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 $\xi \in \mathbb{R}^N$,

$$ \limsup_{\varepsilon \to 0^+} f_\varepsilon(x + \varepsilon y, u, \xi) \leq f(x, u, \xi) \quad \text{and} \quad f_\varepsilon(y, u, \xi) \leq C(f(x, u, \xi) + u) \quad \forall \varepsilon > 0. $$
**PROPOSITION 4.2.** — Assume that $f, (f_\varepsilon)_{\varepsilon>0}$ satisfy (U) and (H$_3$). If $\mu \in \mathcal{M}(\mathbb{R}^N)$, then there exists $(u_\varepsilon)_{\varepsilon>0} \in W_{\text{loc}}^{1,1}(\mathbb{R}^N)$ such that $u_\varepsilon \mathcal{L}^N \xrightarrow{\mathcal{C}'_b} \mu$ when $\varepsilon \to 0$ and which satisfies

$$ \limsup_{\varepsilon \to 0^+} \mathcal{E}_\varepsilon(u_\varepsilon) \leq \mathbf{M}^{H_f, \text{lsc}}(\mu), $$

where $H_f, \text{lsc} \leq H_f$ stands for the lower semicontinuous envelope of $H_f$, defined in (2.5). In other words, we have $\Gamma(\mathcal{C}'_b)$- $\limsup_{\varepsilon \to 0} \mathcal{E}_\varepsilon \leq \mathbf{M}^{H_f, \text{lsc}}$.

**Proof of Proposition 4.2.** — Let $F = \Gamma(\mathcal{C}'_b)$- $\limsup_{\varepsilon \to 0} \mathcal{E}_\varepsilon$. As an upper $\Gamma$-limit, $F$ is sequentially lower semicontinuous in the narrow topology. Hence, by Proposition 2.4, it is enough to prove that $F(\mu) \leq \mathbf{M}^{H_f}(\mu)$ whenever $\mu$ is finitely atomic. Let $\mu = \sum_{i=1}^k m_i \delta_{x_i}$ with $k \in \mathbb{N}$, $m_i \in \mathbb{R}$, $x_i \in \mathbb{R}^N$, and assume without loss of generality that $x_i \neq x_j$ when $i \neq j$ and $\mathbf{M}^{H_f}(\mu) < +\infty$. Fix $\eta > 0$. For each $i = 1, \dots, k$, there exists $u_i \in W_{\text{loc}}^{1,1}(\mathbb{R}^N)$ such that $\int_{\mathbb{R}^N} u_i = m_i$ and $\int_{\mathbb{R}^N} f(x_i, u_i, \nabla u_i) \leq H(x_i, m_i) + \eta < +\infty$. We define for every $i = 1, \dots, k$,

$$ (4.15) \qquad u_{\varepsilon}^{i}(x) = \varepsilon^{-N} u_{i}(\varepsilon^{-1}(x - x_{i})), \quad x \in \mathbb{R}^{N}, $$

and

$$ (4.16) \qquad u_{\varepsilon} = \max\{u_{\varepsilon}^{i} : i = 1, \dots, k\}, $$

which converge narrowly as measures to $u$ as $\varepsilon \to 0$. We have by change of variables:

$$ \begin{aligned} \mathcal{E}_{\varepsilon}(u_{\varepsilon}) &\le \sum_{i=1}^{k} \int_{\{u_{\varepsilon}=u_{\varepsilon}^{i}\}} f_{\varepsilon}(x, \varepsilon^{N} u_{\varepsilon}^{i}(x), \varepsilon^{N+1} \nabla u_{\varepsilon}^{i}(x)) \varepsilon^{-N} dx \\ &\le \sum_{i=1}^{k} \mathcal{E}_{\varepsilon}(u_{\varepsilon}^{i}) = \sum_{i=1}^{k} \int_{\mathbb{R}^{N}} f_{\varepsilon}(x_i + \varepsilon x, u_i, \nabla u_i). \end{aligned} $$

Using our assumption (U) and the dominated convergence theorem, one gets as $\varepsilon \to 0$:

$$ F(\mu) \le \limsup_{\varepsilon \to 0} \mathcal{E}_\varepsilon(u_\varepsilon) \le \sum_{i=1}^k \int_{\mathbb{R}^N} f(x_i, u_i, \nabla u_i) \le \sum_{i=1}^k H(x_i, m_i) + k\eta = \mathbf{M}^H(\mu) + k\eta. $$

The conclusion follows by arbitrariness of $\eta > 0$. $\square$

4.3. **PROOF OF THE MAIN $\Gamma$-CONVERGENCE RESULT.** — We now explain how Theorem 1.2 follows from Proposition 4.1 and Proposition 4.2.

**Proof of Theorem 1.2.** — The lower bound $\Gamma(\mathcal{C}'_0)$- $\liminf_{\varepsilon \to 0} \mathcal{E}_\varepsilon \geq \mathbf{M}^{H_f^-}$ follows from Proposition 4.1, and the upper bound $\Gamma(\mathcal{C}'_b)$- $\limsup_{\varepsilon \to 0} \mathcal{E}_\varepsilon \leq \mathbf{M}^{H_f, \text{lsc}}$ from Proposition 4.2, where the assumption (U) is a consequence of (H$_4$), (H$_7$) and (H$_8$). By (H$_6$) and Theorem 1.1, $H_f^- = H_f$, and since $H_f \geq H_f, \text{lsc}$ by definition, both $\Gamma$-lim inf and $\Gamma$-lim sup (for weak and narrow topologies) coincide with $\mathbf{M}^{H_f}$. $\square$
**PROPOSITION 4.2.** — Assume that $f, (f_\varepsilon)_{\varepsilon>0}$ satisfy (U) and (H$_3$). If $\mu \in \mathcal{M}(\mathbb{R}^N)$, then there exists $(u_\varepsilon)_{\varepsilon>0} \in W_{\text{loc}}^{1,1}(\mathbb{R}^N)$ such that $u_\varepsilon \mathcal{L}^N \xrightarrow{\mathcal{C}'_b} \mu$ when $\varepsilon \to 0$ and which satisfies

$$ \limsup_{\varepsilon \to 0^+} \mathcal{E}_\varepsilon(u_\varepsilon) \leq \mathbf{M}^{H_f, \text{lsc}}(\mu), $$

where $H_f, \text{lsc} \leq H_f$ stands for the lower semicontinuous envelope of $H_f$, defined in (2.5). In other words, we have $\Gamma(\mathcal{C}'_b)$- $\limsup_{\varepsilon \to 0} \mathcal{E}_\varepsilon \leq \mathbf{M}^{H_f, \text{lsc}}$.

**Proof of Proposition 4.2.** — Let $F = \Gamma(\mathcal{C}'_b)$- $\limsup_{\varepsilon \to 0} \mathcal{E}_\varepsilon$. As an upper $\Gamma$-limit, $F$ is sequentially lower semicontinuous in the narrow topology. Hence, by Proposition 2.4, it is enough to prove that $F(\mu) \leq \mathbf{M}^{H_f}(\mu)$ whenever $\mu$ is finitely atomic. Let $\mu = \sum_{i=1}^k m_i \delta_{x_i}$ with $k \in \mathbb{N}$, $m_i \in \mathbb{R}$, $x_i \in \mathbb{R}^N$, and assume without loss of generality that $x_i \neq x_j$ when $i \neq j$ and $\mathbf{M}^{H_f}(\mu) < +\infty$. Fix $\eta > 0$. For each $i = 1, \dots, k$, there exists $u_i \in W_{\text{loc}}^{1,1}(\mathbb{R}^N)$ such that $\int_{\mathbb{R}^N} u_i = m_i$ and $\int_{\mathbb{R}^N} f(x_i, u_i, \nabla u_i) \leq H(x_i, m_i) + \eta < +\infty$. We define for every $i = 1, \dots, k$,

$$ (4.15) \qquad u_{\varepsilon}^{i}(x) = \varepsilon^{-N} u_{i}(\varepsilon^{-1}(x - x_{i})), \quad x \in \mathbb{R}^{N}, $$

and

$$ (4.16) \qquad u_{\varepsilon} = \max\{u_{\varepsilon}^{i} : i = 1, \dots, k\}, $$

which converge narrowly as measures to $u$ as $\varepsilon \to 0$. We have by change of variables:

$$ \begin{aligned} \mathcal{E}_\varepsilon(u_\varepsilon) &\leq \sum_{i=1}^k \int_{\{u_\varepsilon=u_\varepsilon^i\}} f_\varepsilon(x, \varepsilon^N u_\varepsilon^i(x), \varepsilon^{N+1} \nabla u_\varepsilon^i(x)) \varepsilon^{-N} \,dx \\ &\leq \sum_{i=1}^k \mathcal{E}_\varepsilon(u_\varepsilon^i) = \sum_{i=1}^k \int_{\mathbb{R}^N} f_\varepsilon(x_i + \varepsilon x, u_i, \nabla u_i). \end{aligned} $$

Using our assumption (U) and the dominated convergence theorem, one gets as $\varepsilon \to 0$:

$$ F(\mu) \leq \limsup_{\varepsilon \to 0} \mathcal{E}_\varepsilon(u_\varepsilon) \leq \sum_{i=1}^k \int_{\mathbb{R}^N} f(x_i, u_i, \nabla u_i) \leq \sum_{i=1}^k H(x_i, m_i) + k\eta = \mathbf{M}^H(\mu) + k\eta. $$

The conclusion follows by arbitrariness of $\eta > 0$. $\square$

**4.3. PROOF OF THE MAIN $\Gamma$-CONVERGENCE RESULT.** — We now explain how Theorem 1.2 follows from Proposition 4.1 and Proposition 4.2.

**Proof of Theorem 1.2.** — The lower bound $\Gamma(\mathcal{C}'_0)$- $\liminf_{\varepsilon \to 0} \mathcal{E}_\varepsilon \geq \mathbf{M}^{H_f^-}$ follows from Proposition 4.1, and the upper bound $\Gamma(\mathcal{C}'_b)$- $\limsup_{\varepsilon \to 0} \mathcal{E}_\varepsilon \leq \mathbf{M}^{H_f, \text{lsc}}$ from Proposition 4.2, where the assumption (U) is a consequence of (H$_4$), (H$_7$) and (H$_8$). By (H$_6$) and Theorem 1.1, $H_f^- = H_f$, and since $H_f \geq H_f, \text{lsc}$ by definition, both $\Gamma$-lim inf and $\Gamma$-lim sup (for weak and narrow topologies) coincide with $\mathbf{M}^{H_f}$. $\square$
5. EXAMPLES, COUNTEREXAMPLES AND APPLICATIONS

5.1. SCALE-INVARIANT LAGRANGIANS AND NECESSITY OF THE SLOPE ASSUMPTION

Our assumption (H$_6$) is not very standard, but we need a condition of this type
in order to get Γ-convergence of the rescaled energies E_ε towards M^H_f, as shown by
the following class of scale-invariant Lagrangians:

$$
(5.1) \quad f_{\varepsilon}(x, u, \xi) = f(u, \xi) \quad \text{with} \quad f(u, \xi) = \begin{cases} u^{p(1/p^*-1)}|\xi|^p & \text{if } u > 0, \\ 0 & \text{else,} \end{cases}
$$

where $p \in (1, N)$, $N \in \mathbb{N}^*$ and $p^* := pN/(N-p)$. By straightforward computations, $\mathcal{E}_\epsilon(u) = \mathcal{E}_f(u) := \int_{\mathbb{R}^N} f(u, \nabla u)$ for every $\epsilon > 0$ and $u \in W_{\text{loc}}^{1,p}(\mathbb{R}^N)$ in that case.

