^2([\eta, +\infty)) &= \frac{1}{\theta_n} \int_{\{u_n \ge \eta\}} u_n = \frac{1}{\theta_n} \int_{\{u_n \ge \eta\}} u_n^{1-r} u_n^r \\
&\le \frac{1}{\theta_n \eta^{r-1}} \int_{\Omega_n} u_n^r \\
&\le \frac{C}{\theta_n \eta^{r-1}} (\|\nabla u_n\|_{L^p(\Omega_n)} + \|u_n\|_{L^1(\Omega_n)})^{r\alpha} \|u_n\|_{L_{\text{uloc}}^{1}(\Omega_n)}^{r(1-\alpha)} \\
&\le \frac{C'}{\eta^{r-1}} (1 + \theta_n^{p-1}) \|u_n\|_{L_{\text{uloc}}^{1}(\Omega_n)}^{r(1-\alpha)},
\end{aligned}
$$

where in the last inequality, we have used the identity $\alpha r = p$ and (3.19), and $C, C'$ depend only on $N, r, p$ and $\inf_n$ reach $(\partial\Omega_n)$.

Since $(u_n)_{n \in \mathbb{N}}$ is vanishing and $(\theta_n)_{n \in \mathbb{N}}$ is bounded, the last term in the previous inequality goes to zero as $n \to \infty$ and it follows that $\nu_n^2 \xrightarrow{\mathcal{C}'_b} \delta_0$.

Step 3: $\nu_n^3 \xrightarrow{\mathcal{C}'_b} \delta_0$. — Fix $M > 0$ and $\eta > 0$. One has by (3.19),

$$ 
\begin{aligned}
\nu_n^3([M, +\infty)) &= \frac{1}{\theta_n} \int_{\{|\nabla u_n| \ge M\}} u_n \\
&\le \frac{1}{\theta_n} \int_{\{u_n < \eta\} \cap \{|\nabla u_n| \ge M\}} u_n + \frac{1}{\theta_n} \int_{\{u_n > \eta\}} u_n \\
&\le \frac{\eta}{\theta_n} \mathcal{L}^N(\{|\nabla u_n| \ge M\}) + \nu_n^2([\eta, +\infty)) \\
&\le \frac{\eta}{\theta_n} \frac{1}{M^p} \int_{\Omega_n} |\nabla u_n|^p + \nu_n^2([\eta, +\infty)) \\
&\le \frac{\eta K_1}{M^p} + \nu_n^2([\eta, +\infty)).
\end{aligned}
$$

By the previous step, we know that $\lim_{n \to +\infty} \nu_n^2([\eta, +\infty)) = 0$, hence taking the superior limit as $n \to +\infty$ then $\eta \to 0$ we get $\lim_{n \to +\infty} \nu_n^3([M, +\infty)) = 0$. Since this is true for every $M > 0$ we obtain $\nu_n^3 \xrightarrow{\mathcal{C}'_b} \delta_0$.

Step 4: conclusion. — By the previous steps, we deduce that $\nu_n \xrightarrow{\mathcal{C}'_b} \delta_{(x_0, 0, 0)}$ as $n \to +\infty$. We define $g: \mathbb{R}^N \times \mathbb{R}_+ \times \mathbb{R}^N \to [0, +\infty]$ as the lower semicontinuous envelope of $\mathbb{R}^N \times \mathbb{R}_+^* \times \mathbb{R}^N \ni (x, u, \xi) \mapsto \frac{1}{u}f(x, u, \xi)$. By (H1), we have $g(x, u, \xi) = \frac{1}{u}f(x, u, \xi)$ if $u > 0$, and by definition of $f_-$ (see (2.10)), we have $g(x, 0, 0) = f_-(x, 0^+, 0)$ for every $x \in \mathbb{R}^N$. Hence, by lower semicontinuity of $g$ and weak convergence of $(\nu_n)$, we get

$$ 
\begin{aligned}
\liminf_{n \to \infty} \int_{\Omega_n} f(\Phi_n, u_n, \nabla u_n) &\ge \liminf_{n \to \infty} \int_{\{u_n > 0\}} \frac{f(\Phi_n, u_n, \nabla u_n)}{u_n} u_n \\
&= \liminf_{n \to \infty} \int_{\mathbb{R}^N \times \mathbb{R} \times \mathbb{R}^N} g(x, u, \xi) \, d\nu_n(x, u, \xi) \\
&\ge \int_{\mathbb{R}^N} g(x, u, \xi) \, d\delta_{(x_0, 0, 0)} = f_-(x_0, 0^+, 0),
\end{aligned}
$$

which ends the proof of the lemma.

□
We deduce from Markov's inequality, and Lemma 3.7 applied with $r = p(1 + 1/N)$, corresponding to $\alpha = N/(N + 1)$, that

$$
\begin{aligned}
\nu_n^2([\eta, +\infty)) &= \frac{1}{\theta_n} \int_{\{u_n \ge \eta\}} u_n = \frac{1}{\theta_n} \int_{\{u_n \ge \eta\}} u_n^{1-r} u_n^r \\
&\le \frac{1}{\theta_n \eta^{r-1}} \int_{\Omega_n} u_n^r \\
&\le \frac{C}{\theta_n \eta^{r-1}} (\|\nabla u_n\|_{L^p(\Omega_n)} + \|u_n\|_{L^1(\Omega_n)})^{r\alpha} \|u_n\|_{L_{\text{uloc}}^{1}(\Omega_n)}^{r(1-\alpha)} \\
&\le \frac{C'}{\eta^{r-1}} (1 + \theta_n^{p-1}) \|u_n\|_{L_{\text{uloc}}^{1}(\Omega_n)}^{r(1-\alpha)}
\end{aligned}
$$

where in the last inequality, we have used the identity $\alpha r = p$ and (3.19), and $C, C'$ depend only on $N, r, p$ and $\inf_n \text{reach}(\partial\Omega_n)$.

Since $(u_n)_{n \in \mathbb{N}}$ is vanishing and $(\theta_n)_{n \in \mathbb{N}}$ is bounded, the last term in the previous inequality goes to zero as $n \to \infty$ and it follows that $\nu_n^2 \xrightarrow{\mathscr{C}'_b} \delta_0$.

Step 3: $\nu_n^3 \xrightarrow{\mathscr{C}'_b} \delta_0$. — Fix $M > 0$ and $\eta > 0$. One has by (3.19),

$$
\begin{aligned}
\nu_n^3([M, +\infty)) &= \frac{1}{\theta_n} \int_{\{|\nabla u_n| \ge M\}} u_n \\
&\le \frac{1}{\theta_n} \int_{\{u_n < \eta\} \cap \{|\nabla u_n| \ge M\}} u_n + \frac{1}{\theta_n} \int_{\{u_n > \eta\}} u_n \\
&\le \frac{\eta}{\theta_n} \mathcal{L}^N(\{|\nabla u_n| \ge M\}) + \nu_n^2([\eta, +\infty)) \\
&\le \frac{\eta}{\theta_n} \frac{1}{M^p} \int_{\Omega_n} |\nabla u_n|^p + \nu_n^2([\eta, +\infty)) \\
&\le \frac{\eta K_1}{M^p} + \nu_n^2([\eta, +\infty)).
\end{aligned}
$$

By the previous step, we know that $\lim_{n \to +\infty} \nu_n^2([\eta, +\infty)) = 0$, hence taking the superior limit as $n \to +\infty$ then $\eta \to 0$ we get $\lim_{n \to +\infty} \nu_n^3([M, +\infty)) = 0$. Since this is true for every $M > 0$ we obtain $\nu_n^3 \xrightarrow{\mathscr{C}'_b} \delta_0$.

