# Weighted Sobolev Spaces and an Eigenvalue Problem for an Elliptic Equation with $L^1$ Data

Juan A. Apaza

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Juan A. Apaza. Weighted Sobolev Spaces and an Eigenvalue Problem for an Elliptic Equation with $L^1$ Data. 2024. hal-04717561

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Distributed under a Creative Commons Attribution 4.0 International License
WEIGHTED SOBOLEV SPACES AND AN EIGENVALUE PROBLEM FOR AN ELLIPTIC EQUATION WITH L¹ DATA

JUAN A. APAZA

ABSTRACT. The aim of this work is to study the continuity and compactness of the operators $W^{1,q}(\Omega; \mathbf{V}_0, \mathbf{V}_1) \to L^{q_0}(\Omega; \mathbf{V}_2)$ and $W^{1,q}(\Omega; \mathbf{V}_0, \mathbf{V}_1) \to L^{q_1}(\partial\Omega; \mathbf{W})$ in weighted Sobolev spaces. To study additional properties of these Sobolev spaces, we will also study the equation:

$$ egin{cases} -\operatorname{div}(\mathbf{V}_1\nabla u) + \mathbf{V}_0u = \lambda\mathbf{V}_2\tau u + \mathbf{V}_2f_0 & \text{in } \Omega, \\ \mathbf{V}_1\frac{\partial u}{\partial\nu} = \mathbf{W}_1f_1 & \text{on } \partial\Omega, \end{cases} $$

where $\Omega$ is an open subset of a Riemannian manifold, $\lambda$ is a real number, $f_0 \in L^1(\Omega; \mathbf{V}_0)$, $f_1 \in L^1(\partial\Omega; \mathbf{W})$, $\tau$ is a function that changes sign, and $\mathbf{V}_i, \mathbf{W}, \mathbf{W}_1$ are weight functions satisfying suitable conditions. We aim to obtain existence results similar to those for the case where the data are given in $L^2(\Omega; \mathbf{V}_0)$ and $L^2(\partial\Omega; \mathbf{W})$. For the case where $f_0 = 0$ and $f_1 = 0$, we are also interested in studying the limit ess sup$_{\Omega\setminus\Omega_m}$ $|u| \to 0$, where $\Omega_m$ is a sequence of open sets such that $\Omega_m \subset \Omega_{m+1}$.

# 1. INTRODUCTION AND MAIN RESULTS

Weighted Sobolev spaces are utilized as solution spaces for degenerate elliptic equations, leading to extensive research in this area. Authors such as Kufner [38], Triebel [56], and Schmeisser & Triebel [53], among others, have contributed significantly to the study of weighted Sobolev and related function spaces.

In the study of partial differential equations in a domain $\Sigma \subset \mathbb{R}^n$, it is often useful to know that embeddings of the Sobolev space $W^{m,q}(\Sigma)$ into the Lebesgue space $L^{q_0}(\Sigma)$ are compact. For example, for nonlinear equations of variational form, such a compactness property is often used to show that the energy functional for this equation satisfies the Palais-Smale condition. When $\Sigma$ is unbounded, the compactness of the embedding generally fails. For suitable weight functions where compact embeddings can be obtained, see, e.g., [1, 29, 39].

In this paper, we provide an existence result for problems with the form:

$$ egin{cases} -\operatorname{div}(\mathbf{V}_1\nabla u) + \mathbf{V}_0u = \lambda\mathbf{V}_2\tau u + \mathbf{V}_2f_0 & \text{in } \Omega, \\ \mathbf{V}_1\frac{\partial u}{\partial\nu} = \mathbf{W}_1f_1 & \text{on } \partial\Omega, \end{cases} \quad (1.1) $$

where $\Omega$ is an open subset of a noncompact Riemannian manifold $M$ without boundary such that $\bar{\Omega}$ is a smooth manifold with boundary $\partial\Omega$. Additionally, $\nu$ is the outward unit normal vector to $\partial\Omega$, $\lambda$ is a real number, $f_0 \in L^1(\Omega; \mathbf{V}_0)$, $f_1 \in L^1(\partial\Omega; \mathbf{W})$, $\tau: M \to \mathbb{R}$ is a function that changes sign, and $\mathbf{V}_i, \mathbf{W}, \mathbf{W}_1$ are weight functions satisfying suitable conditions.

To achieve this objective, we will study the problem:

$$ egin{cases} -\operatorname{div}(\mathbf{V}_1\nabla u) + \mathbf{V}_0u = \lambda\mathbf{V}_2\tau u & \text{in } \Omega, \\ \mathbf{V}_1\frac{\partial u}{\partial\nu} = 0 & \text{on } \partial\Omega. \end{cases} \quad (1.2) $$

We are also interested in knowing under what conditions the solutions to this problem are bounded, and

$$ \lim_{m \to \infty} \operatorname{ess\,sup}_{\Omega \setminus \bar{\Omega}_m} |u| = 0, \quad (1.3) $$

<footnote n="2020">2020 Mathematics Subject Classification. 35J25; 46E35; 93B60; 35R05</footnote>

<footnote>Keywords and phrases. Sobolev spaces, Riemannian manifolds, weight function, eigenvalue, Neumann problem</footnote>
where $D_m, m \in \mathbb{N}$, is an increasing sequence of open bounded domains in $M$ such that $M = \cup_m D_m$, $\bar{D}_m \subset D_{m+1}$, and $D_m \cap \Omega$ have Lipschitz boundary for all $m \in \mathbb{N}$.

Concerned with the problem of the discreteness of the spectrum of $\Delta_M$, it is well known that when $M$ is a compact Riemannian manifold, or when $M$ is an open subset of $\mathbb{R}^n$ with finite measure and a sufficiently regular boundary, the spectrum is discrete. However, the spectrum of $\Delta_M$ need not be discrete in general. Special situations, which are not included in this standard framework, have been considered in the literature. For instance, conditions for the discreteness of the spectrum of the Laplacian on noncompact complete Riemannian manifolds with a peculiar structure are the subject of several contributions, including [7, 12, 16, 25, 26, 35].

Cianchi & Maz'ya [20], for a noncompact Riemannian manifold $M^n$ with $n \ge 2$ and $\mathcal{H}^n(M) < \infty$, prove that the embedding $W^{1,2}(M) \to L^2(M)$ is compact if and only if

$$ \lim_{s \to 0} \frac{s}{\mu_M(s)} = 0 $$

holds, which in turn is equivalent to the spectrum of $\Delta_M$ being discrete. Here, the isocapacitary function $\mu_M: [0, \mathcal{H}^n(M)/2] \to [0, \infty]$ is given by $\mu_M(s) := \inf\{C(E, G) \mid E \text{ and } G \text{ are measurable subsets of } M \text{ such that } E \subset G \subset M \text{ and } s \le \mathcal{H}^n(E), \mathcal{H}^n(G) \le \mathcal{H}^n(M)/2\}$, and $C(E, G) := \inf\{\int_M |\nabla u|^2 dx \mid u \in W^{1,2}(M), u \ge 1 \text{ in } E \text{ and } u \le 0 \text{ in } M \setminus G \text{ (up to a set of standard capacity zero)}\}$.

Furthermore, Cianchi & Maz'ya [19] prove that if $\mathcal{H}^n(M) < \infty$,

$$ \int_0^1 \frac{ds}{\mu_M(s)} < \infty, $$

then for any eigenvalue $\gamma$ of $\Delta_M$,

$$ \|u\|_{L^\infty(M)} \le C(\mu_M, \gamma) \|u\|_{L^2(M)} $$

for every eigenfunction $u$ of the Laplacian on $M$ associated with $\gamma$.