Moreover, the associated minimal cost function $H_f$ is not trivial. Indeed, applying
the Gagliardo–Nirenberg–Sobolev inequality,

$$
\left( \int_{\mathbb{R}^N} |v|^{p^*} \right)^{1/p^*} \le C \left( \int_{\mathbb{R}^N} |\nabla v|^p \right)^{1/p}, \quad \forall v \in L^{p^*} \cap W_{\text{loc}}^{1,1}(\mathbb{R}^N),
$$

to the function$^{(8)}$ $v = u^{1/p^*}$, we obtain that for every $u \in W_{\text{loc}}^{1,1}(\mathbb{R}^N, \mathbb{R}_+) \cap L^1(\mathbb{R}^N)$,

$$
\left( \int_{\mathbb{R}^N} u \right)^{p/p^*} \le \left( \frac{C}{p^*} \right)^p \int_{\{u>0\}} u^{p/p^*-p} |\nabla u|^p = \left( \frac{C}{p^*} \right)^p \varepsilon_f(u).
$$

Hence, for every $m > 0$, we have $H_f(m) > 0$, and even $H_f(m) < +\infty$ since any function $u = v^{p^*}$, with $v \in W^{1,p}(\mathbb{R}^N, \mathbb{R}_+)$, has finite energy. Replacing $u$ by $mu$ in the infimum defining $H_f$ in (1.2), we actually obtain

$$
(5.2) \qquad H_f(m) = m^{1-p/N} H_f(1), \quad 0 < H_f(1) < +\infty.
$$

In that case, it is clear that the Γ-limit of $\mathcal{E}_\varepsilon \equiv \mathcal{E}$ in the weak or narrow topology of $\mathcal{M}_+(\mathbb{R}^N)$, that is the lower semicontinuous relaxation of $\mathcal{E}_f$, does not coincide with $\mathcal{M}^{H_f}$; indeed, the first functional is finite on diffuse measures whose density has finite energy, while the second functional is always infinite for non-trivial diffuse measures since $H_f'(0^+) = +\infty$.

These scaling invariant Lagrangians are ruled out by our assumption (H$_6$). All the
other assumptions are satisfied except (H$_5$). Note that the following perturbation of f,

$$
\tilde{f}(u, \xi) = (1 + u^{p(1/p^*-1)}) |\xi|^p
$$

satisfies all the assumptions except (H$_6$), and provides a counterexample to the Γ-con-
vergence. Indeed, $M_{H\tilde{f}} \ge M_{H_f}$ is still infinite on diffuse measures, while (the relax-
ation of) $\mathcal{E}_{\tilde{f}}$ is finite for any diffuse measure whose density has finite energy.

We stress that an assumption like (H$_6$) is actually needed, even for the lower semi-
continuity of the function H_f – recall that if M_{H_f} is a Γ-limit, then it must be lower

<footnote>(8)Actually, we apply it to $v_\varepsilon = \phi_\varepsilon(u)$ where $\phi_\varepsilon$ is a suitable approximation of $(\cdot)^{1/p^*}$ and take $\varepsilon \to 0$.</footnote>
# 5. EXAMPLES, COUNTEREXAMPLES AND APPLICATIONS

## 5.1. SCALE-INVARIANT LAGRANGIANS AND NECESSITY OF THE SLOPE ASSUMPTION

Our assumption $(H_6)$ is not very standard, but we need a condition of this type in order to get $\Gamma$-convergence of the rescaled energies $\mathcal{E}_\varepsilon$ towards $M^{H_f}$, as shown by the following class of scale-invariant Lagrangians:

$$ (5.1) \qquad f_{\varepsilon}(x, u, \xi) = f(u, \xi) \quad \text{with} \quad f(u, \xi) = \begin{cases} u^{p(1/p^*-1)}|\xi|^p & \text{if } u > 0, \\ 0 & \text{else,} \end{cases} $$

where $p \in (1, N)$, $N \in \mathbb{N}^*$ and $p^* := pN/(N - p)$. By straightforward computations, $\mathcal{E}_\varepsilon(u) = \mathcal{E}_f(u) := \int_{\mathbb{R}^N} f(u, \nabla u)$ for every $\varepsilon > 0$ and $u \in W_{\text{loc}}^{1,p}(\mathbb{R}^N)$ in that case.

Moreover, the associated minimal cost function $H_f$ is not trivial. Indeed, applying the Gagliardo–Nirenberg–Sobolev inequality,

$$ \left( \int_{\mathbb{R}^N} |v|^{p^*} \right)^{1/p^*} \le C \left( \int_{\mathbb{R}^N} |\nabla v|^p \right)^{1/p}, \quad \forall v \in L^{p^*} \cap W_{\text{loc}}^{1,1}(\mathbb{R}^N), $$

to the function$^{(8)}$ $v = u^{1/p^*}$, we obtain that for every $u \in W_{\text{loc}}^{1,1}(\mathbb{R}^N, \mathbb{R}_+) \cap L^1(\mathbb{R}^N)$,

$$ \left( \int_{\mathbb{R}^N} u \right)^{p/p^*} \le \left( \frac{C}{p^*} \right)^p \int_{\{u>0\}} u^{p/p^*-p} |\nabla u|^p = \left( \frac{C}{p^*} \right)^p \mathcal{E}_f(u). $$

Hence, for every $m > 0$, we have $H_f(m) > 0$, and even $H_f(m) < +\infty$ since any function $u = v^{p^*}$, with $v \in W^{1,p}(\mathbb{R}^N, \mathbb{R}_+)$, has finite energy. Replacing $u$ by $mu$ in the infimum defining $H_f$ in (1.2), we actually obtain

$$ (5.2) \qquad H_f(m) = m^{1-p/N} H_f(1), \quad 0 < H_f(1) < +\infty. $$

In that case, it is clear that the $\Gamma$-limit of $\mathcal{E}_\varepsilon \equiv \mathcal{E}$ in the weak or narrow topology of $\mathcal{M}_+(\mathbb{R}^N)$, that is the lower semicontinuous relaxation of $\mathcal{E}_f$, does not coincide with $M^{H_f}$; indeed, the first functional is finite on diffuse measures whose density has finite energy, while the second functional is always infinite for non-trivial diffuse measures since $H_f'(0^+) = +\infty$.

These scaling invariant Lagrangians are ruled out by our assumption $(H_6)$. All the other assumptions are satisfied except $(H_5)$. Note that the following perturbation of $f$,

$$ \tilde{f}(u, \xi) = (1 + u^{p(1/p^*-1)}) |\xi|^p $$

satisfies all the assumptions except $(H_6)$, and provides a counterexample to the $\Gamma$-convergence. Indeed, $M_{H_f} \ge M_{H_f}$ is still infinite on diffuse measures, while (the relaxation of) $\mathcal{E}_f$ is finite for any diffuse measure whose density has finite energy.

We stress that an assumption like $(H_6)$ is actually needed, even for the lower semi-continuity of the function $H_f$ – recall that if $M_{H_f}$ is a $\Gamma$-limit, then it must be lower

<footnote>$^{(8)}$Actually, we apply it to $v_\varepsilon = \phi_\varepsilon(u)$ where $\phi_\varepsilon$ is a suitable approximation of $(\cdot)^{1/p^*}$ and take $\varepsilon \to 0$.</footnote>
semicontinuous by [Bra02, Prop. 1.28], which in turn implies that the function $H_f$ is lower semicontinuous by Proposition 2.4. Indeed, consider the Lagrangians

$$f(x, u, \xi) = (1 + u^{p(1/p^*-1)})|\xi|^{p(x)},$$

with $p \in \mathcal{C}^0(\mathbb{R}^N, (1, N))$ such that $p(0) = p \in (1, N)$ and $p(x) > p$ when $x \neq 0$. Then, we have $H_f(0, m) = m^{1-p/N}H(1)$, but $H_f(x, \cdot) \equiv 0$ if $x \neq 0$ as can be easily seen via the change of function $\varepsilon^N u(\varepsilon \cdot)$, with $\varepsilon > 0$ small.

## 5.2. GENERAL CONCAVE COSTS IN DIMENSION ONE.
It has been proved in [Wir19] that for any continuous concave function $H: \mathbb{R}_+ \to \mathbb{R}_+$ with $H(0) = 0$, there exists a function $c: \mathbb{R}_+ \to \mathbb{R}_+$ such that $c(0) = 0$, $u \mapsto c(u)/u$ is lower semicontinuous and non-increasing on $(0, +\infty)$, and for every $m \ge 0$,

$$H(m) = \inf \left\{ \int_{\mathbb{R}} |u'|^2 + c(u) : u \in W_{\text{loc}}^{1,1}(\mathbb{R}, \mathbb{R}_+), \int_{\mathbb{R}} u = m \right\}.$$

The Lagrangians of the form $f_{\varepsilon}(x, u, \xi) = |\xi|^2 + c(u)$, in dimension $N=1$, satisfy all our assumptions $(H_1)-(H_8)$, hence our $\Gamma$-convergence result stated in Theorem 1.2 yields the $\Gamma$-convergence of the functionals

$$\mathcal{E}_{\varepsilon}(u) = \int_{\mathbb{R}} \varepsilon^3 |u'|^2 + \frac{c(\varepsilon u)}{\varepsilon}, \quad u \in W^{1,2}(\mathbb{R}, \mathbb{R}_+),$$

towards $M^H$ for both the weak and narrow convergence of measures. Therefore, we may find an elliptic approximation of any concave $H$-mass. Let us stress that $c$ is determined in [Wir19] from $H$ through several operations including a deconvolution problem, but no closed form solution is given in general; nonetheless, an explicit solution is provided if $c$ is affine by parts.

In higher dimension $N \ge 2$, Proposition 2.8 tells us that the class of functions $H = H_f$ with $f$ satisfying $(H_1)-(H_8)$ is smaller, namely, $H$ must satisfy:

$$(5.3) \qquad \exists m_* \ge 0, \quad \begin{cases} H \text{ is linear on } [0, m_*], \\ H \text{ is strictly concave } (m_*, +\infty). \end{cases}$$

We have no positive or negative answer to the inverse problem, consisting in finding $f$ satisfying our assumptions such that $H_f = H$, for a given continuous concave function $H: \mathbb{R}_+ \to \mathbb{R}_+$ satisfying (5.3).