Step 4: conclusion. — By the previous steps, we deduce that $\nu_n \xrightarrow{\mathscr{C}'_b} \delta_{(x_0, 0, 0)}$ as $n \to +\infty$. We define $g: \mathbb{R}^N \times \mathbb{R}_+ \times \mathbb{R}^N \to [0, +\infty]$ as the lower semicontinuous envelope of $\mathbb{R}^N \times \mathbb{R}_+^* \times \mathbb{R}^N \ni (x, u, \xi) \mapsto \frac{1}{u}f(x, u, \xi)$. By (H1), we have $g(x, u, \xi) = \frac{1}{u}f(x, u, \xi)$ if $u > 0$, and by definition of $f_-$ (see (2.10)), we have $g(x, 0, 0) = f_-(x, 0^+, 0)$ for every $x \in \mathbb{R}^N$. Hence, by lower semicontinuity of $g$ and weak convergence of $(\nu_n)$, we get

$$
\begin{aligned}
\liminf_{n \to \infty} \int_{\Omega_n} f(\Phi_n, u_n, \nabla u_n) &\ge \liminf_{n \to \infty} \int_{\{u_n > 0\}} \frac{f(\Phi_n, u_n, \nabla u_n)}{u_n} u_n \\
&= \liminf_{n \to \infty} \int_{\mathbb{R}^N \times \mathbb{R} \times \mathbb{R}^N} g(x, u, \xi) \, d\nu_n(x, u, \xi) \\
&\ge \int_{\mathbb{R}^N} g(x, u, \xi) \, d\delta_{(x_0, 0, 0)} = f'_-(x_0, 0^+, 0),
\end{aligned}
$$

which ends the proof of the lemma.

<figure><img src="image_7.png" /></figure>
As a corollary, we may now relate the slope at 0 of $H_f$ to that of $f$.

**COROLLARY 3.9.** — Assume that $f : \mathbb{R}^N \times \mathbb{R} \times \mathbb{R}^N \to [0, +\infty]$ satisfies (H$_1$), (H$_5$) for some $p \in (1, +\infty)$ and (H$_6$). Fix $x \in \mathbb{R}^N$. If either $N \ge 2$ or $(N = 1$ and $H_f(x, \cdot) \ne +\infty$ on $\mathbb{R}_\pm)$, then $H'_f(x, 0^\pm) = f'_-(x, 0^\pm, 0)$.

*Proof.* — The inequality $H'_f(x, 0^\pm) \le f'_-(x, 0^\pm, 0)$ is precisely (H$_6$), and the converse inequality $H'_f(x, 0^\pm) \ge f'_-(x, 0^\pm, 0)$ comes from Proposition 3.8. Indeed, if $(u_n)_{n \in \mathbb{N}} \in W_{\text{loc}}^{1,1}(\mathbb{R}^N, \mathbb{R}_\pm)$ is a sequence of functions of mass $\theta_n = \int_{\mathbb{R}^N} u_n$ going to 0 and which is almost minimizing in the sense that $\lim_{n \to \infty} \varepsilon_f^x(u_n)/|\theta_n| = \liminf_{n \to \infty} H(x, \theta_n)/\theta_n$ then $(u_n)_{n \in \mathbb{N}}$ is vanishing and Proposition 3.8 yields

$$ \liminf_{n \to \infty} \frac{\varepsilon_f^x(u_n)}{|\theta_n|} \ge f'_-(x, 0^\pm, 0). \quad \square $$

We now establish our main energy lower bound along sequences with bounded mass (not necessarily vanishing):

**PROPOSITION 3.10.** — Assume that $(f_\varepsilon)_{\varepsilon>0}$ is a family of functions $f_\varepsilon : \mathbb{R}^N \times \mathbb{R} \times \mathbb{R}^N \to [0, +\infty]$ satisfying (H$_1$), (H$_2$), (H$_5$) and (H$_8$) where $f = \lim_\varepsilon f_\varepsilon$. Let $(\varepsilon_n)_{n \in \mathbb{N}}$ be a sequence of positive numbers going to zero, $(R_n)_{n \in \mathbb{N}}$ and $(r_n)_{n \in \mathbb{N}}$ be two sequences in $(0, +\infty]$ such that $\lim_{n \to \infty} r_n = \lim_{n \to \infty} R_n - r_n = +\infty$, $(u_n)_{n \in \mathbb{N}}$ be a sequence of functions $u_n \in W_{\text{loc}}^{1,1}(B_{R_n}, \mathbb{R}_\pm)$ with finite limit mass $m := \lim_{n \to \infty} \int_{B_{R_n}} u_n$, and $(\Phi_n)_{n \in \mathbb{N}}$ be a sequence of Borel maps $\Phi_n : B_{R_n} \to \mathbb{R}^N$ such that

$$ (3.20) \qquad \sup_{y \in B_{R_n}} |\Phi_n(y) - x_0| \xrightarrow{n \to \infty} 0 \quad \text{for some } x_0 \in \mathbb{R}^N. $$

Then there exists a family $(u^i)_{0 \le i < k}$ of functions in $W_{\text{loc}}^{1,1}(\mathbb{R}^N, \mathbb{R}_\pm)$ with $k \in \mathbb{N} \cup \{+\infty\}$, such that $m_i := \int_{\mathbb{R}^N} u^i \in \mathbb{R}_\pm^*$ for every $i$, and

$$ (3.21) \qquad m = m_v + \sum_{0 \le i < k} m_i \quad \text{with } \pm m_v \ge 0, $$

$$ (3.22) \qquad \liminf_{n \to \infty} \int_{B_{R_n}} f_{\varepsilon_n}(\Phi_n, u_n, \nabla u_n) \ge |m_v| f'_-(x_0, 0^\pm, 0) + \sum_{0 \le i < k} \int_{\mathbb{R}^N} f(x_0, u^i, \nabla u^i). $$

*Proof.* — Suppose for example that $u_n \ge 0$ a.e. for every $n$. We first assume, up to subsequence, that the left hand side of (3.22) is a finite limit. We apply the profile decomposition Theorem 3.2 to the sequence of positive measures $\mu_n = u_n \mathcal{L}^N \llcorner B_{r_n}$ where, we assume the extraction $\sigma$ to be the identity for convenience, and we use the same notation as in Theorem 3.2. In particular, for each bubble $B_n^i = B_{r_n^i}(x_n^i)$, with $0 \le i < k$, we have $x_n^i \in \text{supp } \mu_n \subseteq \bar{B}_{r_n}$. By assumption, we have $\lim_{n \to \infty} (R_n - r_n) = +\infty$; hence, up to reducing the radii of the balls $B_n^i$ if necessary, in such a way that their radii still diverge (see Remark 3.4), we can assume that

$$ (3.23) \qquad B_n^i \subseteq B_{R_{n-1}}, \quad 0 \le i < k_n. $$

For each $0 \le i < k_n$, we let $u_n^i := u_n(\cdot + x_n^i)$. Since (3.22) is assumed to be finite, we get that the sequence $(u_n^i)_n$ is bounded in $W_{\text{loc}}^{1,p}(\mathbb{R}^N)$ by (H$_5$). Hence, after a further
As a corollary, we may now relate the slope at 0 of $H_f$ to that of $f$.