For a complete Riemannian manifold $M_0$ and a Schrödinger operator $-\Delta_{M_0} + m$ acting on $L^q(M_0)$, Ouhabaz [45] studies related problems on the spectrum of $-\Delta_{M_0} + m$ concerning the positivity of the $L^2$ spectral lower bound $\inf \sigma(-\Delta_{M_0} + m)$. He proves that if $M_0$ satisfies $L^2$-Poincaré inequalities (for some $R > 0$), that is,

$$ \int_{B(p,r)} |u - \overline{u}_{B(p,r)}|^2 dv_g \le C(R) r^2 \int_{B(p,r)} |\nabla u|^2 dv_g \quad \forall u \in C^\infty(B(p,r)), p \in M_0, 0 < r \le R, $$

and a local doubling property, then $\inf \sigma(-\Delta_{M_0} + m) > 0$, provided that $m$ satisfies the mean condition

$$ \inf_{p \in M_0} \frac{1}{|B(p,r)|} \int_{B(p,r)} m dv_g > 0, $$

for some $r > 0$. Ouhabaz also show that this condition is necessary under some additional geometrical assumptions on $M_0$.

If $\Sigma_0$ is a bounded domain in $\mathbb{R}^n$ with a smooth boundary $\partial\Sigma_0$, the eigenvalue problem

$$ \begin{cases} -\Delta u = \lambda u & \text{in } \Sigma_0, \\ u = 0 & \text{on } \partial\Sigma_0, \end{cases} $$

possesses an infinite sequence of positive eigenvalues:

$$ 0 < \lambda_1 < \lambda_2 \le \cdots \lambda_k \le \cdots; \quad \lambda_k \to \infty, \text{ as } k \to \infty $$

with finite multiplicity. The same properties also hold for the following problem:

$$ \begin{cases} -\Delta u + a_0(x)u = \lambda m(x)u & \text{in } \Sigma_0, \\ u = 0 & \text{on } \partial\Sigma_0, \end{cases} $$

where $a_0$ and $m$ are positive and sufficiently smooth on $\bar{\Sigma}_0$. For a detailed study on the existence of principal eigenvalues for second-order differential operators that are not necessarily in divergence form, refer to Fleckinger, Hernández, & Thélin [28], where results are obtained regarding the multiplicity of principal eigenvalues in both variational and general cases.
Existence of eigenvalues with positive eigenfunctions is important in the context of nonlinear problems where positive solutions are of interest. This is particularly relevant in many areas such as reaction-diffusion systems in population dynamics, chemical reactions, combustion, etc. (see [54]).

The classical reference for this theory is the book by Courant & Hilbert [21], where the theory is developed for continuous coefficients and also applies to bounded coefficients. The variational characterization of the eigenvalues establishes their continuous and monotone dependence with respect to both the coefficients and the domain $\Sigma_0$. The main tool employed in this context is the abstract theory of linear compact self-adjoint operators in Hilbert spaces.

This theory can be extended to the case of unbounded coefficients and also when $a_0$ changes sign in $\Sigma_0$. If $a_0 > 0$ in $\Sigma_0$, with $m, a_0 \in L^r(\Sigma_0)$ for $r > N/2$ and $m > 0$ (or $m < 0$) on a subdomain of positive measure, then there exists exactly one principal eigenvalue $\lambda_1^+ > 0$ (or $\lambda_1^- < 0$), with a positive eigenfunction; see [23, 13, 58].

Regarding elliptic problems involving $L^1$ or measure data, the main mathematical difficulty consists in the fact that the classical variational formulation is not possible. Existence results have been obtained using a non-variational framework for these problems. The first approach is due to Stampacchia [55], who obtained solutions with a duality method. However, a limitation of this approach is that it applies only to linear equations. In [9], Boccardo and Gallouët have developed a method based on the approximation of the data by smoother functions. Applications of these two methods in the study of several types of elliptic problems with Dirichlet boundary conditions and $L^1$ or measure data may be found in [11, 43, 44, 49, 50].

In this work, we first extend the results of Pflüger [47] to the case of noncompact Riemannian manifolds. Under certain conditions on the weight functions, we will achieve continuity and compactness of embeddings and traces in $W^{1,q}(\Omega; V_0, V_1)$. For this, we will additionally assume conditions on the geometry of $\Omega$ (see $(U_1) - (U_4)$ below). For different conditions on the geometry of $\Omega$ and on the weights, see, e.g., [3] for a study of Sobolev-Slobodeckii and Besel potential spaces in singular manifolds.

We will find a sequence of eigenvalues of problem (1.2) using the technique from Allegreto [2]. The challenge here is to obtain the limit (1.3) since the domain $\Omega$ is unbounded. For other studies on Fredholm properties of elliptic operators on noncompact manifolds, see Lockhart & McOwen [42] and Lockhart [41].

Following the approaches of Bocea & Redulescu [10] and Orsina [44], we will then seek weak solutions of problem (1.1) that admit $L^1$ data, after having studied problem (1.2).

In the sequel for simplicity we write $\Gamma = \partial\Omega$. We assume that there exists a locally finite covering of $M$ with open subsets $U_{k,i}, \hat{U}_{k,i} \subset M$, $(k, i) \in \{0,1\} \times \mathbb{N}$, having the following properties:

(U1) $\bar{\Omega} \subset (\cup_i U_{0,i}) \cup (\cup_j U_{1,j})$, $\cup_i \hat{U}_{0,i} \subset \Omega$, $\Gamma \subset \cup_j U_{1,j}$, $M = \cup_{(k,i) \in \{0,1\} \times \mathbb{N}} \hat{U}_{k,i}$, and $U_{k,i} \subset \hat{U}_{k,i}$, for all $(k,i) \in \{0,1\} \times \mathbb{N}$.

(U2) There exist $0 < r_{k,i} < \hat{r}_{k,i}$ and charts $\psi_{k,i} : B(0, \hat{r}_{k,i}) \subset \mathbb{R}^n \to \hat{U}_{k,i} \subset M$, for $(k,i) \in \{0,1\} \times \mathbb{N}$, such that: (a) $\psi_{k,i} : B(0, r_{k,i}) \to U_{k,i}$, $(k,i) \in \{0,1\} \times \mathbb{N}$, are diffeomorphism; (b) $\psi_{1,j}(B(0, \hat{r}_{1,j}) \cap \{x_n > 0\}) \subset \Omega$ and $\psi_{1,j}(B(0, \hat{r}_{1,j}) \cap \{x_n = 0\}) = \hat{U}_{1,j} \cap \Gamma$, for $i,j \in \mathbb{N}$; (c) $\psi_{\Gamma,1,j} : B(0, \hat{r}_{1,j}) \cap \{x_n = 0\} \to \hat{U}_{1,j} \cap \Gamma$, $j \in \mathbb{N}$, defined by $\psi_{\Gamma,1,j}(x_1, \dots, x_{n-1}) = \psi_{1,j}(x_1, \dots, x_{n-1}, 0)$, are charts of $\Gamma$.

(U3) There is a global constant $R_1 > 0$ such that $\sum_{k,i} \chi_{\hat{U}_{k,i}} \le R_1$ in $M$.

(U4) There is a constant $R_2 > 0$ such that $(\sup_{|x|\delta_{\mathbb{R}^n}=1} |d(\psi_{1,j})xz|_g)(\sup_{|v|\delta_{\mathbb{R}^n}=1} |d(\psi_{1,j}^{-1})p|_{\delta_{\mathbb{R}^n}}) \le R_2$ and $(\sup_{\hat{U}_{1,j}} \det[g_{ab}])(\sup_{\hat{U}_{1,j}} \det[g^{ab}]) \le R_2^2$, for $j \in \mathbb{N}$, where $\delta_{\mathbb{R}^n}$ is the Euclidean metric, $[g^{ab}] := [g_{ab}]^{-1}$ and $g_{ab} = g(d\psi_{1,j}e_a, d\psi_{1,j}e_b)$.

The number $R_1$ is related to the Besicovitch Covering Theorem, see [24, 33]. Examples of coverings that satisfy condition (U4) can be found in $\mathbb{R}^n$ and in the hyperbolic ball $\mathbb{B}^n$.

Let $V_i, i = 0, \dots, 3$, and $W$ be weights on $M$, i.e., locally integrable functions on $M$ such that $V_i, W > 0$ almost everywhere on $M$. We will also assume that $W$ is continuous and that all weights are bounded from above and from below by positive constants on each compact subset of $M$.