## 5.3. HOMOGENEOUS COSTS IN ANY DIMENSION.
In this section, we provide Lagrangians $f$ to obtain the $\alpha$-mass $M^\alpha := M^{t \to t^\alpha}$ in any dimension $N$ for a wide range of exponents, including exponents $\alpha \in (1 - \frac{1}{N}, 1]$. We consider for every $p \in [1, +\infty)$, $s \in (-\infty, 1]$ and $N \in \mathbb{N}^*$, the energy defined for every $u \in W_{\text{loc}}^{1,1}(\mathbb{R}^N, \mathbb{R}_+)$ by

$$(5.4) \qquad \mathcal{E}_f(u) := \int_{\mathbb{R}^N} f(u, \nabla u) := \int_{\mathbb{R}^N} |\nabla u|^p + u^s \mathbf{1}_{u>0}.$$

Notice that for $p > 1$, $f$ satisfies all our hypotheses $(H_1)-(H_6)$; in particular, $(H_6)$ holds as a consequence of the stronger condition (S) (see Corollary 2.7) which is satisfied with $\rho(t) = t$ in dimension $N \ge 2$. Thus by Theorem 1.2 the re-scaled
semicontinuous by [Bra02, Prop. 1.28], which in turn implies that the function $H_f$ is lower semicontinuous by Proposition 2.4. Indeed, consider the Lagrangians

$$f(x, u, \xi) = (1 + u^{p(1/p^*-1)})|\xi|^{p(x)},$$

with $p \in \mathcal{C}^0(\mathbb{R}^N, (1, N))$ such that $p(0) = p \in (1, N)$ and $p(x) > p$ when $x \neq 0$. Then, we have $H_f(0, m) = m^{1-p/N}H(1)$, but $H_f(x, \cdot) \equiv 0$ if $x \neq 0$ as can be easily seen via the change of function $\varepsilon^N u(\varepsilon \cdot)$, with $\varepsilon > 0$ small.

## 5.2. GENERAL CONCAVE COSTS IN DIMENSION ONE.
It has been proved in [Wir19] that for any continuous concave function $H: \mathbb{R}_+ \to \mathbb{R}_+$ with $H(0) = 0$, there exists a function $c: \mathbb{R}_+ \to \mathbb{R}_+$ such that $c(0) = 0$, $u \mapsto c(u)/u$ is lower semicontinuous and non-increasing on $(0, +\infty)$, and for every $m \ge 0$,

$$H(m) = \inf \left\{ \int_{\mathbb{R}} |u'|^2 + c(u) : u \in W_{\text{loc}}^{1,1}(\mathbb{R}, \mathbb{R}_+), \int_{\mathbb{R}} u = m \right\}.$$

The Lagrangians of the form $f_\varepsilon(x, u, \xi) = |\xi|^2 + c(u)$, in dimension $N=1$, satisfy all our assumptions $(H_1)-(H_8)$, hence our $\Gamma$-convergence result stated in Theorem 1.2 yields the $\Gamma$-convergence of the functionals

$$\mathcal{E}_\varepsilon(u) = \int_{\mathbb{R}} \varepsilon^3 |u'|^2 + \frac{c(\varepsilon u)}{\varepsilon}, \quad u \in W^{1,2}(\mathbb{R}, \mathbb{R}_+),$$

towards $M^H$ for both the weak and narrow convergence of measures. Therefore, we may find an elliptic approximation of any concave $H$-mass. Let us stress that $c$ is determined in [Wir19] from $H$ through several operations including a deconvolution problem, but no closed form solution is given in general; nonetheless, an explicit solution is provided if $c$ is affine by parts.

In higher dimension $N \ge 2$, Proposition 2.8 tells us that the class of functions $H = H_f$ with $f$ satisfying $(H_1)-(H_8)$ is smaller, namely, $H$ must satisfy:

$$(5.3) \qquad \exists m_* \ge 0, \quad \begin{cases} H \text{ is linear on } [0, m_*], \\ H \text{ is strictly concave } (m_*, +\infty). \end{cases}$$

We have no positive or negative answer to the inverse problem, consisting in finding $f$ satisfying our assumptions such that $H_f = H$, for a given continuous concave function $H: \mathbb{R}_+ \to \mathbb{R}_+$ satisfying (5.3).

## 5.3. HOMOGENEOUS COSTS IN ANY DIMENSION.
In this section, we provide Lagrangians $f$ to obtain the $\alpha$-mass $M^\alpha := M^{t \to t^\alpha}$ in any dimension $N$ for a wide range of exponents, including exponents $\alpha \in (1 - 1/N, 1]$. We consider for every $p \in [1, +\infty)$, $s \in (-\infty, 1]$ and $N \in \mathbb{N}^*$, the energy defined for every $u \in W_{\text{loc}}^{1,1}(\mathbb{R}^N, \mathbb{R}_+)$ by

$$(5.4) \qquad \mathcal{E}_f(u) := \int_{\mathbb{R}^N} f(u, \nabla u) := \int_{\mathbb{R}^N} |\nabla u|^p + u^s \mathbf{1}_{u>0}.$$

Notice that for $p > 1$, $f$ satisfies all our hypotheses $(H_1)-(H_6)$; in particular, $(H_6)$ holds as a consequence of the stronger condition (S) (see Corollary 2.7) which is satisfied with $\rho(t) = t$ in dimension $N \ge 2$. Thus by Theorem 1.2 the re-scaled
energies Γ-converge to the $H_f$-mass. In this case, we may show that $H_f(m) = cm^\alpha$ for some $\alpha \in (0, 1)$, $c \in [0, +\infty]$, and the constant $c$ belongs to $(0, +\infty)$ if and only if $s \in (-p', 1)$. Details are given hereafter.

*Homogeneity of $H_f$.* — In order to compute $H_f$, one may first express $H_f(m)$ as the minimum of a scaling invariant expression by optimizing $\mathcal{E}_f$ over all mass-invariant rescalings $u_\lambda = \lambda^N v(\lambda)$ of a given function $u \in W_{\text{loc}}^{1,1}(\mathbb{R}^N, \mathbb{R}_+)$; it yields

$$ (5.5) \quad H_f(m) = \inf \left\{ \inf_{0<\lambda<+\infty} \lambda^{Np+p-N} \int_{\mathbb{R}^N} |\nabla u|^p + \lambda^{Ns-N} \int_{\{u>0\}} u^s : \right. \\ \qquad \qquad \qquad \qquad \left. u \in W_{\text{loc}}^{1,1}(\mathbb{R}^N, \mathbb{R}_+), \int_{\mathbb{R}^N} u = m \right\}. $$

Computing the infimum w.r.t. $\lambda$, we obtain

$$ (5.6) \quad H_f(m) = c_{N,p,s} \inf \left\{ \left( \int_{\mathbb{R}^N} |\nabla u|^p \right)^{\alpha_1} \left( \int_{\{u>0\}} u^s \right)^{\alpha_2} : \right. \\ \qquad \qquad \qquad \qquad \left. u \in W_{\text{loc}}^{1,1}(\mathbb{R}^N, \mathbb{R}_+), \int_{\mathbb{R}^N} u = m \right\}, $$

with the two exponents

$$ \alpha_1 := \frac{N(1-s)}{Np+p-Ns}, \quad \alpha_2 := \frac{Np+p-N}{Np+p-Ns} $$

and the constant

$$ c_{N,p,s} := \frac{(Np+p-Ns)(N-Ns)^{N(s-1)/(Np+p-Ns)}}{(Np+p-N)^{(Np+p-N)/(Np+p-Ns)}}. $$

In particular, substituting $u$ with $mu$ in (5.6) gives

$$ (5.7) \quad H_f(m) = m^\alpha H_f(1), \quad \text{with } \alpha := p\alpha_1 + s\alpha_2 = \frac{Np + sp - Ns}{Np + p - Ns}. $$

It remains to ensure that this function is not trivial, i.e., $0 < H_f(1) < +\infty$.

*Upper bound: $H_f(1) < +\infty$.* — In the case $s \in [0, 1]$, any $u \in \mathcal{C}_c^1(\mathbb{R}^N)$ has finite energy, thus $H_f$ is non-trivial for every $p \in [1, +\infty)$. In the case $s < 0$, consider the competitor $u: x \mapsto (1-|x|)^\gamma$ for $\gamma > 0$ to be fixed later. Then $\int_{\mathbb{R}^N} |\nabla u|^p < +\infty$ if and only if $t \mapsto (1-t)^{(\gamma-1)p}$ is integrable at $1^-$, that is if $(\gamma-1)p > -1$ or, equivalently, $\gamma > 1 - 1/p$. Similarly, $\int_{\{u>0\}} u^s < +\infty$ if and only if $\gamma s > -1$ or, equivalently, $\gamma < -1/s$. Therefore, one may find $\gamma > 0$ satisfying both conditions, and ensure that $H_f$ is non-trivial, if

$$ -p' < s. $$

*Lower bound: $H_f(1) > 0$.* — Here, we can assume w.l.o.g. that $H_f(1) < +\infty$. In the case $s = 1$, taking any competitor $u$ is with finite energy and $\lambda \to 0^+$ in (5.5) yields $H_f(m) = m$ for every $m \ge 0$. Hence, we may also assume that $s < 1$. Note that in
energies Γ-converge to the $H_f$-mass. In this case, we may show that $H_f(m) = cm^\alpha$ for some $\alpha \in (0, 1)$, $c \in [0, +\infty]$, and the constant $c$ belongs to $(0, +\infty)$ if and only if $s \in (-p', 1)$. Details are given hereafter.

*Homogeneity of $H_f$.* — In order to compute $H_f$, one may first express $H_f(m)$ as the minimum of a scaling invariant expression by optimizing $\mathcal{E}_f$ over all mass-invariant rescalings $u_\lambda = \lambda^N v(\lambda)$ of a given function $u \in W_{\text{loc}}^{1,1}(\mathbb{R}^N, \mathbb{R}_+)$; it yields

$$ (5.5) \quad H_f(m) = \inf \left\{ \inf_{0<\lambda<+\infty} \lambda^{Np+p-N} \int_{\mathbb{R}^N} |\nabla u|^p + \lambda^{Ns-N} \int_{\{u>0\}} u^s : \right. \\ \qquad \qquad \qquad \qquad \qquad \qquad \left. u \in W_{\text{loc}}^{1,1}(\mathbb{R}^N, \mathbb{R}_+), \int_{\mathbb{R}^N} u = m \right\}. $$

Computing the infimum w.r.t. $\lambda$, we obtain

$$ (5.6) \quad H_f(m) = c_{N,p,s} \inf \left\{ \left( \int_{\mathbb{R}^N} |\nabla u|^p \right)^{\alpha_1} \left( \int_{\{u>0\}} u^s \right)^{\alpha_2} : \right. \\ \qquad \qquad \qquad \qquad \qquad \qquad \left. u \in W_{\text{loc}}^{1,1}(\mathbb{R}^N, \mathbb{R}_+), \int_{\mathbb{R}^N} u = m \right\}, $$

with the two exponents

$$ \alpha_1 := \frac{N(1-s)}{Np+p-Ns}, \quad \alpha_2 := \frac{Np+p-N}{Np+p-Ns} $$

and the constant

$$ c_{N,p,s} := \frac{(Np+p-Ns)(N-Ns)^{N(s-1)/(Np+p-Ns)}}{(Np+p-N)^{(Np+p-N)/(Np+p-Ns)}}. $$

In particular, substituting $u$ with $mu$ in (5.6) gives

$$ (5.7) \quad H_f(m) = m^\alpha H_f(1), \quad \text{with } \alpha := p\alpha_1 + s\alpha_2 = \frac{Np + sp - Ns}{Np + p - Ns}. $$

It remains to ensure that this function is not trivial, i.e., $0 < H_f(1) < +\infty$.