**COROLLARY 3.9.** — Assume that $f : \mathbb{R}^N \times \mathbb{R} \times \mathbb{R}^N \to [0, +\infty]$ satisfies (H$_1$), (H$_5$) for some $p \in (1, +\infty)$ and (H$_6$). Fix $x \in \mathbb{R}^N$. If either $N \ge 2$ or $(N = 1$ and $H_f(x, \cdot) \ne +\infty$ on $\mathbb{R}_\pm)$, then $H'_f(x, 0^\pm) = f'_-(x, 0^\pm, 0)$.

**Proof.** — The inequality $H'_f(x, 0^\pm) \le f'_-(x, 0^\pm, 0)$ is precisely (H$_6$), and the converse inequality $H'_f(x, 0^\pm) \ge f'_-(x, 0^\pm, 0)$ comes from Proposition 3.8. Indeed, if $(u_n)_{n \in \mathbb{N}} \in W_{\text{loc}}^{1,1}(\mathbb{R}^N, \mathbb{R}_\pm)$ is a sequence of functions of mass $\theta_n = \int_{\mathbb{R}^N} u_n$ going to 0 and which is almost minimizing in the sense that $\lim_{n \to \infty} \varepsilon_f^x(u_n)/|\theta_n| = \liminf_{n \to \infty} H(x, \theta_n)/\theta_n$ then $(u_n)_{n \in \mathbb{N}}$ is vanishing and Proposition 3.8 yields

$$ \liminf_{n \to \infty} \frac{\varepsilon_f^x(u_n)}{|\theta_n|} \ge f'_-(x, 0^\pm, 0). \quad \square $$

We now establish our main energy lower bound along sequences with bounded mass (not necessarily vanishing):

**PROPOSITION 3.10.** — Assume that $(f_\varepsilon)_{\varepsilon>0}$ is a family of functions $f_\varepsilon : \mathbb{R}^N \times \mathbb{R} \times \mathbb{R}^N \to [0, +\infty]$ satisfying (H$_1$), (H$_2$), (H$_5$) and (H$_8$) where $f = \lim_\varepsilon f_\varepsilon$. Let $(\varepsilon_n)_{n \in \mathbb{N}}$ be a sequence of positive numbers going to zero, $(R_n)_{n \in \mathbb{N}}$ and $(r_n)_{n \in \mathbb{N}}$ be two sequences in $(0, +\infty]$ such that $\lim_{n \to \infty} r_n = \lim_{n \to \infty} R_n - r_n = +\infty$, $(u_n)_{n \in \mathbb{N}}$ be a sequence of functions $u_n \in W_{\text{loc}}^{1,1}(B_{R_n}, \mathbb{R}_\pm)$ with finite limit mass $m := \lim_{n \to \infty} \int_{B_{R_n}} u_n$, and $(\Phi_n)_{n \in \mathbb{N}}$ be a sequence of Borel maps $\Phi_n : B_{R_n} \to \mathbb{R}^N$ such that

$$ (3.20) \qquad \sup_{y \in B_{R_n}} |\Phi_n(y) - x_0| \xrightarrow{n \to \infty} 0 \quad \text{for some } x_0 \in \mathbb{R}^N. $$

Then there exists a family $(u^i)_{0 \le i < k}$ of functions in $W_{\text{loc}}^{1,1}(\mathbb{R}^N, \mathbb{R}_\pm)$ with $k \in \mathbb{N} \cup \{+\infty\}$, such that $m_i := \int_{\mathbb{R}^N} u^i \in \mathbb{R}_\pm^*$ for every $i$, and

$$ (3.21) \qquad m = m_v + \sum_{0 \le i < k} m_i \quad \text{with } \pm m_v \ge 0, $$

$$ (3.22) \qquad \liminf_{n \to \infty} \int_{B_{R_n}} f_{\varepsilon_n}(\Phi_n, u_n, \nabla u_n) \ge |m_v| f'_-(x_0, 0^\pm, 0) + \sum_{0 \le i < k} \int_{\mathbb{R}^N} f(x_0, u^i, \nabla u^i). $$

**Proof.** — Suppose for example that $u_n \ge 0$ a.e. for every $n$. We first assume, up to subsequence, that the left hand side of (3.22) is a finite limit. We apply the profile decomposition Theorem 3.2 to the sequence of positive measures $\mu_n = u_n \mathcal{L}^N \llcorner B_{r_n}$ where, we assume the extraction $\sigma$ to be the identity for convenience, and we use the same notation as in Theorem 3.2. In particular, for each bubble $B_n^i = B_{r_n^i}(x_n^i)$, with $0 \le i < k$, we have $x_n^i \in \text{supp } \mu_n \subseteq \bar{B}_{r_n}$. By assumption, we have $\lim_{n \to \infty} (R_n - r_n) = +\infty$; hence, up to reducing the radii of the balls $B_n^i$ if necessary, in such a way that their radii still diverge (see Remark 3.4), we can assume that

$$ (3.23) \qquad B_n^i \subseteq B_{R_n-1}, \quad 0 \le i < k_n. $$

For each $0 \le i < k_n$, we let $u_n^i := u_n(\cdot + x_n^i)$. Since (3.22) is assumed to be finite, we get that the sequence $(u_n^i)_n$ is bounded in $W_{\text{loc}}^{1,p}(\mathbb{R}^N)$ by (H$_5$). Hence, after a further
extraction if needed, we get that $(u_n^i)_{n \in \mathbb{N}} \to u^i$ weakly in $W_{\text{loc}}^{1,p}(\mathbb{R}^N)$ for some limit $u^i$, for every $0 \le i < k = \lim k_n$. Setting $m_i = \int_{\mathbb{R}^N} u^i$ for every $i$, by (D) in Theorem 3.2, we have

$$m_v := m - \sum_{0 \le i < k} m_i = \lim_{n \to \infty} \int_{B_{r_n} \setminus \bigcup_{0 \le i < k_n} B_n^i} u_n.$$

Fix $\varepsilon > 0$. We decompose the energy as

$$(3.24) \quad \int_{B_{R_n}} f_{\varepsilon}(\Phi_n, u_n, \nabla u_n) = \int_{B_{R_n} \setminus \bigcup_{0 \le i < k_n} B_n^i} f_{\varepsilon}(\Phi_n, u_n, \nabla u_n) + \sum_{0 \le i < k_n} \int_{B_{r_n}^i} f_{\varepsilon}(\Phi_n(\cdot + x_n^i), u_n^i, \nabla u_n^i).$$

Note that the domains $\Omega_n := B_{R_n} \setminus \bigcup_{0 \le i < k} B_n^i$ satisfy $\inf_{n \in \mathbb{N}} \text{reach}(\partial \Omega_n) > 0$ as noticed in Example 3.5, thanks to (3.23) and (B), (C) in Theorem 3.2. Hence, applying Proposition 3.8 to the Lagrangian $f_{\varepsilon}$, we obtain

$$(3.25) \quad \liminf_{n \to \infty} \int_{B_{R_n} \setminus \bigcup_{0 \le i < k_n} B_n^i} f_{\varepsilon}(\Phi_n, u_n, \nabla u_n) \ge m_v (f_{\varepsilon})'_-(x_0, 0^+, 0).$$

Moreover, by the lower semicontinuity of integral functionals (see [But89, Th. 4.1.1]), in view of (3.20), we have for each $i$ with $0 \le i < k$,

$$(3.26) \quad \liminf_{n \to \infty} \int_{B_{r_n}^i} f_{\varepsilon}(\Phi_n(\cdot + x_n^i), u_n^i, \nabla u_n^i) \ge \int_{\mathbb{R}^N} f_{\varepsilon}(x_0, u^i, \nabla u^i).$$