We assume that there exist positive, continuous functions $b_i, i = 1, 2, 3$, defined on $M$, and a constant $K_q > 0$, such that for some fixed $m_* \in \mathbb{N}$:
If $(k, i) \in \{0, 1\} \times \mathbb{N}$ and $\psi_{k,i}(0) \in D^{m*}$, then

(W1) $\mathcal{R}_{k,i}^q(p)\mathbf{V}_1(p) \leq K_q\mathbf{V}_0(p)$ for a.e. $p \in \hat{U}_{k,i}$. And $\hat{r}_{k,i}^{-q} \|d\psi_{k,i}\|^{-1}\mathbf{V}_1(p) \leq K_q\mathbf{V}_0(p)$ for a.e. $p \in \hat{U}_{k,i}$.

(W2) $\mathbf{V}_2(p) \leq b_2(\psi_{k,i}(0))$ and $\|d\psi_{k,i}\|b_1(\psi_{k,i}(0)) \leq \mathbf{V}_1(p)$ for a.e. $p \in \hat{U}_{k,i}$.

If $j \in \mathbb{N}$ and $\psi_{1,j}(0) \in D^{m*}$, then

(W3) $\mathbf{W}(p) \leq b_3(\psi_{1,j}(0))$ and $\|d\psi_{1,j}\|b_1(\psi_{1,j}(0)) \leq \mathbf{V}_1(p)$ for a.e. $p \in \hat{U}_{1,j}$.

Here, $\mathcal{R}_{k,i} := \sum_{(a,b) \in \{0,1\} \times \mathbb{N}} \|d(\psi_{a,b}^{-1})\|(\hat{r}_{a,b} - r_{a,b})^{-1}\chi_{\hat{U}_{a,b} \cap \hat{U}_{k,i}}$, $\|d(\psi_{k,i})\| := \sup\{|d(\psi_{k,i})xz|_g \mid x \in B(0, \hat{r}_{k,i}), |z|_{\delta_{\mathbb{R}n}} = 1\}$, and $\|d(\psi_{k,i}^{-1})\| := \sup\{|d(\psi_{k,i}^{-1})pv|_{\delta_{\mathbb{R}n}} \mid (p, v) \in T\hat{U}_{k,i}, |v|_g = 1\}$.

We define

$$ \mathcal{B}_{q,q_0}^m := \sup_{\substack{(k,i) \in \{0,1\} \times \mathbb{N} \\ \psi_{k,i}(0) \in D^m}} \frac{b_2^{1/q_0}(\psi_{k,i}(0))}{b_1^{1/q}(\psi_{k,i}(0))} \|G_{k,i}\|^{\frac{1}{q_0}} \|G_{k,i}^{-1}\|^{\frac{1}{q}} \hat{r}_{k,i}^{\frac{n}{q_0} - \frac{n}{q} + 1}, $$

where $\|G_{k,i}\| := \sup_{\hat{U}_{k,i}} \{\sqrt{\det[g_{ab}]}\} \text{ and } \|G_{k,i}^{-1}\| := \sup_{\hat{U}_{k,i}} \{\sqrt{\det[g^{ab}]}\}$.

Our first main result are Propositions 1.1 and 1.2. For simplicity, we write $\Omega_m := D_m \cap \Omega$, $\Omega^m := \Omega \setminus \bar{\Omega}_m$, $\Gamma_m := D_m \cap \Gamma$, and $\Gamma^m := \Gamma \setminus \bar{\Gamma}_m$.

**Proposition 1.1.** Assume that $(U_1) - (U_4)$, $(\mathbf{W}_1)$ and $(\mathbf{W}_2)$ are verified. Suppose $1 \leq q < n$ and $nq/(n-q) \geq q_0 \geq q$.

(i) If $\lim_{m \to \infty} \mathcal{B}_{q,q_0}^m < \infty$, then $W^{1,q}(\Omega; \mathbf{V}_0, \mathbf{V}_1) \to L^{q_0}(\Omega; \mathbf{V}_2)$ is continuous.

(ii) If $\lim_{m \to \infty} \mathcal{B}_{q,q_0}^m = 0$, then $W^{1,q}(\Omega; \mathbf{V}_0, \mathbf{V}_1) \to L^{q_0}(\Omega; \mathbf{V}_2)$ is compact.

Set

$$ \mathcal{B}_{\Gamma,q,q_0}^m := \sup_{\substack{j \in \mathbb{N} \\ \psi_{1,j}(0) \in \Gamma^m}} \frac{b_3^{1/q_1}(\psi_{1,j}(0))}{b_1^{1/q}(\psi_{1,j}(0))} \|G_{\Gamma,1,j}\|^{\frac{1}{q_1}} \|G_{1,j}^{-1}\|^{\frac{1}{q}} \hat{r}_{1,j}^{\frac{n-1}{q_1} - \frac{n}{q} + 1}, $$

where $\|G_{\Gamma,1,j}\| := \sup_{\Gamma} \{\sqrt{\det[g_{\Gamma,ab}]}\}$ and $g_{\Gamma,ab} := g(d\psi_{\Gamma,1,j}e_a, d\psi_{\Gamma,1,j}e_b)$.

**Proposition 1.2.** Assume that $(U_1) - (U_4)$, $(\mathbf{W}_1)$ and $(\mathbf{W}_3)$ are verified. Suppose $1 \leq q < n$ and $(n-1)q/(n-q) \geq q_1 \geq q$.

(i) If $\lim_{m \to \infty} \mathcal{B}_{\Gamma,q,q_0}^m < \infty$, then there exists a continuous trace operator $W^{1,q}(\Omega; \mathbf{V}_0, \mathbf{V}_1) \to L^{q_1}(\Gamma; \mathbf{W})$.

(ii) If $\lim_{m \to \infty} \mathcal{B}_{\Gamma,q,q_0}^m = 0$, then the trace operator $W^{1,q}(\Omega; \mathbf{V}_0, \mathbf{V}_1) \to L^{q_1}(\Gamma; \mathbf{W})$ is compact.

In Corollaries 3.7 and 3.9, we additionally obtain results analogous to Propositions 1.1 and 1.2 for the case of weighted Sobolev spaces $W^{1,q}(TM; \mathbf{V}_0, \mathbf{V}_1)$ for the tangent bundle.

**Corollary 1.3.** Let $q \in \mathbb{R}^+ \mapsto q_V \in \mathbb{R}^+$ and $q \in \mathbb{R}^+ \mapsto q_W \in \mathbb{R}^+$ be functions satisfying $nq/(n-q) \geq q_V > q$ and $(n-1)q/(n-q) \geq q_W > q$ if $1 \leq q < n$. Suppose that $n \geq 3$ and $(U_1) - (U_4)$, $(\mathbf{W}_1)$, and $(\mathbf{W}_2)$ are verified for every $q \in [1, n)$. Assume further that $\lim_{m \to \infty} \mathcal{B}_{q,q_V}^m < \infty$ and $\lim_{m \to \infty} \mathcal{B}_{\Gamma,q,q_W}^m < \infty$. Then there exist constants $C_{q_V} > 0$ and $C_{q_W} > 0$ such that:

$$ \|u\|_{q_V, \Omega, \mathbf{V}_2} \leq C_{q_V} \|u\|_{1, q, \Omega, \mathbf{V}_0, \mathbf{V}_1}, \quad (1.4) $$

$$ \|u\|_{q_W, \Gamma, \mathbf{W}} \leq C_{q_W} \|u\|_{1, q, \Omega, \mathbf{V}_0, \mathbf{V}_1}, \quad (1.5) $$

for all $u \in W^{1,q}(\Omega; \mathbf{V}_0, \mathbf{V}_1)$

Other conditions that guarantee the continuity of the embeddings are given as follows:

**Remark 1.4.** See [46]. Let $(M_1^n, g)$, $n \geq 2$, be a Riemannian manifold endowed with weight functions $\rho$ and $w$ on $M_1$. Assume there exist constants $c_1 > 0$, $c_2 \in \mathbb{R}$, $c_3 > 1$, and $c_4, c > 0$ such that $\forall p \in M_1$, $\forall p_1 \in B(p, c\rho(p))$:

(a) $i_p(g) \geq c_1\rho(p)$, where $i_p(g)$ is the injectivity radius at $p$.