*Upper bound: $H_f(1) < +\infty$.* — In the case $s \in [0, 1]$, any $u \in \mathcal{C}_c^1(\mathbb{R}^N)$ has finite energy, thus $H_f$ is non-trivial for every $p \in [1, +\infty)$. In the case $s < 0$, consider the competitor $u: x \mapsto (1-|x|)^\gamma$ for $\gamma > 0$ to be fixed later. Then $\int_{\mathbb{R}^N} |\nabla u|^p < +\infty$ if and only if $t \mapsto (1-t)^{(\gamma-1)p}$ is integrable at $1^-$, that is if $(\gamma-1)p > -1$ or, equivalently, $\gamma > 1 - 1/p$. Similarly, $\int_{\{u>0\}} u^s < +\infty$ if and only if $\gamma s > -1$ or, equivalently, $\gamma < -1/s$. Therefore, one may find $\gamma > 0$ satisfying both conditions, and ensure that $H_f$ is non-trivial, if

$$ -p' < s. $$

*Lower bound: $H_f(1) > 0$.* — Here, we can assume w.l.o.g. that $H_f(1) < +\infty$. In the case $s = 1$, taking any competitor $u$ is with finite energy and $\lambda \to 0^+$ in (5.5) yields $H_f(m) = m$ for every $m \ge 0$. Hence, we may also assume that $s < 1$. Note that in
view of (5.6) and (5.7), we know that $H_f(1)$ is related to the greatest constant $c \ge 0$ such that the Gagliardo-Nirenberg-Sobolev (GNS) type inequality$^{(9)}$

$$ (5.8) \quad c\left(\int_{\mathbb{R}^N} u\right) \le \left(\int_{\mathbb{R}^N} |\nabla u|^p\right)^{Np(1-s)/(Np+sp-Ns)} \left(\int_{\{u>0\}} u^s\right)^{(Np+p-N)/(Np+sp-Ns)}, \quad \forall u \in W_{\text{loc}}^{1,1}(\mathbb{R}^N) $$

holds true, namely, we have $c = (H_f(1)/c_{N,p,s})^{1/\alpha}$; hence, the inequality (5.8) holds with $c > 0$ if and only if $H_f(1) > 0$. This is actually equivalent to proving existence of a minimizer of $\mathcal{E}_f$ over functions $u \in W_{\text{loc}}^{1,1}(\mathbb{R}^N, \mathbb{R}_+)$ with $\int_{\mathbb{R}^N} u = 1$ in view of Proposition 3.12. We prove this, together with uniqueness and basic properties of optimal profiles in the next paragraph.

**EXISTENCE, UNIQUENESS AND PROPERTIES OF OPTIMAL PROFILES.** — It is well-known that optimal profiles in the classical Gagliardo-Nirenberg-Sobolev inequalities do exist; besides, up to rescalings and translations they are unique, radially decreasing and compactly supported. Existence of radially decreasing solutions is a consequence of Pólya-Szegö inequality (see for instance [BZ88]), compactness of the support follows for example from the compact support principle of Pucci, Serrin and Zou [PSZ99] or Pohozaev-type identities, and uniqueness from the work of Serrin and Tang [ST00] for instance. All these techniques may be adapted to our case for exponents $s \in [0, 1)$, but we did not find a comprehensive reference, even more so when $s \in (-p', 0)$, except for $p=2$ and $s < 0$ which is treated in [Dub98]. For this reason and for self-containedness, we provide a sketch of proof for the existence, symmetry, compactness of the support for $s \in (-p', 1]$) of optimal profiles in (5.4). We are only able to justify uniqueness for $s \in (0, 1]$.

1. Existence of an optimal profile which is radially symmetric. — The radially symmetric decreasing rearrangement $u^*$ of an admissible function $u \in W_{\text{loc}}^{1,1}(\mathbb{R}^N, \mathbb{R}_+)$ satisfies $\|\nabla u^*\|_p \le \|\nabla u\|_p$ by the Pólya-Szegö inequality, and we have $\int_{\mathbb{R}^N} |u^*|^s \le \int_{\mathbb{R}^N} |u|^s$ by equimeasurability. Hence,

$$ \mathcal{E}_f(u^*) \le \mathcal{E}_f(u). $$

Thus we can restrict the minimization to radially symmetric non-increasing functions. Take a minimizing sequence $(u_n)_n$ in this class: $u_n(\cdot) = v_n(|\cdot|)$ with $v_n: [0, +\infty)$ non-increasing, $\int_{\mathbb{R}^N} u_n = m \in \mathbb{R}_+$, and $\mathcal{E}_f(u_n) \to H_f(m)$. $(u_n)_n$ is weakly precompact in $L_{\text{loc}}^1(\mathbb{R}^N)$, by compact Sobolev embedding, and even globally precompact in $L^1(\mathbb{R}^N)$. Indeed, when $s > 0$, the upper bound

$$ |\mathbb{S}^{N-1}| v_n(r)^s \frac{r^N}{N} \le |\mathbb{S}^{N-1}| \int_0^r v_n(t)^s t^{N-1} dt \le \mathcal{E}_f(u_n) \le C < +\infty, \quad \forall r \in \mathbb{R}_+, $$

<footnote>$^{(9)}$The notable difference with the classical GNS inequality is that the exponent $s$ is smaller than 1, and may even be negative.</footnote>
view of (5.6) and (5.7), we know that $H_f(1)$ is related to the greatest constant $c \ge 0$ such that the Gagliardo-Nirenberg-Sobolev (GNS) type inequality$^{(9)}$

$$ (5.8) \quad c\left(\int_{\mathbb{R}^N} u\right) \le \left(\int_{\mathbb{R}^N} |\nabla u|^p\right)^{Np(1-s)/(Np+sp-Ns)} \left(\int_{\{u>0\}} u^s\right)^{(Np+p-N)/(Np+sp-Ns)}, \quad \forall u \in W_{\text{loc}}^{1,1}(\mathbb{R}^N) $$

holds true, namely, we have $c = (H_f(1)/c_{N,p,s})^{1/\alpha}$; hence, the inequality (5.8) holds with $c > 0$ if and only if $H_f(1) > 0$. This is actually equivalent to proving existence of a minimizer of $\mathcal{E}_f$ over functions $u \in W_{\text{loc}}^{1,1}(\mathbb{R}^N, \mathbb{R}_+)$ with $\int_{\mathbb{R}^N} u = 1$ in view of Proposition 3.12. We prove this, together with uniqueness and basic properties of optimal profiles in the next paragraph.

**EXISTENCE, UNIQUENESS AND PROPERTIES OF OPTIMAL PROFILES.** — It is well-known that optimal profiles in the classical Gagliardo-Nirenberg-Sobolev inequalities do exist; besides, up to rescalings and translations they are unique, radially decreasing and compactly supported. Existence of radially decreasing solutions is a consequence of Pólya-Szegö inequality (see for instance [BZ88]), compactness of the support follows for example from the compact support principle of Pucci, Serrin and Zou [PSZ99] or Pohozaev-type identities, and uniqueness from the work of Serrin and Tang [ST00] for instance. All these techniques may be adapted to our case for exponents $s \in [0, 1)$, but we did not find a comprehensive reference, even more so when $s \in (-p', 0)$, except for $p=2$ and $s < 0$ which is treated in [Dub98]. For this reason and for self-containedness, we provide a sketch of proof for the existence, symmetry, compactness of the support for $s \in (-p', 1]$) of optimal profiles in (5.4). We are only able to justify uniqueness for $s \in (0, 1]$.

1. Existence of an optimal profile which is radially symmetric. — The radially symmetric decreasing rearrangement $u^*$ of an admissible function $u \in W_{\text{loc}}^{1,1}(\mathbb{R}^N, \mathbb{R}_+)$ satisfies $\|\nabla u^*\|_p \le \|\nabla u\|_p$ by the Pólya-Szegö inequality, and we have $\int_{\mathbb{R}^N} |u^*|^s \le \int_{\mathbb{R}^N} |u|^s$ by equimeasurability. Hence,

$$ \mathcal{E}_f(u^*) \le \mathcal{E}_f(u). $$

Thus we can restrict the minimization to radially symmetric non-increasing functions. Take a minimizing sequence $(u_n)_n$ in this class: $u_n(\cdot) = v_n(|\cdot|)$ with $v_n: [0, +\infty)$ non-increasing, $\int_{\mathbb{R}^N} u_n = m \in \mathbb{R}_+$, and $\mathcal{E}_f(u_n) \to H_f(m)$. $(u_n)_n$ is weakly precompact in $L_{\text{loc}}^1(\mathbb{R}^N)$, by compact Sobolev embedding, and even globally precompact in $L^1(\mathbb{R}^N)$. Indeed, when $s > 0$, the upper bound

$$ |\mathbb{S}^{N-1}| v_n(r)^s \frac{r^N}{N} \le |\mathbb{S}^{N-1}| \int_0^r v_n(t)^s t^{N-1} dt \le \mathcal{E}_f(u_n) \le C < +\infty, \quad \forall r \in \mathbb{R}_+, $$

<footnote>$^{(9)}$The notable difference with the classical GNS inequality is that the exponent $s$ is smaller than 1, and may even be negative.</footnote>
gives a uniform integrable decay in $|x|^{-N/s}$ at infinity; when $s \le 0$, we use the elementary inequality $t + t^s \ge 1$ for all $t > 0$, to obtain

$$ (5.9) \qquad |\{u_n > 0\}| \le \int_{\{u_n > 0\}} (u_n + u_n^s) dx \le m + C < +\infty, $$

so that the size of the support of $u_n$ (a ball of radius $R_n > 0$) is uniformly bounded with respect to $n$.

We deduce by lower semicontinuity of $\mathcal{E}_f$, that, extracting a subsequence if necessary, $(u_n)_n$ converges in $L^1$ to a minimizer $u$ of $\mathcal{E}_f$ with mass $m$, i.e., $\mathcal{E}_f(u) = H_f(m)$ and $\int_{\mathbb{R}^N} u = m$. In particular,

$$ H_f(1) > 0. $$

2. *Euler-Lagrange equation and compactness of the support.* — Let $u(\cdot) = v(|\cdot|)$ be a global minimizer of (5.4) with mass $m > 0$, with $v: \mathbb{R}_+ \to \mathbb{R}_+$ non-increasing. Let also $\bar{B}(0, R)$ be the support of $u$, with $0 < R \le +\infty$. (Here, $\bar{B}(0, R) = \mathbb{R}^N$ if $R = +\infty$.) Computing the first order variation of the energy, we obtain that for every test function $w \in C_0^1(\mathbb{R}^N)$ which is compactly supported in $B(0, R)$ and satisfies $\int_{\mathbb{R}^N} w(x) dx = 0$, we have

$$ (5.10) \qquad \langle \delta \mathcal{E}_f(u), w \rangle := \int_{\mathbb{R}^N} (p(\nabla u)^{p-1} \cdot \nabla w + s u^{s-1} w) dx = 0. $$

In other words, $u$ solves in the weak sense the Euler-Lagrange equation

$$ -p\Delta_p u + s u^{s-1} = \lambda, \quad \text{in } B(0, R), $$

where $\lambda \in \mathbb{R}$ is the Lagrange multiplier associated to the mass constraint. One can see that

$$ \lambda = H'_f(m) = \alpha m^{\alpha-1} H_f(1). $$

Indeed, let $u_1$ be a minimizer of $\mathcal{E}_f$ of mass 1; then a minimizer of $\mathcal{E}_f$ with mass $m$ is given by $u_m(\cdot) := m\lambda_m^N u_1(\lambda_m \cdot)$, with $\lambda_m = m^{s-p/(Np+p-Ns)}$ (see (5.5)); hence,

$$ \begin{align} (5.11) \qquad H'_f(m) &= \frac{d}{dm} \mathcal{E}_f(u_m) = \langle \delta \mathcal{E}_f(u_m), \frac{d}{dm} u_m \rangle_{L^2(\mathbb{R}^N)} \nonumber \\ &= \langle \lambda, \frac{d}{dm} u_m \rangle_{L^2(\mathbb{R}^N)} = \frac{d}{dm} \langle \lambda, u_m \rangle_{L^2(\mathbb{R}^N)} = \frac{d}{dm} (\lambda m) = \lambda. \end{align} $$

In particular, $\lambda$ is unique.