Finally, by (3.24), (3.25), (3.26), (H8) together with monotone convergence, we deduce that

$$\liminf_{n \to \infty} \int_{B_{R_n}} f_{\varepsilon}(\Phi_n, u_n, \nabla u_n) \ge \lim_{\varepsilon \to 0^+} \left( m_v (f_{\varepsilon})'_-(x_0, 0^+, 0) + \sum_{0 \le i < k} \int_{\mathbb{R}^N} f_{\varepsilon}(x_0, u^i, \nabla u^i) \right) \\ = m_v f'_-(x_0, 0^+, 0) + \sum_{0 \le i < k} \int_{\mathbb{R}^N} f(x_0, u^i, \nabla u^i).$$

The similar statement for non-positive functions is obtained in the same way. $\square$

### 3.3. EXISTENCE OF OPTIMAL PROFILES.
For the existence of an optimal profile in (1.2), we need a criterion that rules out splitting and vanishing of minimizing sequences:

**LEMMA 3.11.** — Let $H: \mathbb{R}_+ \to \mathbb{R}_+$ be a concave function. Then $H$ is subadditive, and if for some $0 < \theta < m$ one has $H(m) = H(m - \theta) + H(\theta)$, then $H$ is linear on $(0, m)$.

*Proof.* — By concavity, $t \mapsto H(t)/t$ is non-increasing. Hence,

$$H(m) = \theta \frac{H(m)}{m} + (m - \theta) \frac{H(m)}{m} \le \theta \frac{H(\theta)}{\theta} + (m - \theta) \frac{H(m - \theta)}{m - \theta}.$$

But, by assumption, the last inequality is an equality which means that $H(m)/m = H(\theta)/\theta = H(m - \theta)/(m - \theta)$. In particular, the monotone function $t \mapsto H(t)/t$ must
extraction if needed, we get that $(u_n^i)_{n \in \mathbb{N}} \to u^i$ weakly in $W_{\text{loc}}^{1,p}(\mathbb{R}^N)$ for some limit $u^i$, for every $0 \le i < k = \lim k_n$. Setting $m_i = \int_{\mathbb{R}^N} u^i$ for every $i$, by (D) in Theorem 3.2, we have

$$m_v := m - \sum_{0 \le i < k} m_i = \lim_{n \to \infty} \int_{B_{r_n} \setminus \bigcup_{0 \le i < k_n} B_n^i} u_n.$$

Fix $\varepsilon > 0$. We decompose the energy as

$$(3.24) \quad \int_{B_{R_n}} f_{\varepsilon}(\Phi_n, u_n, \nabla u_n) = \int_{B_{R_n} \setminus \bigcup_{0 \le i < k_n} B_n^i} f_{\varepsilon}(\Phi_n, u_n, \nabla u_n) + \sum_{0 \le i < k_n} \int_{B_{r_n}^i} f_{\varepsilon}(\Phi_n(\cdot + x_n^i), u_n^i, \nabla u_n^i).$$

Note that the domains $\Omega_n := B_{R_n} \setminus \bigcup_{0 \le i < k} B_n^i$ satisfy $\inf_{n \in \mathbb{N}} \text{reach}(\partial \Omega_n) > 0$ as noticed in Example 3.5, thanks to (3.23) and (B), (C) in Theorem 3.2. Hence, applying Proposition 3.8 to the Lagrangian $f_{\varepsilon}$, we obtain

$$(3.25) \quad \liminf_{n \to \infty} \int_{B_{R_n} \setminus \bigcup_{0 \le i < k_n} B_n^i} f_{\varepsilon}(\Phi_n, u_n, \nabla u_n) \ge m_v (f_{\varepsilon})'_-(x_0, 0^+, 0).$$

Moreover, by the lower semicontinuity of integral functionals (see [But89, Th. 4.1.1]), in view of (3.20), we have for each $i$ with $0 \le i < k$,

$$(3.26) \quad \liminf_{n \to \infty} \int_{B_{r_n}^i} f_{\varepsilon}(\Phi_n(\cdot + x_n^i), u_n^i, \nabla u_n^i) \ge \int_{\mathbb{R}^N} f_{\varepsilon}(x_0, u^i, \nabla u^i).$$

Finally, by (3.24), (3.25), (3.26), (H8) together with monotone convergence, we deduce that

$$\begin{aligned} \liminf_{n \to \infty} \int_{B_{R_n}} f_{\varepsilon}(\Phi_n, u_n, \nabla u_n) &\ge \lim_{\varepsilon \to 0^+} \left( m_v (f_{\varepsilon})'_-(x_0, 0^+, 0) + \sum_{0 \le i < k} \int_{\mathbb{R}^N} f_{\varepsilon}(x_0, u^i, \nabla u^i) \right) \\ &= m_v f'_-(x_0, 0^+, 0) + \sum_{0 \le i < k} \int_{\mathbb{R}^N} f(x_0, u^i, \nabla u^i). \end{aligned}$$

The similar statement for non-positive functions is obtained in the same way. $\square$

### 3.3. EXISTENCE OF OPTIMAL PROFILES.
For the existence of an optimal profile in (1.2), we need a criterion that rules out splitting and vanishing of minimizing sequences:

**LEMMA 3.11.** — Let $H: \mathbb{R}_+ \to \mathbb{R}_+$ be a concave function. Then $H$ is subadditive, and if for some $0 < \theta < m$ one has $H(m) = H(m-\theta) + H(\theta)$, then $H$ is linear on $(0, m)$.

*Proof.* — By concavity, $t \mapsto H(t)/t$ is non-increasing. Hence,

$$H(m) = \theta \frac{H(m)}{m} + (m - \theta) \frac{H(m)}{m} \le \theta \frac{H(\theta)}{\theta} + (m - \theta) \frac{H(m - \theta)}{m - \theta}.$$

But, by assumption, the last inequality is an equality which means that $H(m)/m = H(\theta)/\theta = H(m-\theta)/(m-\theta)$. In particular, the monotone function $t \mapsto H(t)/t$ must
be constant on $[\theta, m]$, i.e., $H$ must be linear on $[\theta, m]$. By concavity this is only possible if $H$ is linear on $[0, m]$. $\square$

We can now state and prove our existence result:

**PROPOSITION 3.12.** — Assume that $f : \mathbb{R} \times \mathbb{R}^N \ni (u, \xi) \mapsto f(u, \xi) \in [0, +\infty]$ satisfies (H$_1$), (H$_2$), (H$_3$), (H$_5$) and (H$_6$). Let $m \in \mathbb{R}_+$ (resp. $m \in \mathbb{R}_-$). If the cost function $H_f$, defined in (1.3), is not linear on $[0, m]$ (resp. $[m, 0]$), then the minimization problem in (1.3) admits a solution $u \in W_{\text{loc}}^{1,1}(\mathbb{R}^N)$, i.e., $\int_{\mathbb{R}^N} u = m$ and $\int_{\mathbb{R}^N} f(u, \nabla u) = H_f(m)$, such that $u \ge 0$ (resp. $u \le 0$) in $\mathbb{R}^N$.