(b) $\mathrm{Ric}_p(g) \geq c_2\rho(p)^{-2}g_p$.

(c) $(1/c_3)\rho(p) \leq \rho(p_1) \leq c_3\rho(p)$.

(d) $(1/c_4)w(p) \leq w(p_1) \leq c_4w(p)$.
Then there exists a continuous embedding

$$W_{1,w}^q(M_1) \to L_w^{q*}(M_1), \quad q^* := \frac{nq}{n-q},$$

where $1 \le q < n$ and $W_{1,w}^q(M_1)$ is the Banach space completion of $\{u \in C^\infty(M_1) \mid \|u\|_{W_{1,w}^q} < \infty\}$,

and the norm used is $\|u\|_{W_{1,w}^q} := (\int_{M_1} |wu|^q \rho^{-n} dv_g + \int_{M_1} |w\rho\nabla u|^q \rho^{-n} dv_g)^{1/q}$.

For the case of manifolds Euclidean at infinity, see also [17, Weighted Sobolev spaces].

We now assume the following:

(W4) $\mathcal{R}_{k,i}^q(p)V_1(p) \le K_q V_2(p)$ for a.e. $p \in \hat{U}_{k,i}$, and $\int_{\hat{U}_{k,i}} V_2 dv_g \le K_3$ for every $(k, i) \in \{0, 1\} \times \mathbb{N}$.

For weights $V$ and $w$, for simplicity we write $dV = Vdv_g$, $dW = Wd\sigma_g$, $V(U) = \int_U Vdv_g$, $W(U \cap \Gamma) = \int_{U \cap \Gamma} Wd\sigma_g$, for every measurable set $U \subset M$.

**Theorem 1.5.** We assume the hypotheses of Corollary 1.3 hold. Suppose $V_1 \in C^1(M)$, $\tau \in L^{2\nu/(2\nu-2)}(\Omega; V_2) \cap L^\infty(\Omega)$, and $\{\tau > 0\}$ and $\{\tau < 0\}$ do not have empty interiors. Then there exist infinitely many eigenvalues $\cdots \le \lambda_2^- \le \lambda_1^- < 0 < \lambda_1^+ \le \lambda_2^+ \le \cdots$ of (1.2). For a weak solution $u \in W^{1,2}(\Omega; V_0, V_1)$ of (1.2), if, in addition, (W4) holds, then $u \in L^\infty(\Omega)$. Moreover, if $u \in W_0^{1,2}(\bar{\Omega}; V_0, V_1)$, (1.3) is satisfied.

For some conditions that guarantee $u \in W^{2,2}(\Omega; V_0, V_1, V_3)$, see Proposition 4.6 and Remark 4.7 in Section 4.

We now consider an elliptic problem with $L^1$ data. Additionally, we introduce a continuous weight function, denoted by $W_1$, and a positive continuous function $b_4$, both defined on $M$, such that:

(W5) $[\|d\psi_{k,i}\|(\hat{r}_{k,i}-r_{k,i})+1]V_2(p) \le K_3V_0(p)$, $[\|d\psi_{1,i}\|(\hat{r}_{1,i}-r_{1,i})+1]W_1(p) \le K_3W(p)$, $b_4(\psi_{k,i}(0)) \le V_0(p)$, for a.e. $p \in \hat{U}_{k,i}$,

$$\frac{\|G_{k,i}^{-1}\|}{b_1(\psi_{k,i}(0))\|d\psi_{k,i}\|(\hat{r}_{k,i}-r_{k,i})^n} \le K_3 \quad \text{and} \quad \frac{\|G_{k,i}^{-1}\|}{b_4(\psi_{k,i}(0))(\hat{r}_{k,i}-r_{k,i})^n} \le K_3,$$

for every $(k, i) \in \{0, 1\} \times \mathbb{N}$.

Compare this condition to those used by Brown & Opic [14], where continuous and compact embeddings of the weighted Sobolev space into spaces of weighted continuous and Hölder-continuous functions for domains in Euclidean spaces are studied.

Furthermore, we assume that:

(H1) (i) $V_i(\Omega) < \infty$, $i = 0, 1, 2$, and $V_1 \in C^1(M)$.

(ii) There is $1 < \sigma < n/(n-1)$ such that $\frac{\sigma}{\sigma-1} \le \left(\frac{\sigma v}{\sigma v-1}\right)_V$, $\frac{((i-1)\sigma)_v}{((i-1)\sigma)_v-(i-1)} \le \left(\frac{(i\sigma)_v}{(i\sigma)_v-i}\right)_V$, and $(k\sigma)_v \ge 2k$, where $k \in \mathbb{N}$, $2k \le n < 2(k+1)$, and $i = 2, \dots, k$.

(iii) There is $1 < q_2 < n/(n-1)$ such that

$$\lim_{m \to \infty} \sup_{\|u\|_{1,q_2,\Omega, V_0, V_1} \le 1} \|u\|_{1,\Omega^m, V_0} = 0.$$

<figure><img src="image_2.png" /><figcaption>FIGURE 1</figcaption></figure>
**Example 1.6.** Define $\Omega := \{x \in \mathbb{R}^n \mid |(x_{d+1}, \dots, x_n)| \le \rho(|(x_1, \dots, x_d)|)\}$ (see Figure 1), where $\rho: [0, \infty) \to \mathbb{R}^+ \in C^\infty$ with $\rho$ bounded, and $C^{-1} \le |\rho'| \le C$ on $[0, \infty)$ for some positive constant. Also, $d \in \{1, \dots, n-1\}$ with $n-d-1 > 0$. Define $V_i(x) = (1+|x|)^{\alpha_i}$, $i = 0, 1, 2$, where $\alpha_0 \ge 0 \ge \alpha_2 \ge \alpha_1 > -n$, and $\alpha_0 > \alpha_2 + n/2$. For an appropriate function $\rho$ that decays to infinity, we have $V_i(\Omega) < \infty$, $i = 0, 1, 2$. Define

$$q_V = \frac{q(n + \alpha_2)}{n - q + \alpha_0} \quad \text{for} \quad 1 < q < n.$$

With these hypotheses, we have $\lim_{m \to \infty} B_{q,qv}^m < \infty$, where $D_m = \{x \in \mathbb{R}^n \mid |x| < m\}$. And also, conditions in (H$_1$) are verified. In these types of domains, for a study of quasilinear elliptic equations with Robin boundary conditions and $L^1$ data, we can refer to [6], where the solutions are in weighted Sobolev spaces.

Let $f_0 \in L^1(\Omega; V_0)$ and $f_1 \in L^1(\Gamma; W)$. Let $(f_{0,j}) \subset L^2(\Omega, V_0)$ and $(f_{1,j}) \subset L^2(\Gamma, W)$ be sequences such that $f_{0,j} \to f_0$ strongly in $L^1(\Omega, V_0)$ and $f_{1,j} \to f_1$ strongly in $L^1(\Gamma, W)$. Let $u_j$ be the variational solution of (1.1) corresponding to $f_0$ and $f_1$ (see Appendix B). We shall prove that $(u_j)$ converges weakly in each space $W^{1,q}(\Omega; V_0, V_1)$, where $1 \le q < n/(n-1)$, to a solution $u$ (called a *solution obtained by approximation*) of (1.1) which does not depend on the choice of the approximating sequences $(f_{0,j})$ and $(f_{1,j})$. More precisely, we shall show that this limit is a solution of our problem in the sense of the following

**Definition 1.7.** A function $u \in \cap_{1 \le q < \frac{n}{n-1}} W^{1,q}(\Omega; V_0, V_1)$ is a solution of (1.1) provided that

$$\int_{\Omega} g(\nabla u, \nabla v) dV_1 + \int_{\Omega} uv dV_0 = \lambda \int_{\Omega} \tau uv dV_2 + \int_{\Omega} fv dV_2 + \int_{\Gamma} f_1 v dW_1, \quad (1.6)$$

for all $v \in \cup_{q>n} W^{1,q}(\Omega; V_0, V_1)$.