From the Euler-Lagrange equation, we also get that $u$ is smooth in $B(0, R)$ by a bootstrap argument; hence, $u$ solves the Euler-Lagrange equation in the classical sense. In terms of the profile $v$, the Euler-Lagrange equation rewrites

$$ (5.12) \qquad -(pv'(r)^{p-1}r^{N-1})' + sv(r)^{s-1}r^{N-1} = \lambda r^{N-1}, \quad \forall r \in (0, R). $$

For every $r \in (0, R)$, integrating (5.12) on $(0, r)$ yields

$$ (5.13) \qquad pv'(r)^{p-1}r^{N-1} = \int_0^r (sv(\rho)^{s-1} - \lambda)\rho^{N-1} d\rho. $$

The LHS in (5.13) is non-positive as $v' \le 0$ in $(0, R)$. If $s > 0$ and $R = +\infty$, the RHS is $+\infty$ since the integrand goes to $+\infty$ as $\rho \to R$, which is a contradiction. When $s \le 0$, we already saw in (5.9) that the support of $v$ is bounded. In any case, we have thus proved that

$$ R < +\infty. $$
gives a uniform integrable decay in $|x|^{-N/s}$ at infinity; when $s \le 0$, we use the elementary inequality $t + t^s \ge 1$ for all $t > 0$, to obtain

$$ (5.9) \qquad |\{u_n > 0\}| \le \int_{\{u_n > 0\}} (u_n + u_n^s) dx \le m + C < +\infty, $$

so that the size of the support of $u_n$ (a ball of radius $R_n > 0$) is uniformly bounded with respect to $n$.

We deduce by lower semicontinuity of $\mathcal{E}_f$, that, extracting a subsequence if necessary, $(u_n)_n$ converges in $L^1$ to a minimizer $u$ of $\mathcal{E}_f$ with mass $m$, i.e., $\mathcal{E}_f(u) = H_f(m)$ and $\int_{\mathbb{R}^N} u = m$. In particular,

$$ H_f(1) > 0. $$

2. *Euler-Lagrange equation and compactness of the support.* — Let $u(\cdot) = v(|\cdot|)$ be a global minimizer of (5.4) with mass $m > 0$, with $v: \mathbb{R}_+ \to \mathbb{R}_+$ non-increasing. Let also $\bar{B}(0, R)$ be the support of $u$, with $0 < R \le +\infty$. (Here, $\bar{B}(0, R) = \mathbb{R}^N$ if $R = +\infty$.) Computing the first order variation of the energy, we obtain that for every test function $w \in C_0^1(\mathbb{R}^N)$ which is compactly supported in $B(0, R)$ and satisfies $\int_{\mathbb{R}^N} w(x) dx = 0$, we have

$$ (5.10) \qquad \langle \delta \mathcal{E}_f(u), w \rangle := \int_{\mathbb{R}^N} (p(\nabla u)^{p-1} \cdot \nabla w + s u^{s-1} w) dx = 0. $$

In other words, $u$ solves in the weak sense the Euler-Lagrange equation

$$ -p\Delta_p u + s u^{s-1} = \lambda, \quad \text{in } B(0, R), $$

where $\lambda \in \mathbb{R}$ is the Lagrange multiplier associated to the mass constraint. One can see that

$$ \lambda = H'_f(m) = \alpha m^{\alpha-1} H_f(1). $$

Indeed, let $u_1$ be a minimizer of $\mathcal{E}_f$ of mass 1; then a minimizer of $\mathcal{E}_f$ with mass $m$ is given by $u_m(\cdot) := m\lambda_m^N u_1(\lambda_m \cdot)$, with $\lambda_m = m^{s-p/(Np+p-Ns)}$ (see (5.5)); hence,

$$ (5.11) \qquad H'_f(m) = \frac{d}{dm} \mathcal{E}_f(u_m) = \langle \delta \mathcal{E}_f(u_m), \frac{d}{dm} u_m \rangle_{L^2(\mathbb{R}^N)} \\ = \langle \lambda, \frac{d}{dm} u_m \rangle_{L^2(\mathbb{R}^N)} = \frac{d}{dm} \langle \lambda, u_m \rangle_{L^2(\mathbb{R}^N)} = \frac{d}{dm} (\lambda m) = \lambda. $$

In particular, $\lambda$ is unique.

From the Euler-Lagrange equation, we also get that $u$ is smooth in $B(0, R)$ by a bootstrap argument; hence, $u$ solves the Euler-Lagrange equation in the classical sense. In terms of the profile $v$, the Euler-Lagrange equation rewrites

$$ (5.12) \qquad -(pv'(r)^{p-1}r^{N-1})' + sv(r)^{s-1}r^{N-1} = \lambda r^{N-1}, \quad \forall r \in (0, R). $$

For every $r \in (0, R)$, integrating (5.12) on $(0, r)$ yields

$$ (5.13) \qquad pv'(r)^{p-1}r^{N-1} = \int_0^r (sv(\rho)^{s-1} - \lambda)\rho^{N-1} d\rho. $$

The LHS in (5.13) is non-positive as $v' \le 0$ in $(0, R)$. If $s > 0$ and $R = +\infty$, the RHS is $+\infty$ since the integrand goes to $+\infty$ as $\rho \to R$, which is a contradiction. When $s \le 0$, we already saw in (5.9) that the support of $v$ is bounded. In any case, we have thus proved that

$$ R < +\infty. $$
3. Uniqueness when $s > 0$. — We justify that there is a unique minimizer of the energy with mass $m$. Once we know that $v'$ does not vanish on $(0, R)$, the case of equality in Pólya–Szegő inequality (see [BZ88, Th. 1.1]) implies that any minimizer $u$ satisfies $u = u^*$. Then, applying [ST00, Th. 1] yields the uniqueness of radial solution to $\Delta_p u + f(u) = 0$ for the non-linearity $f(t) = \lambda - st^{s-1}$, thus we get that $v$ is unique.

We now prove that $v' < 0$ on $(0, R)$. Since the LHS in (5.13) is non-positive, the non-decreasing function $g: \rho \mapsto sv(\rho)^{s-1} - \lambda$, which tends to $+\infty$ as $\rho \to R$, must be negative near 0. If we had $v'(r) = 0$ for some $r \in (0, R)$, then the non-positive function $t \mapsto pv'(t)^{p-1}t^{N-1}$ would be maximal and its derivative would vanish at $t = r$. By (5.13), this means that $g(r) = sv(r)^{s-1} - \lambda = 0$ so that $g \le 0$ on $[0, r]$, but also that $g(\rho)\rho^{N-1}$ integrates to 0 on $[0, r]$. Hence, $g \equiv 0$ on $[0, r]$, a contradiction.

CONCLUSION AND RANGE OF $\alpha$-MASSES OBTAINED IN THIS WAY. — To summarize, we have shown that if $-p' < s \le 1$ then $H_f$ is non-trivial. The converse is true. Indeed, if $H_f(1) < +\infty$ then, by the preceding, there exists a radial decreasing minimizer $u(\cdot) = v(|\cdot|)$ with $v: [0, +\infty) \to \mathbb{R}_+$ non-increasing and compactly supported on $[0, R]$, $0 < R < +\infty$. But, by the Young inequality,

$$ \mathcal{E}_f(u) \ge c \int_{R/2}^{R} v(r)^{s/p'} |v'(r)| r^{N-1} dr \ge c_R \int_{0}^{v(R/2)} t^{s/p'} dt $$

with $c_R > 0$, the latter being finite if and only if $s/p' > -1$.

Since $\alpha$, in (5.7), is a monotone function of $s$, one may easily compute the range of parameters $\alpha$ that we obtain. If $p$ and $N$ are fixed, $\alpha$ ranges over $((N-1)/(N+1-1/p), 1]$ when $s \in (-p', 1]$. Hence, when $N = 1$ we obtain the whole range $\alpha \in (0, 1]$, and at least the range $\alpha \in (1 - 2/(N+1), 1]$ when $p$ ranges over $(1 + \infty)$ in dimension $N \ge 2$.

## 5.4. BRANCHED TRANSPORT APPROXIMATION: $H$-MASSES OF NORMAL 1-CURRENTS

Branched transport is a variant of classical optimal transport (see [San15] and Section 4.4.2 therein for a brief presentation of branched transport, and [BCM09] for a vast exposition) where the transport energy concentrates on a network, i.e., a 1-dimensional subset of $\mathbb{R}^d$, which has a graph structure when optimized with prescribed source and target measures. It can be formulated as a minimal flow problem,

$$ \min \{ \mathbf{M}_1^H(w) : \operatorname{div}(w) = \mu^- - \mu^+ \}, $$

where $\mu^\pm$ are probability measures on $\mathbb{R}^d$, $H: \mathbb{R}^d \times \mathbb{R}_+ \to \mathbb{R}_+$ is mass-concave, and the $H$-mass $\mathbf{M}_1^H$ is this time defined for finite vector measures $w \in \mathcal{M}(\mathbb{R}^d, \mathbb{R}^d)$ whose distributional divergence is also a finite measure; in the language of currents, it is called a 1-dimensional normal current. Any such measure may be decomposed into a 1-rectifiable part $\theta\xi \cdot \mathcal{H}^1 \llcorner \Sigma$ where $\theta(x) \ge 0$ and $\xi(x)$ is a unit tangent vector to $\Sigma$ for $\mathcal{H}^1$-a.e. $x \in \Sigma$, and a 1-diffuse part $w^\perp$ satisfying $|w^\perp|(A) = 0$ for every 1-rectifiable set $A$:

$$ w = \theta\xi \cdot \mathcal{H}^1 \llcorner M + w^\perp. $$
3. Uniqueness when $s > 0$. — We justify that there is a unique minimizer of the energy with mass $m$. Once we know that $v'$ does not vanish on $(0, R)$, the case of equality in Pólya-Szegő inequality (see [BZ88, Th. 1.1]) implies that any minimizer $u$ satisfies $u = u^*$. Then, applying [ST00, Th. 1] yields the uniqueness of radial solution to $\Delta_p u + f(u) = 0$ for the non-linearity $f(t) = \lambda - st^{s-1}$, thus we get that $v$ is unique.