*Proof.* — We consider the case $m \ge 0$, the case $m < 0$ can then be deduced by considering $\tilde{f}(u, \xi) = f(-u, -\xi)$. We assume without loss of generality that $H_f$ is finite on $(0, +\infty)$, otherwise by Theorem 1.1 there is nothing to prove. By Remark 2.1, the admissible class in (1.3) can be reduced to non-negative functions. In particular, if $m = 0$, then $u = 0$ is the only non-negative solution. If $m > 0$, we apply Proposition 3.10 in the following situation: $f_\varepsilon(x, u, \xi) = f(u, \xi)$ for every $(x, u, \xi) \in \mathbb{R}^N \times \mathbb{R} \times \mathbb{R}^N$, $\varepsilon > 0$, $R_n \equiv +\infty$, $\Phi_n \equiv x_0 \in \mathbb{R}^N$, $(u_n)_{n \in \mathbb{N}}$ is a minimizing sequence for the minimization problem in (1.3), and $(r_n)_{n \in \mathbb{N}}$ is a sequence of positive radii going to $+\infty$ such that $\lim_{n \to \infty} \int_{B_{r_n}} u_n = m$. We obtain

$$H_f(m) \ge m_v f'_-(0^+, 0) + \sum_{0 \le i < k} \int_{\mathbb{R}^N} f(u^i, \nabla u^i),$$

with $k \in \mathbb{N} \cup \{+\infty\}$, $u^i \in W_{\text{loc}}^{1,p}(\mathbb{R}^N, \mathbb{R}_+)$ and $m = \sum_{0 \le i < k} m_i + m_v$, where $m_i := \int_{\mathbb{R}^N} u^i$. By Proposition 2.5 and Proposition 2.6, in view of our assumption (H$_6$), and since $H_f$ is assumed to be finite on $(0, +\infty)$ (for the case $N = 1$), we have $f'_-(0^+, 0) \ge H'_f(0^+)$. Moreover, by Theorem 1.1, we have $m_v H'_f(0^+) \ge H_f(m_v)$. Hence, by definition of $H_f$,

$$H_f(m) \ge H_f(m_v) + \sum_{0 \le i < k} H_f(m_i).$$

Since the concave function $H_f$ is not linear on $[0, m]$, by Lemma 3.11, we have either $k = 1$ and $m_v = 0$, and we are done, or $k = 0$ and $m = m_v$. But in the latter case, we would have $H_f(m) = m H'_f(0^+)$ which implies that the monotone function $t \mapsto H_f(t)/t$ is constant on $[0, m]$, i.e., that $H_f$ is linear on $[0, m]$. This contradicts our assumption. $\square$

**REMARK 3.13.** — Notice that the end of the proof actually shows, under the given assumptions, that the set of minimizers for a given mass $m$ is compact in $L^1$ modulo translations.

## 4. $\Gamma$-CONVERGENCE OF THE RESCALED ENERGIES TOWARDS THE $H$-MASS

We establish lower and upper bounds for the $\Gamma$-lim inf and $\Gamma$-lim sup respectively, from which we deduce the proof of our main $\Gamma$-convergence result. The upper bound on the $\Gamma$-lim sup holds under more general assumptions and will be needed in Section 5.5.
be constant on $[\theta, m]$, i.e., $H$ must be linear on $[\theta, m]$. By concavity this is only possible if $H$ is linear on $[0, m]$. $\square$

We can now state and prove our existence result:

**PROPOSITION 3.12.** — Assume that $f: \mathbb{R} \times \mathbb{R}^N \ni (u, \xi) \mapsto f(u, \xi) \in [0, +\infty]$ satisfies (H$_1$), (H$_2$), (H$_3$), (H$_5$) and (H$_6$). Let $m \in \mathbb{R}_+$ (resp. $m \in \mathbb{R}_-$). If the cost function $H_f$, defined in (1.3), is not linear on $[0, m]$ (resp. $[m, 0]$), then the minimization problem in (1.3) admits a solution $u \in W_{\text{loc}}^{1,1}(\mathbb{R}^N)$, i.e., $\int_{\mathbb{R}^N} u = m$ and $\int_{\mathbb{R}^N} f(u, \nabla u) = H_f(m)$, such that $u \ge 0$ (resp. $u \le 0$) in $\mathbb{R}^N$.

*Proof.* — We consider the case $m \ge 0$, the case $m < 0$ can then be deduced by considering $\tilde{f}(u, \xi) = f(-u, -\xi)$. We assume without loss of generality that $H_f$ is finite on $(0, +\infty)$, otherwise by Theorem 1.1 there is nothing to prove. By Remark 2.1, the admissible class in (1.3) can be reduced to non-negative functions. In particular, if $m = 0$, then $u = 0$ is the only non-negative solution. If $m > 0$, we apply Proposition 3.10 in the following situation: $f_\varepsilon(x, u, \xi) = f(u, \xi)$ for every $(x, u, \xi) \in \mathbb{R}^N \times \mathbb{R} \times \mathbb{R}^N$, $\varepsilon > 0$, $R_n \equiv +\infty$, $\Phi_n \equiv x_0 \in \mathbb{R}^N$, $(u_n)_{n \in \mathbb{N}}$ is a minimizing sequence for the minimization problem in (1.3), and $(r_n)_{n \in \mathbb{N}}$ is a sequence of positive radii going to $+\infty$ such that $\lim_{n \to \infty} \int_{B_{r_n}} u_n = m$. We obtain

$$H_f(m) \ge m_v f'_-(0^+, 0) + \sum_{0 \le i < k} \int_{\mathbb{R}^N} f(u^i, \nabla u^i),$$

with $k \in \mathbb{N} \cup \{+\infty\}$, $u^i \in W_{\text{loc}}^{1,p}(\mathbb{R}^N, \mathbb{R}_+)$ and $m = \sum_{0 \le i < k} m_i + m_v$, where $m_i := \int_{\mathbb{R}^N} u^i$. By Proposition 2.5 and Proposition 2.6, in view of our assumption (H$_6$), and since $H_f$ is assumed to be finite on $(0, +\infty)$ (for the case $N = 1$), we have $f'_-(0^+, 0) \ge H'_f(0^+)$. Moreover, by Theorem 1.1, we have $m_v H'_f(0^+) \ge H_f(m_v)$. Hence, by definition of $H_f$,

$$H_f(m) \ge H_f(m_v) + \sum_{0 \le i < k} H_f(m_i).$$

Since the concave function $H_f$ is not linear on $[0, m]$, by Lemma 3.11, we have either $k = 1$ and $m_v = 0$, and we are done, or $k = 0$ and $m = m_v$. But in the latter case, we would have $H_f(m) = m H'_f(0^+)$ which implies that the monotone function $t \mapsto H_f(t)/t$ is constant on $[0, m]$, i.e., that $H_f$ is linear on $[0, m]$. This contradicts our assumption. $\square$

**REMARK 3.13.** — Notice that the end of the proof actually shows, under the given assumptions, that the set of minimizers for a given mass $m$ is compact in $L^1$ modulo translations.

## 4. $\Gamma$-CONVERGENCE OF THE RESCALED ENERGIES TOWARDS THE $H$-MASS

We establish lower and upper bounds for the $\Gamma$-lim inf and $\Gamma$-lim sup respectively, from which we deduce the proof of our main $\Gamma$-convergence result. The upper bound on the $\Gamma$-lim sup holds under more general assumptions and will be needed in Section 5.5.
4.1. LOWER BOUND FOR THE Γ-lim inf. — Given a Borel function $f: \mathbb{R}^N \times \mathbb{R} \times \mathbb{R}^N \to [0, +\infty]$, we define for every $(x, m) \in \mathbb{R}^N \times \mathbb{R}$,

$$ (4.1) \qquad H_f^{-}(x, m) := H_f(x, m) \wedge (f_{-}^{\prime}(x, 0^{\pm}, 0)|m|), \quad \text{if } \pm m \ge 0, $$

recalling that $H_f$ is defined in (1.2) and $f_{-}^{\prime}(x, 0^{\pm}, 0)$ in (2.10), with the usual convention $(\pm\infty) \times 0 = 0$. Notice that it is concave on $\mathbb{R}_+$ and $\mathbb{R}_-$ by Theorem 1.1, and under $(H_6)$ we have $H_f^{-}(x, m) = H_f(x, m)$.