**Theorem 1.8.** Assume that $(W_5)$, $(H_1)$, and the hypotheses of Corollary 1.3 are satisfied. Suppose $\tau \in L^\infty(\Omega)$, and $\{\tau > 0\}$ and $\{\tau < 0\}$ do not have an empty interior. Additionally, let $f_0 \in L^1(\Omega; V_0)$ and $f_1 \in L^1(\Gamma; W_1)$. Under these conditions:

(i) If $\lambda$ is not an eigenvalue of (1.2), then (1.1) has a solution $u$ obtained by approximation.

(ii) Suppose $(W_4)$ holds. If $\lambda$ is an eigenvalue of equation (1.2), then equation (1.1) has a solution $u$ obtained by approximation if and only if

$$\int_{\Omega} f_0 v dV_2 + \int_{\Gamma} f_1 v dW_1 = 0,$$

for all $v \in E_\lambda := \{u \in W^{1,2}(\Omega; V_0, V_1) \mid u \text{ solves (1.2)}\}$.

Next, we proceed to find an estimate of the first eigenvalue $\lambda_1(\Omega)$ of problem (1.2):

$$\begin{cases} -\operatorname{div}(V_1 \nabla \phi_{\Omega}) + V_0 \phi_{\Omega} = \lambda_1(\Omega) V_2 \tau \phi_{\Omega} & \text{in } \Omega, \\ V_1 \frac{\partial \phi_{\Omega}}{\partial \nu} = 0 & \text{on } \Gamma, \end{cases}$$

where $\tau \ge \delta$ in $\Omega$ for some $\delta > 0$, $\phi_\Omega \in W^{1,2}(\Omega; V_0, V_1)$, and $\int_\Omega \tau \phi_\Omega^2 dV_2 = 1$. For this objective, we will additionally consider an open set $N$ such that $\bar{\Omega} \subset N$ and $M \setminus \bar{N} \neq \emptyset$. We will denote by $\{\lambda_i(N)\}$ the infinite sequence of eigenvalues of

$$\begin{cases} -\operatorname{div}(V_1 \nabla \kappa_i) + V_0 \kappa_i = \lambda_i(N) V_2 \tau \kappa_i & \text{in } N, \\ \kappa_i = 0 & \text{on } \partial N, \end{cases}$$

where $\tau \ge \delta$ in $N$, $\kappa_i \in W^{1,2}(N; V_0, V_1)$, and $\int_N \tau \kappa_i^2 dV_2 = 1$.

We also denote by $\{\lambda_i^{\pm}(M)\}$ the double sequence of eigenvalues of

$$-\operatorname{div}(V_1 \nabla \phi_i^{\pm}) + V_0 \phi_i^{\pm} = \lambda_i^{\pm}(M) V_2 \tau \phi_i^{\pm} \quad \text{in } M, \quad (1.7)$$

where $\{\tau < 0\} \cap (M \setminus \bar{N})$ has nonempty interior, $\cdots \le \lambda_2^-(M) \le \lambda_1^-(M) < 0 < \lambda_1^+(M) \le \lambda_2^+(M) \le \cdots$, $\phi_i^{\pm} \in W^{1,2}(M; V_0, V_1)$, and $\int_M \tau \phi_i^{\pm} dV_2 = \operatorname{sign}(\lambda_i^{\pm}(M))1$.
We will use the Dirichlet capacity of $M\setminus N$, see [8, 22]:

$$ \mathrm{Cap}^{\pm}(M\setminus N) := \inf \left\{ \int_M u^2 \mathrm{d}\mathbf{V}_0 + \int_M |\nabla u|^2 \mathrm{d}\mathbf{V}_1 \mid u \in W^{1,2}(M; \mathbf{V}_0, \mathbf{V}_1), u - \phi_1^{\pm} = 0 \text{ in } M\setminus N \right\}. $$

Motivated by this definition of capacity, we define:

**Definition 1.9.** Let $\ell \in (0, 1)$ and $\bar{\nu}$ be an extension of $\nu$ in some neighborhood of $\Gamma$. Define $\mathcal{F}_{\Omega} := \{U \subset \Omega \mid U \text{ is an open, bounded set}\}$,

$$ \mathcal{A} := \left\{ \begin{array}{l} u \in W^{1,2}(N; \mathbf{V}_0, \mathbf{V}_1) \cap W_{\mathrm{loc}}^{2,2}(\bar{\Omega}) \mid \|u - \kappa_1\|_{W^{2,2}(U)} \le \|\phi_{\Omega}\|_{W^{2,2}(U)} \forall U \in \mathcal{F}_{\Omega}, \\ \qquad \int_{N \setminus \Omega} \tau(u - \kappa_1)^2 \mathrm{d}\mathbf{V}_2 \le \ell \text{ and } g(\nabla(u - \kappa_1), \bar{\nu})|_{\Gamma} = 0 \end{array} \right\}. $$

and

$$ C_{\nu}(\Gamma) := \inf_{u \in \mathcal{A}} \left\{ \int_N u^2 \mathrm{d}\mathbf{V}_0 + \int_N |\nabla u|^2 \mathrm{d}\mathbf{V}_1 \right\}. $$

We prove the following theorem.

**Theorem 1.10.** Suppose the hypotheses of Corollary 1.3 hold, $\tau \in L^{2\nu/(2\nu-2)}(\Omega, \mathbf{V}_2) \cap L^{\infty}(\Omega)$, $\tau \ge \delta > 0$ in $N$, and $\{\tau < 0\} \cap (M\setminus \bar{N})$ has nonempty interior. Assume further that $\mathbf{V}_1 \in C^1(M)$, $\mathbf{V}_2(\Omega) < \infty$ and $C_{\nu}(\Gamma) > 0$.

(i) Let $\phi \in W^{1,2}(M; \mathbf{V}_0, \mathbf{V}_1)$ be any extension of $\phi_{\Omega} \in W^{1,2}(\Omega; \mathbf{V}_0, \mathbf{V}_1)$ such that $\mathrm{supp} \phi \subset N$ and $\int_{N \setminus \Omega} \tau \phi^2 \mathrm{d}\mathbf{V}_2 \le \ell$, then

$$ \left\{ \begin{array}{l} \|\phi\|_{1,2,M,\mathbf{V}_0,\mathbf{V}_1}^2 - \lambda_1(N) \ge C_{\nu}(\Gamma) \frac{\lambda_2(N) - \lambda_1(N)}{\lambda_2(N) + \lambda_1(N)}, \\ \lambda_1(N) - \Lambda_1 \ge C(M \setminus N) \frac{\Lambda_2 - \Lambda_1}{\Lambda_2 + \Lambda_1}, \end{array} \right. $$

where

$$ \left\{ \begin{array}{l} \Lambda_1 = |\lambda_1^-(M)|, \Lambda_2 = \lambda_1^+(M), |\lambda_1^-(M)| \le \lambda_1^+(M) \le |\lambda_2^-(M)|, C(M\setminus N) = \mathrm{Cap}^-(M\setminus N), \\ \Lambda_1 = |\lambda_1^-(M)|, \Lambda_2 = |\lambda_2^-(M)|, |\lambda_1^-(M)| \le |\lambda_2^-(M)| \le \lambda_1^+(M), C(M\setminus N) = \mathrm{Cap}^-(M\setminus N), \\ \Lambda_1 = \lambda_1^+(M), \Lambda_2 = |\lambda_1^-(M)|, \lambda_1^+(M) \le |\lambda_1^-(M)| \le \lambda_2^+(M), C(M\setminus N) = \mathrm{Cap}^+(M\setminus N), \\ \Lambda_1 = \lambda_1^+(M), \Lambda_2 = \lambda_2^+(M), \lambda_1^+(M) \le \lambda_2^+(M) \le |\lambda_1^-(M)|, C(M\setminus N) = \mathrm{Cap}^+(M\setminus N). \end{array} \right. \quad (1.8) $$