We now prove that $v' < 0$ on $(0, R)$. Since the LHS in (5.13) is non-positive, the non-decreasing function $g : \rho \mapsto sv(\rho)^{s-1} - \lambda$, which tends to $+\infty$ as $\rho \to R$, must be negative near 0. If we had $v'(r) = 0$ for some $r \in (0, R)$, then the non-positive function $t \mapsto pv'(t)^{p-1}t^{N-1}$ would be maximal and its derivative would vanish at $t = r$. By (5.13), this means that $g(r) = sv(r)^{s-1} - \lambda = 0$ so that $g \le 0$ on $[0, r]$, but also that $g(\rho)\rho^{N-1}$ integrates to 0 on $[0, r]$. Hence, $g \equiv 0$ on $[0, r]$, a contradiction.

CONCLUSION AND RANGE OF $\alpha$-MASSES OBTAINED IN THIS WAY. — To summarize, we have shown that if $-p' < s \le 1$ then $H_f$ is non-trivial. The converse is true. Indeed, if $H_f(1) < +\infty$ then, by the preceding, there exists a radial decreasing minimizer $u(\cdot) = v(|\cdot|)$ with $v : [0, +\infty) \to \mathbb{R}_+$ non-increasing and compactly supported on $[0, R]$, $0 < R < +\infty$. But, by the Young inequality,

$$ \mathcal{E}_f(u) \ge c \int_{R/2}^R v(r)^{s/p'} |v'(r)| r^{N-1} dr \ge c_R \int_0^{v(R/2)} t^{s/p'} dt $$

with $c_R > 0$, the latter being finite if and only if $s/p' > -1$.

Since $\alpha$, in (5.7), is a monotone function of $s$, one may easily compute the range of parameters $\alpha$ that we obtain. If $p$ and $N$ are fixed, $\alpha$ ranges over $((N-1)/(N+1-1/p), 1]$ when $s \in (-p', 1]$. Hence, when $N = 1$ we obtain the whole range $\alpha \in (0, 1]$, and at least the range $\alpha \in (1 - 2/(N+1), 1]$ when $p$ ranges over $(1 + \infty)$ in dimension $N \ge 2$.

## 5.4. BRANCHED TRANSPORT APPROXIMATION: $H$-MASSES OF NORMAL 1-CURRENTS

Branched transport is a variant of classical optimal transport (see [San15] and Section 4.4.2 therein for a brief presentation of branched transport, and [BCM09] for a vast exposition) where the transport energy concentrates on a network, i.e., a 1-dimensional subset of $\mathbb{R}^d$, which has a graph structure when optimized with prescribed source and target measures. It can be formulated as a minimal flow problem,

$$ \min \{ \mathbf{M}_1^H(w) : \operatorname{div}(w) = \mu^- - \mu^+ \}, $$

where $\mu^\pm$ are probability measures on $\mathbb{R}^d$, $H : \mathbb{R}^d \times \mathbb{R}_+ \to \mathbb{R}_+$ is mass-concave, and the $H$-mass $\mathbf{M}_1^H$ is this time defined for finite vector measures $w \in \mathcal{M}(\mathbb{R}^d, \mathbb{R}^d)$ whose distributional divergence is also a finite measure; in the language of currents, it is called a 1-dimensional normal current. Any such measure may be decomposed into a 1-rectifiable part $\theta\xi \cdot \mathcal{H}^1 \llcorner \Sigma$ where $\theta(x) \ge 0$ and $\xi(x)$ is a unit tangent vector to $\Sigma$ for $\mathcal{H}^1$-a.e. $x \in \Sigma$, and a 1-diffuse part $w^\perp$ satisfying $|w^\perp|(A) = 0$ for every 1-rectifiable set $A$:

$$ w = \theta\xi \cdot \mathcal{H}^1 \llcorner M + w^\perp. $$
The H-mass is then defined by:

$$ (5.14) \qquad M_1^H(w) := \int_{\Sigma} H(x, \theta(x)) \, d\mathcal{H}^1(x) + \int_{\mathbb{R}^d} H'(x, 0) \, d|w^{\perp}|. $$

In the case $H(x, m) = m^\alpha$ with $0 < \alpha < 1$, a family of approximations of these functional has been introduced in [OS11]:

$$ (5.15) \qquad \mathcal{E}_{\varepsilon}(w) = \begin{cases} \int_{\mathbb{R}^d} \varepsilon^{\gamma_1} |\nabla v|^2 + \varepsilon^{-\gamma_2} |v|^{\beta} & \text{if } w = v \mathcal{L}^d, v \in W_{\text{loc}}^{1,2}(\mathbb{R}^d, \mathbb{R}^d), \\ +\infty & \text{otherwise,} \end{cases} $$

with $\beta = (2 - 2d + 2\alpha d)/(3 - d + \alpha(d-1))$, $\gamma_1 = (d-1)(1-\alpha)$ and $\gamma_2 = 3 - d + \alpha(d-1)$. It has been shown in [OS11, Mon17] that the functionals $\mathcal{E}_\varepsilon$ $\Gamma$-converge as $\varepsilon \to 0^+$, in the topology of weak convergence of $u$ and its divergence, to a non-trivial multiple of the $\alpha$-mass $M_1^\alpha := M_1^H$ with $H(x, m) = m^\alpha$ in dimension $d=2$. The result extends to any dimension $d$, by [Mon15], thanks to a slicing method that relates the energy $\mathcal{E}_\varepsilon$ with the energy of the sliced measures $u = (w \cdot \nu)_+$ supported on the slices $V_a = \{x \in \mathbb{R}^d : x \cdot \nu = a\} \simeq \mathbb{R}^N$, for any given unit vector $\nu \in \mathbb{R}^d$, defined by

$$ \bar{\mathcal{E}}_{\varepsilon}(u) = \int_{\mathbb{R}^N} \varepsilon^{\gamma_1} |\nabla u|^2 + \varepsilon^{-\gamma_2} |u|^{\beta}. $$

The functionals $\bar{\mathcal{E}}_\varepsilon$ $\Gamma$-converge as $\varepsilon \to 0^+$, in the narrow topology, to $cM^\alpha$ for some non-trivial $c$, as shown in Section 5.3, and one may recover every $\alpha$-mass in this way for $\alpha \in ((2d-4)/(2d+1), 1]$, and in particular every so-called super-critical exponents for branched transport in dimension $d$, that is $\alpha \in (1 - 1/d, 1]$.

The same slicing method would allow to extend our $\Gamma$-convergence result stated in Theorem 1.2 to functionals defined on vector measure

$$ (5.16) \qquad \mathcal{E}_{\varepsilon}(w) = \begin{cases} \int_{\mathbb{R}^d} f_{\varepsilon}(x, \varepsilon^{d-1}|v|(x), \varepsilon^d|\nabla v|(x)) \varepsilon^{1-d} \, dx & \text{if } w = v \mathcal{L}^d, v \in W_{\text{loc}}^{1,1}(\mathbb{R}^d, \mathbb{R}^d), \\ +\infty & \text{otherwise,} \end{cases} $$

for Lagrangians $f_\varepsilon \to f$ fitting the framework of Theorem 1.2. The expected $\Gamma$-limit, for the weak topology of measures and their divergence measure, would be the functional $M_1^{H_f}$, with $H_f$ defined in (1.2). Note that this approach would provide approximations of H-masses for more general continuous and concave cost functions $H: \mathbb{R}_+ \to \mathbb{R}_+$ satisfying $H(0) = 0$. By [Wir19], we would obtain all such H-masses when $N=1$ (corresponding to $d=2$).

5.5. A CAHN-HILLIARD MODEL FOR DROPLETS. — Following the works [BDS96] in the one-dimensional case and [Dub98] in higher dimension, we consider functionals on $M_+(\mathbb{R}^N)$ of the form:

$$ (5.17) \qquad W_{\varepsilon}(\mu) = \begin{cases} \int_{\mathbb{R}^N} \varepsilon^{-\rho} (W(u) + \varepsilon |\nabla u|^2) & \text{if } \mu = u \mathcal{L}^N, u \in W_{\text{loc}}^{1,1}(\mathbb{R}^N, \mathbb{R}_+), \\ +\infty & \text{otherwise,} \end{cases} $$

where $W: \mathbb{R}_+ \to \mathbb{R}_+$ is a Borel function satisfying $W(t) \sim_{u \to +\infty} u^s$ for some exponent $s \in (-\infty, 1)$. In [BDS96, Dub98], it is in particular proved, under some assumptions
The H-mass is then defined by:

$$ (5.14) \qquad \mathbf{M}_1^H(w) := \int_{\Sigma} H(x, \theta(x)) \, d\mathcal{H}^1(x) + \int_{\mathbb{R}^d} H'(x, 0) \, d|w^{\perp}|. $$

In the case $H(x, m) = m^\alpha$ with $0 < \alpha < 1$, a family of approximations of these functional has been introduced in [OS11]:

$$ (5.15) \qquad \mathcal{E}_{\varepsilon}(w) = \begin{cases} \int_{\mathbb{R}^d} \varepsilon^{\gamma_1} |\nabla v|^2 + \varepsilon^{-\gamma_2} |v|^{\beta} & \text{if } w = v \mathcal{L}^d, v \in W_{\text{loc}}^{1,2}(\mathbb{R}^d, \mathbb{R}^d), \\ +\infty & \text{otherwise,} \end{cases} $$

with $\beta = (2 - 2d + 2\alpha d)/(3 - d + \alpha(d-1))$, $\gamma_1 = (d-1)(1-\alpha)$ and $\gamma_2 = 3 - d + \alpha(d-1)$. It has been shown in [OS11, Mon17] that the functionals $\mathcal{E}_\varepsilon$ $\Gamma$-converge as $\varepsilon \to 0^+$, in the topology of weak convergence of $u$ and its divergence, to a non-trivial multiple of the $\alpha$-mass $\mathbf{M}_1^\alpha := \mathbf{M}_1^H$ with $H(x, m) = m^\alpha$ in dimension $d=2$. The result extends to any dimension $d$, by [Mon15], thanks to a slicing method that relates the energy $\mathcal{E}_\varepsilon$ with the energy of the sliced measures $u = (w \cdot \nu)_+$ supported on the slices $V_a = \{x \in \mathbb{R}^d : x \cdot \nu = a\} \simeq \mathbb{R}^N$, for any given unit vector $\nu \in \mathbb{R}^d$, defined by

$$ \bar{\mathcal{E}}_{\varepsilon}(u) = \int_{\mathbb{R}^N} \varepsilon^{\gamma_1} |\nabla u|^2 + \varepsilon^{-\gamma_2} |u|^{\beta}. $$

The functionals $\bar{\mathcal{E}}_\varepsilon$ $\Gamma$-converge as $\varepsilon \to 0^+$, in the narrow topology, to $c\mathbf{M}^\alpha$ for some non-trivial $c$, as shown in Section 5.3, and one may recover every $\alpha$-mass in this way for $\alpha \in ((2d-4)/(2d+1), 1]$, and in particular every so-called super-critical exponents for branched transport in dimension $d$, that is $\alpha \in (1 - 1/d, 1]$.

The same slicing method would allow to extend our $\Gamma$-convergence result stated in Theorem 1.2 to functionals defined on vector measure

(5.16)

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

for Lagrangians $f_\varepsilon \to f$ fitting the framework of Theorem 1.2. The expected $\Gamma$-limit, for the weak topology of measures and their divergence measure, would be the functional $\mathbf{M}_1^{H_f}$, with $H_f$ defined in (1.2). Note that this approach would provide approximations of H-masses for more general continuous and concave cost functions $H: \mathbb{R}_+ \to \mathbb{R}_+$ satisfying $H(0) = 0$. By [Wir19], we would obtain all such H-masses when $N=1$ (corresponding to $d=2$).