PROPOSITION 4.1. — Assume that $(f_{\varepsilon})_{\varepsilon>0}$ is a family of functions $f_{\varepsilon}: \mathbb{R}^N \times \mathbb{R} \times \mathbb{R}^N \to [0, +\infty]$ satisfying $(H_1), (H_2), (H_3), (H_5)$ and $(H_8)$ where $f = \lim_{\varepsilon \to 0} f_{\varepsilon}$. Let $(\varepsilon_n)_{n \in \mathbb{N}}$ be a sequence of positive numbers going to zero, $(u_n)_{n \in \mathbb{N}}$ be a sequence in $W_{\text{loc}}^{1,1}(\mathbb{R}^N)$, and let

$$ e_n := f_{\varepsilon_n}(\cdot, \varepsilon_n^N u_n, \varepsilon_n^{N+1} \nabla u_n) \varepsilon_n^{-N} \mathcal{L}^N $$

be the energy measure associated with $u_n$. If $u_n \mathcal{L}^N \xrightarrow{\mathcal{L}'_0} \mu \in \mathcal{M}(\mathbb{R}^N)$ and $e_n \xrightarrow{\mathcal{L}'_0} e \in \mathcal{M}(\mathbb{R}^N)$, then

$$ (4.2) \qquad e \ge H_f^{-}(\mu). $$

In particular, $\Gamma(\mathcal{L}'_0)$-liminf$_{\varepsilon \to 0} \varepsilon_\varepsilon \ge M^{H_f^{-}}$.

Proof of Proposition 4.1. — Set $H := H_f^{-}$ and recall that it is concave on $\mathbb{R}_+$ and $\mathbb{R}_-$ by Theorem 1.1. Let us assume first that $u_n \ge 0$ a.e. for every $n$. To obtain (4.2), it is enough to prove that for every $x_0 \in \mathbb{R}^N$,

$$ (4.3) \qquad e(\{x_0\}) \ge H(x_0, \mu(\{x_0\})). $$

and that if $x_0 \in \text{supp } \mu$ is not an atom of $\mu$, then

$$ (4.4) \qquad \limsup_{R \to 0^+} \frac{e(B_R(x_0))}{\mu(B_R(x_0))} \ge H'(x_0, 0^+), $$

Indeed (4.3) implies that $e \ge (H(\mu))^a$ (the atomic part of the measure $H(\mu)$ defined in Definition 2.2) while (4.4) implies that $e \ge H'(\cdot, 0^+)\mu^d = (H(\mu))^d$, by Radon-Nikodým theorem (see [AFP00, Th. 2.22]); these two relations yield $e \ge (H(\mu))^a + (H(\mu))^d = H(\mu)$ as required.

We fix $x_0 \in \text{supp } \mu$ and proceed in several steps.

Step 1: blow-up near $x_0$. — We first take two sequences of positive radii $(R_\ell)_{\ell \in \mathbb{N}} \to 0$ and $(r_\ell)_{\ell \in \mathbb{N}}$ such that for every $\ell \in \mathbb{N}$, $r_\ell \in (0, R_\ell)$,

$$ (4.5) \qquad e(\partial B_{R_\ell}(x_0)) = \mu(\partial B_{r_\ell}(x_0)) = 0, $$

and

$$ (4.6) \qquad \lim_{\ell \to \infty} \frac{e(B_{R_\ell}(x_0))}{\mu(B_{r_\ell}(x_0))} = \limsup_{R \to 0^+} \frac{e(B_R(x_0))}{\mu(B_R(x_0))}. $$

This last property is obtained by taking first a sequence $(\rho_\ell)_\ell$ such that

$$ \limsup_{R \to 0^+} \frac{e(B_R(x_0))}{\mu(B_R(x_0))} = \lim_{\ell \to \infty} \frac{e(B_{\rho_\ell}(x_0))}{\mu(B_{\rho_\ell}(x_0))}, $$
4.1. LOWER BOUND FOR THE $\Gamma$-lim inf. — Given a Borel function $f: \mathbb{R}^N \times \mathbb{R} \times \mathbb{R}^N \to [0, +\infty]$, we define for every $(x, m) \in \mathbb{R}^N \times \mathbb{R}$,

$$ (4.1) \qquad H_f^{-}(x, m) := H_f(x, m) \wedge (f'_{-}(x, 0^{\pm}, 0)|m|), \quad \text{if } \pm m \ge 0, $$

recalling that $H_f$ is defined in (1.2) and $f'_{-}(x, 0^{\pm}, 0)$ in (2.10), with the usual convention $(\pm\infty) \times 0 = 0$. Notice that it is concave on $\mathbb{R}_+$ and $\mathbb{R}_-$ by Theorem 1.1, and under $(H_6)$ we have $H_f^{-}(x, m) = H_f(x, m)$.

**PROPOSITION 4.1.** — Assume that $(f_{\varepsilon})_{\varepsilon>0}$ is a family of functions $f_{\varepsilon}: \mathbb{R}^N \times \mathbb{R} \times \mathbb{R}^N \to [0, +\infty]$ satisfying $(H_1), (H_2), (H_3), (H_5)$ and $(H_8)$ where $f = \lim_{\varepsilon \to 0} f_{\varepsilon}$. Let $(\varepsilon_n)_{n \in \mathbb{N}}$ be a sequence of positive numbers going to zero, $(u_n)_{n \in \mathbb{N}}$ be a sequence in $W_{\text{loc}}^{1,1}(\mathbb{R}^N)$, and let

$$ e_n := f_{\varepsilon_n}(\cdot, \varepsilon_n^N u_n, \varepsilon_n^{N+1} \nabla u_n) \varepsilon_n^{-N} \mathcal{L}^N $$

be the energy measure associated with $u_n$. If $u_n \mathcal{L}^N \xrightarrow{\mathcal{L}'_0} \mu \in \mathcal{M}(\mathbb{R}^N)$ and $e_n \xrightarrow{\mathcal{L}'_0} e \in \mathcal{M}(\mathbb{R}^N)$, then

$$ (4.2) \qquad e \ge H_f^{-}(\mu). $$

In particular, $\Gamma(\mathcal{L}'_0)$-liminf$_{\varepsilon \to 0} \varepsilon_{\varepsilon} \ge M^{H_f^{-}}$.

**Proof of Proposition 4.1.** — Set $H := H_f^{-}$ and recall that it is concave on $\mathbb{R}_+$ and $\mathbb{R}_-$ by Theorem 1.1. Let us assume first that $u_n \ge 0$ a.e. for every $n$. To obtain (4.2), it is enough to prove that for every $x_0 \in \mathbb{R}^N$,

$$ (4.3) \qquad e(\{x_0\}) \ge H(x_0, \mu(\{x_0\})). $$

and that if $x_0 \in \text{supp } \mu$ is not an atom of $\mu$, then

$$ (4.4) \qquad \limsup_{R \to 0^+} \frac{e(B_R(x_0))}{\mu(B_R(x_0))} \ge H'(x_0, 0^+), $$

Indeed (4.3) implies that $e \ge (H(\mu))^a$ (the atomic part of the measure $H(\mu)$ defined in Definition 2.2) while (4.4) implies that $e \ge H'(\cdot, 0^+)\mu^d = (H(\mu))^d$, by Radon-Nikodým theorem (see [AFP00, Th. 2.22]); these two relations yield $e \ge (H(\mu))^a + (H(\mu))^d = H(\mu)$ as required.