(ii) Suppose that

$$ C_{\nu}(\Gamma) < \frac{\lambda_1(N)}{16}(1-\ell)^2 \quad \text{and} \quad \mathrm{Cap}^{\pm}(M\setminus N) < \frac{\min\{|\lambda_1^-(M)|, \lambda_1^+(M)\}}{16}. \quad (1.9) $$

Then

$$ \left\{ \begin{array}{l} \lambda_1(\Omega) - \lambda_1(N) \le \frac{16}{7(1-\ell)} \left[ \lambda_1(N)\ell + \frac{9}{2}\sqrt{\lambda_1(N) C_{\nu}(\Gamma)} \right], \\ \lambda_1(N) - |\lambda_1^{\pm}(M)| \le \frac{4}{7} \left[ 17 + \frac{|\lambda_1^{\pm}(M)|}{\min\{|\lambda_1^-(M)|, \lambda_1^+(M)\}} \right] \sqrt{|\lambda_1^{\pm}(M)|} \sqrt{\mathrm{Cap}^{\pm}(M\setminus N)}. \end{array} \right. $$

Let us observe that if $\cup_{(k,i) \in \{0,1\} \times N} \psi_{k,i}(B(0, r_{k,i} + \delta^{-1}(\hat{r}_{k,i} - r_{k,i})) \subset N$ for some $\delta \ge 1$, then by Proposition 3.2, $\|\phi\|_{1,2,N,\mathbf{V}_0,\mathbf{V}_1}^2 \le \max\{(1+R_2)(1+\delta^2\mathbf{K}_2), 1+R_2^3\} \lambda_1(\Omega)$ for $\phi = \mathcal{E}\phi_{\Omega}$.

This paper is organized as follows. In Section 2, we gather preliminary definitions and results, which are used several times in the paper. In Section 3, we study the operators $W^{1,q}(\Omega; \mathbf{V}_0, \mathbf{V}_1) \to L^{q_0}(\Omega; \mathbf{V}_2)$ and $W^{1,q}(\Omega; \mathbf{V}_0, \mathbf{V}_1) \to L^{q_1}(\partial\Omega; \mathbf{W})$ in weighted Sobolev spaces. We show that under certain conditions on the weight functions $\mathbf{V}_i$ and $\mathbf{W}$, these operators are continuous or compact. In Section 4, as an application of the results from the previous section, we will study the existence of the eigenvalues of the elliptic problem (1.2) with Neumann boundary conditions and $\tau$ changing sign. Additionally, we will examine the behavior at infinity of its associated eigenfunctions. In Section 5, using the spectrum associated with problem (1.2), we provide an existence result for a linear eigenvalue problem (1.1) with an indefinite weight and data in $L^1$. Our approach is not variational and uses the notion of a solution obtained by
approximation. In Section 6, the objective is to compare the first eigenvalue $\lambda_1(\Omega)$ of (1.2), for the case $\tau|_{\Omega} \ge \delta > 0$, with the eigenvalue $\lambda_1^{\pm}(M)$ of (1.7). For this, we will introduce a parameter motivated by the Dirichlet capacity.

## 2. PRELIMINARIES

Let $(M, g)$ be a smooth Riemannian manifold. For $k$ integer, and $u : M \to \mathbb{R}$ smooth, we denote by $\nabla^k u$ the $k$th covariant derivative of $u$, and $|\nabla^k u|$ the norm of $\nabla^k u$ defined in a local chart by

$$|\nabla^k u|^2 = g^{i_1 j_1} \cdots g^{i_1 j_k} (\nabla^k u)_{i_1 \dots i_k} (\nabla^k u)_{j_1 \dots j_k}.$$

Recall that $(\nabla u)_i = \partial_i u$, while

$$(\nabla^2 u)_{ij} = \partial_{ij} u - \Gamma^k_{ij} \partial_k u. \qquad (2.1)$$

Now we define $W^{k,q}(M; \mathbf{V}_0, \dots, \mathbf{V}_k)$ and $W^{k,q}(\bar{\Omega}; \mathbf{V}_0, \dots, \mathbf{V}_k)$, $k = 1, 2$, following [34, 57, 32, 37]. Let $(M, g)$ be a Riemannian manifold of dimension $n$. Let $U$ be and open set on $M$ and $q \ge 1$, define

$$L^q(U; \mathbf{V}_i) := \{ u : U \to \mathbb{R} \mid u \text{ is measurable function and } \int_U |u|^q d\mathbf{V}_i < \infty \},$$

where $i = 0, 1, 2$.

We denote

$$C^{k,q}(M; \mathbf{V}_0, \dots, \mathbf{V}_k) := \{ u \in C^k(M) \mid \sum_{i=0}^{k} \int_M |\nabla^i u|^q d\mathbf{V}_i < \infty \} \quad \text{for } k = 1, 2,$$

$$C_0^{\infty}(\bar{\Omega}) := \{ u \in C^{\infty}(\Omega) \mid \text{there is } \bar{u} \in C_0^{\infty}(M) \text{ such that } u = \bar{u}|_{\Omega} \},$$

$$L_{\text{loc}}^{q}(\bar{\Omega}) := \{ u : \Omega \to \mathbb{R} \mid u \in L^{q}(U) \text{ for any open bounded set } U \subset \Omega \}.$$

Recall that $\mathbf{V}_0$ and $\mathbf{V}_1$ are bounded from above on compact subset of $M$.

**Definition 2.1.**

(i) The Sobolev space $W^{k,q}(M; \mathbf{V}_0, \dots, \mathbf{V}_k)$, $k = 1, 2$, is the completion of $C^{k,q}(M; \mathbf{V}_0, \dots, \mathbf{V}_k)$ with respect the norm

$$\|u\|_{k,q,M,\mathbf{V}_0,\dots,\mathbf{V}_k} := \left( \sum_{i=0}^{k} \int_M |\nabla^i u|^q d\mathbf{V}_i \right)^{\frac{1}{q}}.$$

(ii) The Sobolev space $W_0^{k,q}(\bar{\Omega}; \mathbf{V}_0, \dots, \mathbf{V}_k)$, $k = 1, 2$, is the completion of $C_0^{k,q}(\bar{\Omega})$ with respect the norm

$$\|u\|_{k,q,\Omega,\mathbf{V}_0,\dots,\mathbf{V}_k} := \left( \sum_{i=0}^{k} \int_{\Omega} |\nabla^i u|^q d\mathbf{V}_i \right)^{\frac{1}{q}}.$$

(iii) The Sobolev space $W_0^{k,q}(\Omega; \mathbf{V}_0, \dots, \mathbf{V}_k)$, $k = 1, 2$, is the completion of $C_0^{k,q}(\Omega)$ with respect the norm $\|u\|_{k,q,\Omega,\mathbf{V}_0,\dots,\mathbf{V}_k}$.

**Definition 2.2.** We say that $u \in W^{1,2}(\Omega; \mathbf{V}_0, \mathbf{V}_1)$ is a weak solution of (1.2) if

$$\int_{\Omega} g(\nabla u, \nabla v) d\mathbf{V}_1 + \int_{\Omega} uv d\mathbf{V}_0 = \lambda \int_{\Omega} \tau uv d\mathbf{V}_2,$$

for all $v \in W^{1,2}(\Omega; \mathbf{V}_0, \mathbf{V}_1)$.
2.1. **Weighted Sobolev space for the tangent bundle.** For the following definitions, we will follow [3, 4, 36, 46]. Let $\mathbf{V}$ be a weight function on $M$. Given an open set $U \subset M$ and $q \ge 1$, the Lebesgue space $L^q(TU; \mathbf{V})$ is the Banch space of all (equivalence classes of measurable) vector fields $v: U \to TU$ such that

$$ \|v\|_{q, TU, \mathbf{V}} := \left( \int_U |v|^q d\mathbf{V} \right)^{1/q} < \infty. $$

By $C^k(TM)$, $k \in \mathbb{N} \cup \{\infty\}$, we mean the set of all $C^k$ vector fields. We denote

$$ C^{k,q}(TM; \mathbf{V}_0, \mathbf{V}_1) := \{v \in C^k(TM) \mid \int_M |v|^q d\mathbf{V}_0 + \int_M \|\nabla v\|_g^q d\mathbf{V}_1 < \infty\}, $$

where $\|\nabla v\|_g(p) = \sup\{|\nabla_X v|(p) \mid X \in T_pM, |X| \le 1\}$. Here, we will always assume that the connection $\nabla$ is compatible with $g$.