5.5. A CAHN-HILLIARD MODEL FOR DROPLETS. — Following the works [BDS96] in the one-dimensional case and [Dub98] in higher dimension, we consider functionals on $\mathcal{M}_+(\mathbb{R}^N)$ of the form:

$$ (5.17) \qquad W_{\varepsilon}(\mu) = \begin{cases} \int_{\mathbb{R}^N} \varepsilon^{-\rho} (W(u) + \varepsilon |\nabla u|^2) & \text{if } \mu = u \mathcal{L}^N, u \in W_{\text{loc}}^{1,1}(\mathbb{R}^N, \mathbb{R}_+), \\ +\infty & \text{otherwise,} \end{cases} $$

where $W: \mathbb{R}_+ \to \mathbb{R}_+$ is a Borel function satisfying $W(t) \sim_{u \to +\infty} u^s$ for some exponent $s \in (-\infty, 1)$. In [BDS96, Dub98], it is in particular proved, under some assumptions
on the slope of W at 0 and its regularity, that the family $(W_\varepsilon)_{\varepsilon>0}$ $\Gamma$-converges to a non-trivial multiple of the $\alpha$-mass, $\alpha = \frac{1-s/2+s/N}{1-s/2+1/N}$, when $s \in (-2, 1)$ and $\rho = \rho(s, N) := \frac{N(1-s)}{(N+2)+N(1-s)}$. In this section, we recover this $\Gamma$-convergence result using our general model.

Replacing $\varepsilon$ with $\bar{\varepsilon} := \varepsilon^{(N+2)+N(1-s)}$ and noticing that $1 - \rho = \frac{N+2}{(N+2)+N(1-s)}$, one gets for every $u \in W_{\text{loc}}^{1,1}(\mathbb{R}^N, \mathbb{R}_+)$:

$$ 
\begin{aligned}
W_{\bar{\varepsilon}}(u) &= \int_{\mathbb{R}^N} \varepsilon^{-N(1-s)} W(u) + \varepsilon^{N+2} |\nabla u|^2 \\
&= \int_{\mathbb{R}^N} ([\varepsilon^{Ns} W(\varepsilon^{-N} \varepsilon^N u)] + |\varepsilon^{N+1} \nabla u|^2) \varepsilon^{-N} \\
&= \int_{\mathbb{R}^N} f_{\varepsilon}^W (x, \varepsilon^N u, \varepsilon^{N+1} \nabla u) \varepsilon^{-N},
\end{aligned}
$$

where $f_{\varepsilon}^W$ is defined for every $x \in \mathbb{R}^N$, $u \in \mathbb{R}_+$, $\xi \in \mathbb{R}^N$ by

$$ f_{\varepsilon}^{W}(x, u, \xi) := W_{\varepsilon}(u) + |\xi|^2 \quad \text{and} \quad W_{\varepsilon}(u) := \varepsilon^{Ns} W(\varepsilon^{-N} u). $$

Therefore if we take $f_{\varepsilon} = f_{\varepsilon}^W$ in our general model (1.4) we exactly get $W_{\bar{\varepsilon}} = \mathcal{E}_{\varepsilon}$. The fact that $W(u) \sim u^s$ as $u \to +\infty$ implies that $W_{\varepsilon}$ converges pointwise to the map $w_s : u \mapsto u^s$ if $u > 0$, $w_s(0) = 0$, hence $f_{\varepsilon}^W$ converges to $f_s : (x, u, \xi) \mapsto w_s(u) + |\xi|^2$.

**THEOREM 5.1.** — Assume that $W : \mathbb{R}_+ \to \mathbb{R}_+$ satisfies:

(HW$_1$) $W$ is lower semicontinuous,

(HW$_2$) $\{W = 0\} = \{0\}$,

(HW$_3$) $W(u) \sim_{u \to +\infty} u^s$ for some $s \in (-\infty, 1)$,

(HW$_4$) $\sup_{u>0} \frac{W(u)}{u^s} < +\infty$,

(HW$_5$) $0 < \liminf_{u \to 0^+} \frac{W(u)}{u}$.

Then $(W_\varepsilon)_{\varepsilon>0}$ $\Gamma$-converges to $M^{H_{fs}}$, for both topologies $\mathcal{C}'_0$ and $\mathcal{C}'_b$, and if $s \in (-2, 1]$ then $M^{H_{fs}}$ is a non-trivial multiple of $M^\alpha$ where $\alpha = \frac{1-s/2+s/N}{1-s/2+1/N}$.

To prove this theorem, we start with a simple lemma.

**LEMMA 5.2.** — Assume that $W$ satisfies (HW$_1$)-(HW$_5$). Then for every $\delta \in (0, 1)$, there exists $c_\delta \in (0, +\infty)$ such that for every $\varepsilon > 0$ and every $u \in \mathbb{R}_+$,

$$ (5.18) \qquad \delta(u^s \wedge c_\delta \varepsilon^{-N(1-s)} u) \le W_\varepsilon(u). $$

*Proof.* — Fix $\delta \in (0, 1)$. There exists $M > 0$ such that $\delta u^s \le W(u)$ for every $u \ge M$. Besides, the map $w : u \mapsto W(u)/u$ is lower semicontinuous and positive on $(0, M]$ by (HW$_1$) and (HW$_2$), and since $\liminf_{u \to 0} w(u) > 0$ by (HW$_5$), $w$ is necessarily bounded from below on $(0, M]$ by some constant $c > 0$. As a consequence $W_\varepsilon(u) \ge \delta u^s$ if $u \ge \varepsilon^N M$ and $W_\varepsilon(u) \ge c\varepsilon^{N(s-1)}u$ if $u \le \varepsilon^N M$, hence:

$$ \forall u \in \mathbb{R}, \quad W_\varepsilon(u) \ge \delta(u^s \wedge c\varepsilon^{-N(1-s)}u). \quad \square $$

<figure><img src="image_6.png" /></figure>
on the slope of W at 0 and its regularity, that the family $(W_\varepsilon)_{\varepsilon>0}$ Γ-converges to a non-trivial multiple of the $\alpha$-mass, $\alpha = \frac{1-s/2+s/N}{1-s/2+1/N}$, when $s \in (-2, 1)$ and $\rho = \rho(s, N) := \frac{N(1-s)}{(N+2)+N(1-s)}$. In this section, we recover this Γ-convergence result using our general model.

Replacing $\varepsilon$ with $\bar{\varepsilon} := \varepsilon^{(N+2)+N(1-s)}$ and noticing that $1 - \rho = \frac{N+2}{(N+2)+N(1-s)}$, one gets for every $u \in W_{\text{loc}}^{1,1}(\mathbb{R}^N, \mathbb{R}_+)$:

$$ 
\begin{align*} 
W_{\bar{\varepsilon}}(u) &= \int_{\mathbb{R}^N} \varepsilon^{-N(1-s)} W(u) + \varepsilon^{N+2} |\nabla u|^2 \\
&= \int_{\mathbb{R}^N} ([\varepsilon^{Ns} W(\varepsilon^{-N} \varepsilon^N u)] + |\varepsilon^{N+1} \nabla u|^2) \varepsilon^{-N} \\
&= \int_{\mathbb{R}^N} f_{\varepsilon}^W (x, \varepsilon^N u, \varepsilon^{N+1} \nabla u) \varepsilon^{-N}, 
\end{align*} 
$$

where $f_{\varepsilon}^W$ is defined for every $x \in \mathbb{R}^N$, $u \in \mathbb{R}_+$, $\xi \in \mathbb{R}^N$ by

$$ f_{\varepsilon}^{W}(x, u, \xi) := W_{\varepsilon}(u) + |\xi|^2 \quad \text{and} \quad W_{\varepsilon}(u) := \varepsilon^{Ns} W(\varepsilon^{-N} u). $$

Therefore if we take $f_{\varepsilon} = f_{\varepsilon}^W$ in our general model (1.4) we exactly get $W_{\bar{\varepsilon}} = \mathcal{E}_{\varepsilon}$. The fact that $W(u) \sim u^s$ as $u \to +\infty$ implies that $W_{\varepsilon}$ converges pointwise to the map $w_s : u \mapsto u^s$ if $u > 0$, $w_s(0) = 0$, hence $f_{\varepsilon}^W$ converges to $f_s : (x, u, \xi) \mapsto w_s(u) + |\xi|^2$.

**THEOREM 5.1.** — Assume that $W : \mathbb{R}_+ \to \mathbb{R}_+$ satisfies:

(HW$_1$) $W$ is lower semicontinuous,

(HW$_2$) $\{W = 0\} = \{0\}$,

(HW$_3$) $W(u) \sim_{u \to +\infty} u^s$ for some $s \in (-\infty, 1)$,

(HW$_4$) $\sup_{u>0} \frac{W(u)}{u^s} < +\infty$,

(HW$_5$) $0 < \liminf_{u \to 0^+} \frac{W(u)}{u}$.

Then $(W_\varepsilon)_{\varepsilon>0}$ Γ-converges to $M^{H_{fs}}$, for both topologies $C'_0$ and $C'_b$, and if $s \in (-2, 1]$ then $M^{H_{fs}}$ is a non-trivial multiple of $M^\alpha$ where $\alpha = \frac{1-s/2+s/N}{1-s/2+1/N}$.

To prove this theorem, we start with a simple lemma.

**LEMMA 5.2.** — Assume that $W$ satisfies (HW$_1$)-(HW$_5$). Then for every $\delta \in (0, 1)$, there exists $c_\delta \in (0, +\infty)$ such that for every $\varepsilon > 0$ and every $u \in \mathbb{R}_+$,

$$ (5.18) \qquad \delta(u^s \wedge c_\delta \varepsilon^{-N(1-s)} u) \le W_\varepsilon(u). $$

*Proof.* — Fix $\delta \in (0, 1)$. There exists $M > 0$ such that $\delta u^s \le W(u)$ for every $u \ge M$. Besides, the map $w : u \mapsto W(u)/u$ is lower semicontinuous and positive on $(0, M]$ by (HW$_1$) and (HW$_2$), and since $\liminf_{u \to 0} w(u) > 0$ by (HW$_5$), $w$ is necessarily bounded from below on $(0, M]$ by some constant $c > 0$. As a consequence $W_\varepsilon(u) \ge \delta u^s$ if $u \ge \varepsilon^N M$ and $W_\varepsilon(u) \ge c\varepsilon^{N(s-1)}u$ if $u \le \varepsilon^N M$, hence:

$$ \forall u \in \mathbb{R}, \quad W_\varepsilon(u) \ge \delta(u^s \wedge c\varepsilon^{-N(1-s)}u). \quad \square $$

<figure><img src="image_8.png" /></figure>
Proof of Theorem 5.1. — By (HW$_4$), there exists a constant C such that $f_{\varepsilon}^W \le C f_s$ for every $\varepsilon$, and since $f_{\varepsilon}^W$ does not depend on the $x$ variable and converges pointwise to $f_s$, (U) is satisfied and our $\Gamma$-lim sup result stated in Proposition 4.2 yields

$$M^{H_{f_s}} \ge \Gamma(\mathscr{C}'_b)-\limsup_{\varepsilon \to 0} \mathcal{E}_{\varepsilon}.$$