We fix $x_0 \in \text{supp } \mu$ and proceed in several steps.

**Step 1:** blow-up near $x_0$. — We first take two sequences of positive radii $(R_l)_{l \in \mathbb{N}} \to 0$ and $(r_l)_{l \in \mathbb{N}}$ such that for every $l \in \mathbb{N}$, $r_l \in (0, R_l)$,

$$ (4.5) \qquad e(\partial B_{R_l}(x_0)) = \mu(\partial B_{r_l}(x_0)) = 0, $$

and

$$ (4.6) \qquad \lim_{l \to \infty} \frac{e(B_{R_l}(x_0))}{\mu(B_{r_l}(x_0))} = \limsup_{R \to 0^+} \frac{e(B_R(x_0))}{\mu(B_R(x_0))}. $$

This last property is obtained by taking first a sequence $(\rho_l)_l$ such that

$$ \limsup_{R \to 0^+} \frac{e(B_R(x_0))}{\mu(B_R(x_0))} = \lim_{l \to \infty} \frac{e(B_{\rho_l}(x_0))}{\mu(B_{\rho_l}(x_0))}, $$
then using monotone convergence the measures to get first $r_\ell$ then $R_\ell$ such that

$$0 < r_{\ell} < R_{\ell} < \rho_{\ell}, \quad \mu(B_{r_{\ell}}(x_0)) \ge (1 - 2^{-\ell})\mu(B_{\rho_{\ell}}(x_0))$$

$$\text{and} \qquad e(B_{R_\ell}(x_0)) \ge (1 - 2^{-\ell})e(B_{\rho_\ell}(x_0)).$$

By weak convergence and (4.5), according to [AFP00, Prop. 1.62 b)], we have for every $\ell \in \mathbb{N}$,

$$\lim_{n \to \infty} e_n(B_{R_\ell}(x_0)) = e(B_{R_\ell}(x_0)) \quad \text{and} \quad \lim_{n \to \infty} \int_{B_{r_\ell}(x_0)} u_n = \mu(B_{r_\ell}(x_0)).$$

Hence, there exists an extraction $(n_\ell)_{\ell \in \mathbb{N}} \in \Sigma$ such that

$$(4.7) \qquad \lim_{\ell \to \infty} \frac{r_\ell}{\varepsilon_{n_\ell}} = +\infty \quad \text{and} \quad \lim_{\ell \to \infty} \frac{R_\ell - r_\ell}{\varepsilon_{n_\ell}} = +\infty,$$

satisfying the following conditions:

$$(4.8) \qquad \mu(\{x_0\}) = \lim_{\ell \to \infty} \int_{B_{r_\ell}(x_0)} u_{n_\ell}, \quad e(\{x_0\}) = \lim_{\ell \to \infty} e_{n_\ell}(B_{R_\ell}(x_0)),$$

and

$$(4.9) \qquad \limsup_{\ell \to \infty} \frac{e(B_{R_\ell}(x_0))}{\mu(B_{r_\ell}(x_0))} = \lim_{\ell \to \infty} \frac{e_{n_\ell}(B_{R_\ell}(x_0))}{\int_{B_{r_\ell}(x_0)} u_{n_\ell}}.$$

We may rewrite the mass and energy in terms of the re-scaled map $v_\ell$ defined by

$$(4.10) \qquad v_\ell(y) := \varepsilon_{n_\ell}^N u_{n_\ell}(x_0 + \varepsilon_{n_\ell} y), \quad y \in \mathbb{R}^N, \ell \in \mathbb{N},$$

as follows:

$$(4.11) \qquad \int_{B_{r_\ell}(x_0)} u_{n_\ell} = \int_{B_{\varepsilon_{n_\ell}^{-1} r_\ell}} v_\ell,$$

and

$$(4.12) \qquad e_{n_\ell}(B_{R_\ell}(x_0)) = \int_{B_{\varepsilon_{n_\ell}^{-1} R_\ell}} f_{\varepsilon_{n_\ell}}(x_0 + \varepsilon_{n_\ell} y, v_\ell(y), \nabla v_\ell(y)) \, dy.$$

Step 2: proof of (4.3). — By Proposition 3.10, we have

$$(4.13) \qquad e(\{x_0\}) = \lim_{\ell \to \infty} \int_{B_{\varepsilon_{n_\ell}^{-1} R_\ell}} f_{\varepsilon_{n_\ell}}(x_0 + \varepsilon_{n_\ell} y, v_\ell(y), \nabla v_\ell(y)) \, dy \\ \ge m_v f'_-(x_0, 0^+, 0) + \sum_{0 \le i < k} H_f(x_0, m_i).$$

Here $k \in \mathbb{N} \cup \{+\infty\}$ and $m = m_v + \sum_{0 \le i < k} m_i$, with $m_i > 0$, $m_v \ge 0$ and

$$m = \lim_{\ell \to \infty} \int_{B_{\varepsilon_{n_\ell}^{-1} r_\ell}} v_\ell = \mu(\{x_0\}).$$

Since the function $H = H_f^-$, defined in (4.1), is the infimum of two functions which are mass-concave, it is mass-concave hence subadditive. From (4.13) we thus arrive at

$$e(\{x_0\}) \ge H(x_0, m_v) + \sum_{0 \le i < k} H(x_0, m_i) \ge H(x_0, m_v + \sum_{0 \le i < k} m_i) = H(x_0, \mu(\{x_0\})).$$
then using monotone convergence the measures to get first $r_\ell$ then $R_\ell$ such that

$$0 < r_\ell < R_\ell < \rho_\ell, \quad \mu(B_{r_\ell}(x_0)) \ge (1 - 2^{-\ell})\mu(B_{\rho_\ell}(x_0))$$

$$\text{and} \qquad e(B_{R_\ell}(x_0)) \ge (1 - 2^{-\ell})e(B_{\rho_\ell}(x_0)).$$

By weak convergence and (4.5), according to [AFP00, Prop. 1.62 b)], we have for every $\ell \in \mathbb{N}$,

$$\lim_{n \to \infty} e_n(B_{R_\ell}(x_0)) = e(B_{R_\ell}(x_0)) \quad \text{and} \quad \lim_{n \to \infty} \int_{B_{r_\ell}(x_0)} u_n = \mu(B_{r_\ell}(x_0)).$$

Hence, there exists an extraction $(n_\ell)_{\ell \in \mathbb{N}} \in \Sigma$ such that

$$(4.7) \qquad \lim_{\ell \to \infty} \frac{r_\ell}{\varepsilon_{n_\ell}} = +\infty \quad \text{and} \quad \lim_{\ell \to \infty} \frac{R_\ell - r_\ell}{\varepsilon_{n_\ell}} = +\infty,$$

satisfying the following conditions:

$$(4.8) \qquad \mu(\{x_0\}) = \lim_{\ell \to \infty} \int_{B_{r_\ell}(x_0)} u_{n_\ell}, \quad e(\{x_0\}) = \lim_{\ell \to \infty} e_{n_\ell}(B_{R_\ell}(x_0)),$$

and

$$(4.9) \qquad \limsup_{\ell \to \infty} \frac{e(B_{R_\ell}(x_0))}{\mu(B_{r_\ell}(x_0))} = \lim_{\ell \to \infty} \frac{e_{n_\ell}(B_{R_\ell}(x_0))}{\int_{B_{r_\ell}(x_0)} u_{n_\ell}}.$$