The weighted Sobolev Space $W^{1,q}(TM; \mathbf{V}_0, \mathbf{V}_1)$ is the completion of $C^{1,q}(TM, \mathbf{V}_0, \mathbf{V}_1)$ with respect to the norm

$$ \|u\|_{1,q,TM,\mathbf{V}_0,\mathbf{V}_1} := \left( \int_M |u|^q d\mathbf{V}_0 + \int_M \|\nabla u\|_g^q d\mathbf{V}_1 \right)^{1/q}. $$

We have the

**Lemma 2.3.** Let $v \in C^1(TM)$. Then, away from the zero set of $v$,

$$ |\nabla|v|| \le \|\nabla v\|_g \quad \text{on } M. \qquad (2.2) $$

*Proof.* From

$$ |\nabla|v|^2|^2 = g(\nabla|v|^2, \nabla|v|^2) = \nabla_{\nabla|v|^2} g(v, v) = 2g(\nabla_{\nabla|v|^2} v, v) \le 2\|\nabla v\|_g \|\nabla|v|^2\| |v|, $$

we can conclude (2.2).

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

Using this result, we have:

**Lemma 2.4.** Let $v \in C^1(T\bar{\Omega})$ be such that $\text{supp } |v| \subset \hat{U}_{1,j}$ for some $j \in \mathbb{N}$, then

$$ \int_{\hat{U}_{1,j} \cap \Gamma} |v|^q d\sigma_g \le q \|\psi_{1,j}\| \|G_{\Gamma,1,j}\| \|G_{1,j}^{-1}\| \left( \int_{\hat{U}_{1,j}} |v|^q dv_g + \int_{\hat{U}_{1,j}} \|\nabla v\|_g^q dv_g \right). $$

*Proof.* Follows from the arguments in some of the references [15, 27]; see also [40, 31].

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

Set

$$ L_{\text{loc}}^q(T\Omega_\Gamma; \mathbb{W}) := \{v : \Gamma \to T\Omega \mid v \text{ is a measurable vector field and} \\ \int_V |v|^q d\mathbb{W} < \infty \text{ for any open bounded set } V \subset \Gamma\}. $$

Recall that $\mathbf{V}_0$, $\mathbf{V}_1$ and $\mathbb{W}$ are bounded from above and from below by positive constants on each compact subset of $M$. By Lemma 2.4, we can define the trace operator:

$$ \mathcal{T} : W^{1,q}(T\Omega; \mathbf{V}_0, \mathbf{V}_1) \to L_{\text{loc}}^q(T\Omega_\Gamma; \mathbb{W}), $$

satisfying $\mathcal{T}v = v$ on $\Gamma$ for any $v \in C^1(T\bar{\Omega})$.

### 3. EMBEDDINGS OF WEIGHTED SOBOLEV SPACES

To prove Propositions 1.1 and 1.2, we will need an extension operator, which is given in Proposition 3.2.

**Lemma 3.1.** Let $(k, i) \in \{0, 1\} \times \mathbb{N}$ and $\delta \ge 1$. There exists a smooth function $\zeta : \hat{U}_{k,i} \to [0, 1]$ such that $\zeta = 1$ in $U_{k,i}$, $\text{supp } \zeta \subset \psi_{k,i}(B(0, r_{k,i} + \delta^{-1}(\hat{r}_{k,i} - r_{k,i})))$, and $|\nabla \zeta|_g \le C(n) \|d(\psi_{k,i}^{-1})\| \delta(\hat{r}_{k,i} - r_{k,i})^{-1}$ in $\hat{U}_{k,i}$.
**Proof.** There exists a smooth function $\zeta_0 : B(0, \hat{r}_{k,i}) \subset \mathbb{R}^n \to [0, 1]$ such that $\zeta_0 = 1$ in $B(0, r_{k,i} + \delta^{-1}(\hat{r}_{k,i} - r_{k,i}))$, $\text{supp } \zeta_0 \subset B(0, \hat{r}_{k,i})$, and $|\nabla \zeta_0|_{\delta_{\mathbb{R}^n}} \le C_1(n)\delta(\hat{r}_{k,i} - r_{k,i})^{-1}$.

Then $\zeta = \zeta_0 \circ \psi_{k,i}^{-1} : \hat{U}_{k,i} \to [0, 1]$ satisfied $\zeta = 1$ in $U_{k,i}$, $\text{supp } \zeta \subset \psi_{k,i}(B(0, r_{k,i} + \delta^{-1}(\hat{r}_{k,i} - r_{k,i}))$, and

$$|\nabla \zeta|_g \le \|d(\psi_{k,i}^{-1})\|_{\nabla \zeta_0|_{\delta_{\mathbb{R}^n}}} \le C_1 \|d(\psi_{k,i}^{-1})\|_{\delta(\hat{r}_{k,i} - r_{k,i})^{-1}} \quad \text{in } \hat{U}_{k,i}.$$

Which prove the lemma.

□

**Proposition 3.2.** Assume $(U_1)-(U_4)$ is verified for $q \ge 1$, and the weight functions $\mathbf{V}_0, \mathbf{V}_1$ satisfy the condition $(\mathbf{W}_1)$. Let $\delta \ge 1$. There exists a bounded linear extension operator

$$\mathcal{E} : W^{1,q}(\Omega; \mathbf{V}_0, \mathbf{V}_1) \to W^{1,q}(M; \mathbf{V}_0, \mathbf{V}_1)$$

satisfying $\mathcal{E}u = u$ a.e. in $\Omega$, $\text{supp } \mathcal{E}u \subset \cup_{(k,i) \in \{0,1\} \times \mathbb{N}} \psi_{k,i}(B(0, r_{k,i} + \delta^{-1}(\hat{r}_{k,i} - r_{k,i})))$, and

$$\|\mathcal{E}u\|_{1,q,M,\mathbf{V}_0,\mathbf{V}_1} \le C(n,q) \left[ (1+\mathbf{R}_2)(1+\delta^q \mathbf{K}_q) \int_{\Omega} |u|^q d\mathbf{V}_0 + (1+\mathbf{R}_2^{1+q}) \int_{\Omega} |\nabla u|^q d\mathbf{V}_1 \right]^{\frac{1}{q}}. \quad (3.1)$$

**Proof.** Using Lemma 3.1, there exist a partition of unity $\{\zeta_{k,i} : M \to [0,1]\}_{(k,i) \in \{0,1\} \times \mathbb{N}} \subset C^\infty(M)$ subordinate to $\{\hat{U}_{k,i}\}_{(k,i) \in \{0,1\} \times \mathbb{N}}$ such that $\text{supp } \zeta_{k,i} \subset \psi_{k,i}(B(0, r_{k,i} + \delta^{-1}(\hat{r}_{k,i} - r_{k,i})))$, $\sum_{k,i} \zeta_{k,i} = 1$ on $\Omega$, and $|\nabla \zeta_{k,i}|_g \le C_1(n)\delta \mathbf{R}_{k,i}$ in $\hat{U}_{k,i}$.