Fix $\delta \in (0, 1)$. By Lemma 5.2, there exists $c_\delta$ such that

$$\forall x, u, \xi, \quad f_{\varepsilon}^{W}(x, u, \xi) \ge \delta(|\xi|^2 + (u^s \wedge c_{\delta}\varepsilon^{-N(1-s)}u) =: f_{\varepsilon}^{\delta}(x, u, \xi).$$

It is easy to check that $f_{\varepsilon}^{\delta}$ satisfies (H$_1$), (H$_2$) and (H$_5$) for every $\varepsilon > 0$. Moreover $f_{\varepsilon}^{\delta} \uparrow \delta f_s$ and $(f_{\varepsilon}^{\delta})'_-(\cdot, 0^+, 0) = \delta c_{\delta} \varepsilon^{-N(1-s)} \uparrow (+\infty) = (\delta f_s)'_-(\cdot, 0^+, 0)$ as $\varepsilon \to 0$, thus (H$_8$) holds for the family $(f_{\varepsilon}^{\delta})_{\varepsilon>0}$, and by applying our $\Gamma$-lim inf result stated in Proposition 4.1 to the energies $\mathcal{E}_{\varepsilon}^{\delta}$ induced by $f_{\varepsilon}^{\delta}$ we get:

$$\Gamma(\mathscr{C}'_0)-\liminf \mathcal{E}_{\varepsilon} \ge \Gamma(\mathscr{C}'_0)-\liminf \mathcal{E}_{\varepsilon} \ge M^{H_{\bar{\delta}f_s}}.$$

We get the result by taking the limit $\delta \to 1$, noticing that $(f_s)'_-(\cdot, 0^+, 0) = +\infty$, so that $H_{\bar{\delta}f_s} = H_{\delta f_s} = \delta H_{f_s}$ and $M^{H_{\bar{\delta}f_s}} = M^{\delta H_{f_s}} = \delta M^{H_{f_s}}$. $\square$

REMARK 5.3. — We recover the $\Gamma$-convergence results of [BDS96] and [Dub98] when $s \in (-2, 1)$ under slightly more general assumptions: besides (HW$_2$) and (HW$_3$), the authors impose the existence of a non-trivial slope $\lim_{u \to 0} W(u)/u \in (0, +\infty)$ and a regularity condition (either $W$ is of class $\mathscr{C}^1$ or continuous and non-decreasing close to 0), which are stronger than (HW$_1$), (HW$_4$) and (HW$_5$). Let us stress however that these works also tackle the cases $s < -2$ in any dimension, where the exponent $\rho$ has to be fixed to $\rho(-2, N)$, and the case $s = -2$ in dimension one, where a logarithmic factor must be introduced, replacing $\varepsilon^{-\rho}$ with $\varepsilon^{-\rho(-2,1)}|\log \varepsilon|^{-1} = \varepsilon^{-1/2}|\log \varepsilon|^{-1}$. This implies that in our model we get a trivial $\Gamma$-limit when $s \le -2$, namely $H_{f_s} = +\infty$ on $(0, +\infty)$.

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*Proof of Theorem 5.1.* — By (HW$_4$), there exists a constant $C$ such that $f_{\varepsilon}^W \le C f_s$ for every $\varepsilon$, and since $f_{\varepsilon}^W$ does not depend on the $x$ variable and converges pointwise to $f_s$, (U) is satisfied and our $\Gamma$-lim sup result stated in Proposition 4.2 yields

$$M^{H_{f_s}} \ge \Gamma(\mathscr{C}'_b)-\limsup_{\varepsilon \to 0} \mathcal{E}_{\varepsilon}.$$

Fix $\delta \in (0, 1)$. By Lemma 5.2, there exists $c_{\delta}$ such that

$$\forall x, u, \xi, \quad f_{\varepsilon}^{W}(x, u, \xi) \ge \delta(|\xi|^2 + (u^s \wedge c_{\delta}\varepsilon^{-N(1-s)}u) =: f_{\varepsilon}^{\delta}(x, u, \xi).$$

It is easy to check that $f_{\varepsilon}^{\delta}$ satisfies (H$_1$), (H$_2$) and (H$_5$) for every $\varepsilon > 0$. Moreover $f_{\varepsilon}^{\delta} \uparrow \delta f_s$ and $(f_{\varepsilon}^{\delta})'_-(\cdot, 0^+, 0) = \delta c_{\delta} \varepsilon^{-N(1-s)} \uparrow (+\infty) = (\delta f_s)'_-(\cdot, 0^+, 0)$ as $\varepsilon \to 0$, thus (H$_8$) holds for the family $(f_{\varepsilon}^{\delta})_{\varepsilon>0}$, and by applying our $\Gamma$-lim inf result stated in Proposition 4.1 to the energies $\mathcal{E}_{\varepsilon}^{\delta}$ induced by $f_{\varepsilon}^{\delta}$ we get:

$$\Gamma(\mathscr{C}'_0)-\liminf \mathcal{E}_{\varepsilon} \ge \Gamma(\mathscr{C}'_0)-\liminf \mathcal{E}_{\varepsilon} \ge M^{H_{\bar{\delta}f_s}}.$$

We get the result by taking the limit $\delta \to 1$, noticing that $(f_s)'_-(\cdot, 0^+, 0) = +\infty$, so that $H_{\bar{\delta}f_s} = H_{\delta f_s} = \delta H_{f_s}$ and $M^{H_{\bar{\delta}f_s}} = M^{\delta H_{f_s}} = \delta M^{H_{f_s}}$. $\square$

**Remark 5.3.** — We recover the $\Gamma$-convergence results of [BDS96] and [Dub98] when $s \in (-2, 1)$ under slightly more general assumptions: besides (HW$_2$) and (HW$_3$), the authors impose the existence of a non-trivial slope $\lim_{u \to 0} W(u)/u \in (0, +\infty)$ and a regularity condition (either $W$ is of class $\mathscr{C}^1$ or continuous and non-decreasing close to 0), which are stronger than (HW$_1$), (HW$_4$) and (HW$_5$). Let us stress however that these works also tackle the cases $s < -2$ in any dimension, where the exponent $\rho$ has to be fixed to $\rho(-2, N)$, and the case $s = -2$ in dimension one, where a logarithmic factor must be introduced, replacing $\varepsilon^{-\rho}$ with $\varepsilon^{-\rho(-2,1)}|\log \varepsilon|^{-1} = \varepsilon^{-1/2}|\log \varepsilon|^{-1}$. This implies that in our model we get a trivial $\Gamma$-limit when $s \le -2$, namely $H_{f_s} \equiv +\infty$ on $(0, +\infty)$.

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Manuscript received 2nd March 2022
accepted 16th January 2024

ANTONIN MONTEIL, Université Paris-Est Créteil Val-de-Marne, LAMA,
61 avenue du Général de Gaulle, 94010 Créteil, France
E-mail: antonin.monteil@u-pec.fr
Url: https://perso.math.u-pem.fr/monteil.antonin/

PAUL PEGON, Université Paris-Dauphine, CEREMADE & INRIA Paris, MOKAPLAN,
France
E-mail: pegon@ceremade.dauphine.fr
Url: https://www.ceremade.dauphine.fr/~pegon/
[Bra02] A. BRAIDES – *Gamma-convergence for beginners*, Oxford Lect. Series in Math. and its Appl., Oxford University Press, Oxford, 2002.

[BZ88] J. E. BROTHERS & W. P. ZIEMER – “Minimal rearrangements of sobolev functions”, *J. reine angew. Math.* **384** (1988), p. 153–179.

[But89] G. BUTTAZZO – *Semicontinuity, relaxation and integral representation in the calculus of variations*, Longman Scientific & Technical; New York: John Wiley & Sons, Harlow, 1989.

[CDRMS17] M. COLOMBO, A. DE ROSA, A. MARCHESE & S. STUVARD – “On the lower semicontinuous envelope of functionals defined on polyhedral chains”, *Nonlinear Anal.* **163** (2017), p. 201–215.

[DPH03] T. DE PAUW & R. HARDT – “Size minimization and approximating problems”, *Calc. Var. Partial Differential Equations* **17** (2003), no. 4, p. 405–442.

[Dub98] C. DUBS – “Problèmes de perturbations singulières avec un potentiel dégénéré à l’infini”, Thèse de doctorat, Toulon, 1998.

[Fed59] H. FEDERER – “Curvature measures”, *Trans. Amer. Math. Soc.* **93** (1959), no. 3, p. 418–491.

[Fle66] W. H. FLEMING – “Flat chains over a finite coefficient group”, *Trans. Amer. Math. Soc.* **121** (1966), no. 1, p. 160–186.

[Mar14] M. MARIŞ – “Profile decomposition for sequences of Borel measures”, 2014, arXiv: 1410.6125.

[MM77] L. MODICA & S. MORTOLA – “Un esempio di $\gamma^-$-convergenza”, *Boll. Un. Mat. Ital. B* **5** (1974), p. 285–299.

[Mon15] A. MONTEIL – “Elliptic approximations of singular energies under divergence constraint”, PhD Thesis, Université Paris-Saclay, 2015.

[Mon17] —, “Uniform estimates for a Modica–Mortola type approximation of branched transportation”, *ESAIM Control Optim. Calc. Var.* **23** (2017), no. 1, p. 309–335.

[Nir59] L. NIRENBERG – “On elliptic partial differential equations”, *Ann. Scuola Norm. Sup. Pisa Cl. Sci.* **13** (1959), no. 2, p. 115–162.

[OS11] E. OUDET & F. SANTAMBROGIO – “A Modica-Mortola approximation for branched transport and applications”, *Arch. Rational Mech. Anal.* **201** (2011), no. 1, p. 115–142.

[PSZ99] P. PUCCI, J. SERRIN & H. ZOU – “A strong maximum principle and a compact support principle for singular elliptic inequalities”, *J. Math. Pures Appl. (9)* **78** (1999), no. 8, p. 769–789.

[RV73] A. ROBERTS & D. VARBERG – *Convex functions*, Pure and Applied Math., vol. 57, Academic Press, 1973.

[San15] F. SANTAMBROGIO – *Optimal transport for applied mathematicians*, Progress in nonlinear differential equations and their applications, vol. 87, Springer International Publishing, Cham, 2015.

[ST00] J. SERRIN & M. TANG – “Uniqueness of ground states for quasilinear elliptic equations”, *Indiana Univ. Math. J.* **49** (2000), no. 3, p. 897–923.

[Wir19] B. WIRTH – “Phase field models for two-dimensional branched transportation problems”, *Calc. Var. Partial Differential Equations* **58** (2019), no. 5, article no. 164 (31 pages).

Manuscript received 2nd March 2022
accepted 16th January 2024

ANTONIN MONTEIL, Université Paris-Est Créteil Val-de-Marne, LAMA,
61 avenue du Général de Gaulle, 94010 Créteil, France

E-mail: antonin.monteil@u-pec.fr

Url: https://perso.math.u-pem.fr/monteil.antonin/

PAUL PEGON, Université Paris-Dauphine, CEREMADE & INRIA Paris, MOKAPLAN,
France

E-mail: pegon@ceremade.dauphine.fr

Url: https://www.ceremade.dauphine.fr/~pegon/