We may rewrite the mass and energy in terms of the re-scaled map $v_\ell$ defined by

$$(4.10) \qquad v_\ell(y) := \varepsilon_{n_\ell}^N u_{n_\ell}(x_0 + \varepsilon_{n_\ell} y), \quad y \in \mathbb{R}^N, \ell \in \mathbb{N},$$

as follows:

$$(4.11) \qquad \int_{B_{r_\ell}(x_0)} u_{n_\ell} = \int_{B_{\varepsilon_{n_\ell}^{-1} r_\ell}} v_\ell,$$

and

$$(4.12) \qquad e_{n_\ell}(B_{R_\ell}(x_0)) = \int_{B_{\varepsilon_{n_\ell}^{-1} R_\ell}} f_{\varepsilon_{n_\ell}}(x_0 + \varepsilon_{n_\ell} y, v_\ell(y), \nabla v_\ell(y)) \, dy.$$

Step 2: proof of (4.3). — By Proposition 3.10, we have

$$(4.13) \qquad e(\{x_0\}) = \lim_{\ell \to \infty} \int_{B_{\varepsilon_{n_\ell}^{-1} R_\ell}} f_{\varepsilon_{n_\ell}}(x_0 + \varepsilon_{n_\ell} y, v_\ell(y), \nabla v_\ell(y)) \, dy \\ \ge m_v f'_-(x_0, 0^+, 0) + \sum_{0 \le i < k} H_f(x_0, m_i).$$

Here $k \in \mathbb{N} \cup \{+\infty\}$ and $m = m_v + \sum_{0 \le i < k} m_i$, with $m_i > 0$, $m_v \ge 0$ and

$$m = \lim_{\ell \to \infty} \int_{B_{\varepsilon_{n_\ell}^{-1} r_\ell}} v_\ell = \mu(\{x_0\}).$$

Since the function $H = H_f^-$, defined in (4.1), is the infimum of two functions which are mass-concave, it is mass-concave hence subadditive. From (4.13) we thus arrive at

$$e(\{x_0\}) \ge H(x_0, m_v) + \sum_{0 \le i < k} H(x_0, m_i) \ge H(x_0, m_v + \sum_{0 \le i < k} m_i) = H(x_0, \mu(\{x_0\})).$$
Step 3: proof of (4.4). — Fix $\varepsilon > 0$ and assume that $m = \mu(\{x_0\}) = 0$. In that case, we apply Proposition 3.8 to the sequence of functions $(v_\ell)_{\ell \in \mathbb{N}}$ defined on the sets $\Omega_\ell = B_{\varepsilon n_\ell}^{-1} r_\ell$ and the function $f_\varepsilon$ to get, thanks to (H$_8$):

$$ 
\begin{aligned}
\limsup_{R \to 0^+} \frac{e(B_R(x_0))}{\mu(B_R(x_0))} &= \lim_{\ell \to \infty} \frac{e_{n_\ell}(B_{R_\ell}(x_0))}{\int_{B_{r_\ell}(x_0)} u_{n_\ell}} \\
&\geq \liminf_{\ell \to \infty} \inf \frac{1}{\int_{B_{\varepsilon n_\ell}^{-1} r_\ell} v_\ell} \int_{B_{\varepsilon n_\ell}^{-1} r_\ell} f_\varepsilon(x_0 + \varepsilon_{n_\ell} y, v_\ell(y), \nabla v_\ell(y)) \\
&\geq (f_\varepsilon)'_-(x_0, 0^+, 0).
\end{aligned}
$$

Taking the limit $\varepsilon \to 0^+$, we deduce by (H$_8$) and (4.1):

$$ (4.14) \qquad \limsup_{R \to 0^+} \frac{e(B_R(x_0))}{\mu(B_R(x_0))} \geq f'_-(x_0, 0^+, 0) \geq H'(x_0, 0^+). $$

In view of the discussion at the beginning of the proof, we have now proved (4.2).

Step 4: proof of (4.2) for signed $(u_n)_n$. — Notice that the preceding reasoning for non-negative $u_n$ applies also to the case of non-positive $u_n$. Let us handle the case where the $(u_n)$'s may change sign. We simply apply the above cases to the positive and negative parts $((u_n)_\pm)_n$ which converge weakly as measures (up to subsequence) to some measures $\mu^\pm \in \mathcal{M}_+(\mathbb{R}^N)$ which satisfy $\mu = \mu^+ - \mu^-$, so that $e \geq H(\pm \mu^\pm)$. We know that the Jordan decomposition $\mu = \mu_+ - \mu_-$ is minimal, so that $\mu_\pm \leq \mu^\pm$ and $\mu_+ \perp \mu_-$. By monotonicity of the function $H_f$ (see Theorem 1.1), we have $e \geq H(\pm \mu^\pm) \geq H(\pm \mu_\pm)$. Since $H(\mu_+) \perp H(-\mu_-)$, we get

$$ e \geq H(\mu_+) + H(-\mu_-) = H(\mu). $$

Step 5: lower bound for the $\Gamma$-lim inf. — We justify that (4.2) implies the lower bound $\Gamma(\mathcal{C}'_0)$-lim inf$_{\varepsilon \to 0} \mathcal{E}_\varepsilon \geq M^H$. Indeed, fix $\mu \in \mathcal{M}(\mathbb{R}^N)$ and consider a family $(u_\varepsilon)_{\varepsilon>0}$ weakly converging to $\mu$ as $\varepsilon \to 0$. We need to show that $M^H(\mu) \leq \liminf_{\varepsilon \to 0} \mathcal{E}_\varepsilon(u_\varepsilon)$. Assume without loss of generality that the inferior limit is finite and take a sequence of positive numbers $(\varepsilon_n)_{n \in \mathbb{N}} \to 0$ such that this inferior limit is equal to $\lim_{n \to \infty} \mathcal{E}_{\varepsilon_n}(u_{\varepsilon_n})$. Now the energy measure $e_n$ associated with $u_n = u_{\varepsilon_n}$ has bounded mass and up to extracting a subsequence one may assume that it converges weakly to some measure $e \in \mathcal{M}_+(\mathbb{R}^N)$. By the previous steps, $e \geq H(\mu)$, and by lower semicontinuity and monotonicity of the mass:

$$ \liminf_{\varepsilon \to 0^+} \mathcal{E}_\varepsilon(u_\varepsilon) = \liminf_{n \to \infty} \|e_n\| \geq \|e\| \geq \|H(\mu)\| = M^H(\mu). \quad \square $$

4.2. UPPER BOUND FOR THE $\Gamma$-lim sup. — In this section, we introduce the following substitute for (H$_4$), (H$_7$) and (H$_8$), where $f$, $(f_\varepsilon)_{\varepsilon>0}$ are Borel maps from $\mathbb{R}^N \times \mathbb{R} \to \mathbb{R}^N$ to $[0, +\infty]$:

(U) there exists $C < +\infty$ such that for every $x, y \in \mathbb{R}^N$, $u \in \mathbb{R}$ and $\xi \in \mathbb{R}^N$,

$$ \limsup_{\varepsilon \to 0^+} f_\varepsilon(x + \varepsilon y, u, \xi) \leq f(x, u, \xi) \quad \text{and} \quad f_\varepsilon(y, u, \xi) \leq C(f(x, u, \xi) + u) \quad \forall \varepsilon > 0. $$