Define $f : \mathbb{R}^n \to \mathbb{R}^n$ by $f(x) = (x_1, x_2, \dots, |x_n|)$. Let $u \in W^{1,q}(\Omega; \mathbf{V}_0, \mathbf{V}_1)$, for $(k,i) \in \{0,1\} \times \mathbb{N}$ we define the functions $\bar{u}_{k,i}$ by

$$\bar{u}_{0,i} := u\zeta_{0,i} \quad \text{in } \hat{U}_{0,i} \\ \bar{u}_{1,j}(p) := (u \circ \psi_{1,j} \circ f \circ \psi_{1,j}^{-1})(p)\zeta_{1,j}(p) \quad \text{if } p \in \hat{U}_{1,j}.$$

By $(U_4)$, we have

$$\int_M |\bar{u}_{1,j}|^q d\mathbf{V}_0 = \int_{\psi_{1,j}(B(0, \hat{r}_{1,j}) \cap \{x_n > 0\})} |\bar{u}_{1,j}|^q d\mathbf{V}_0 + \int_{\psi_{1,j}(B(0, \hat{r}_{1,j}) \cap \{x_n < 0\})} |\bar{u}_{1,j}|^q d\mathbf{V}_0 \\ \le (1 + \mathbf{R}_2) \int_{\hat{U}_{1,j}} |u|^q d\mathbf{V}_0. \quad (3.2)$$

Furthermore, from the inequality $|a+b| \le 2^{q-1}(|a|^q + |b|^q)$, $(U_4)$, and $(\mathbf{W}_1)$,

$$\int_M |\nabla \bar{u}_{1,j}|^q d\mathbf{V}_1 = \int_{\psi_{1,j}(B(0, \hat{r}_{1,j}) \cap \{x_n > 0\})} |\nabla \bar{u}_{1,j}|^q d\mathbf{V}_1 + \int_{\psi_{1,j}(B(0, \hat{r}_{1,j}) \cap \{x_n < 0\})} |\nabla \bar{u}_{1,j}|^q d\mathbf{V}_1 \\ \le \int_{\psi_{1,j}(B(0, \hat{r}_{1,j}) \cap \{x_n > 0\})} 2^{q-1} (|\zeta_{1,j} \nabla u|^q + |u \nabla \zeta_{1,j}|^q) d\mathbf{V}_1 \\ + \int_{\psi_{1,j}(B(0, \hat{r}_{1,j}) \cap \{x_n > 0\})} 2^{q-1} \mathbf{R}_2 (|\mathbf{R}_2 \zeta_{1,j} \nabla u|^q + |u \nabla \zeta_{1,j}|^q) d\mathbf{V}_1 \\ \le C_2(n,q) \left[ \delta^q \mathbf{K}_q (1 + \mathbf{R}_2) \int_{\hat{U}_{1,j}} |u|^q d\mathbf{V}_0 + (1 + \mathbf{R}_2^{1+q}) \int_{\hat{U}_{1,j}} |\nabla u|^q d\mathbf{V}_1 \right]. \quad (3.3)$$

Hence, the operator $\mathcal{E} : W^{1,q}(\Omega; \mathbf{V}_0, \mathbf{V}_1) \to W^{1,q}(M; \mathbf{V}_0, \mathbf{V}_1)$, defined by

$$\mathcal{E}u = \sum_{(k,i) \in \{0,1\} \times \mathbb{N}} \bar{u}_{k,i},$$

is linear. Also, by $(U_3)$, (3.2), and (3.3), we obtain the estimate (3.1).

□

Now we will define the extension operator for weighted Sobolev spaces for the tangent bundle.

**Corollary 3.3.** Assume $(U_1)-(U_4)$ is verified for $q \ge 1$, and the weight functions $\mathbf{V}_0, \mathbf{V}_1$ satisfy the condition $(\mathbf{W}_1)$. Let $\delta \ge 1$. There exists a bounded linear extension operator

$$\mathcal{E}_T : W^{1,q}(T\Omega; \mathbf{V}_0, \mathbf{V}_1) \to W^{1,q}(TM; \mathbf{V}_0, \mathbf{V}_1)$$
satisfying $\mathcal{E}_T u = u$ a.e. in $\Omega$, supp $\mathcal{E}_T u \subset \cup_{(k,i) \in \{0,1\} \times \mathbb{N}} \psi_{k,i}(B(0, r_{k,i} + \delta^{-1}(\hat{r}_{k,i} - r_{k,i})))$, and

$$ \|\mathcal{E}_T v\|_{1,q,TM,\nu_0,\nu_1} \le c(n,q) \left[ (1+R_2)(1+\delta^q K_q) \int_{\Omega} |v|^q d\nu_0 + (1+R_2^{1+q})(1+R_2^{1-q}) \int_{\Omega} \|\nabla v\|_q^q d\nu_1 \right]^{\frac{1}{q}}. $$

*Proof.* The proof follows the same argument as the proof of Proposition 3.2, while also considering

$$ v = a^j \partial_j \quad \text{and} \quad \nabla_X v = X(a^j) \partial_j + a^j \nabla_X \partial_j \quad \text{in } \hat{U}_{k,i}, $$

where $a^j \in W^{1,q}(\hat{U}_{k,i}; \nu_0, \nu_1)$, $j = 1, \dots, n$, and $X$ is a vector field. $\square$

To obtain the extension operator under other hypotheses, we have the following remarks:

**Remark 3.4.** See [48]. Let $\mathcal{M}_3$ be a Riemannian manifold of dimension $n \ge 2$, $r > 0$ and $H, K \ge 0$. An open subset $U \subset \mathcal{M}_3$ is called $(r, H, K)$-regular if

(a) $\bar{U} \neq \mathcal{M}_3$ is a connected smooth manifold with (smooth) boundary $\partial U$.

(b) For any $q_1 \in \partial U$, there is a point $q \in \mathcal{M}_3 \setminus U$ such that $B(q, r) \subset \mathcal{M}_3 \setminus U$ and $\overline{B(q, r)} \cap \partial U = \{q_1\}$.

(c) For any $q_1 \in \partial U$, there is a point $q \in U$ such that $B(q, r) \subset U$ and $\overline{B(q, r)} \cap \partial U = \{q_1\}$;

(d) The second fundamental form $II$ with respect to the inward pointing normal of $\partial U$ satisfies $-H \le II \le H$.

(e) the sectional curvature satisfies $\text{Sec} \le K$ on the tubular neighborhood $T(\partial U, r)$ of $\partial U$.

Fix $K, H \ge 0$ and a complete Riemannian manifold $\mathcal{M}_3 n$ with $n \ge 2$. There exists $r_0 = r_0(K, H) > 0$ such that for any $r \in (0, r_0]$, there exists $C(r, K, H) > 0$ and an extension operator $E_U : H^1(U) \to H^1(\mathcal{M}_3)$ satisfying

$$ \|E_U\| \le C(K, H, r) $$

for any open $(r, H, K)$-regular subset $U$.

**Remark 3.5.** See [18]. An open set $\Omega_0 \subset \mathbb{R}^n$, $n \ge 2$, is an $(\epsilon, \delta)$ domain if for all $x, y \in \Omega_0$ with $|x - y| < \delta$, there exists a rectifiable curve $\gamma$ connecting $x$ and $y$ such that $\gamma$ lies in $\Omega_0$ and

$$ \ell(\gamma) < \frac{|x - y|}{\epsilon}, \quad d(z) > \frac{\epsilon|x - z||y - z|}{|x - y|} \quad \forall z \in \gamma. $$

Here, $\ell(\gamma)$ is the length of $\gamma$, and $d(z)$ is the distance between $z$ and the boundary of $\Omega_0$. Let us decompose $\Omega_0 = \cup \Omega_{0,\alpha}$ into connected components and define

$$ \mathrm{rad}(\Omega_0) := \inf_{\alpha} \inf_{x \in \Omega_{0,\alpha}} \sup_{y \in \Omega_{0,\alpha}} |x - y|. $$

Let $\Omega_0$ be an $(\epsilon, \delta)$ domain with $\mathrm{rad}(\Omega_0) > 0$. If $1 \le q < \infty$ and $w \in A_q$ (the class of Muckenhoupt weights), then there exists an extension operator $E$ on $\Omega_0$ such that

$$ \|Eu\|_{W^{1,q}(\mathbb{R}^n; w)} \le C \|u\|_{W^{1,q}(\Omega_0; w)} \quad \forall u \in W^{1,q}(\mathbb{R}^n; w), $$

where $\|u\|_{W^{1,q}(\Omega_0; w)} := (\int_{\Omega_0} |u|^q w dx + \int_{\Omega_0} |\nabla u|^q w dx)^{1/q}$ and $C = C(\epsilon, \delta, k, w, p, n, \mathrm{rad}(\Omega_0)) > 0$. Furthermore, $\|E\| \to \infty$ as either $\mathrm{rad}(\Omega_0) \to 0$, $\epsilon \to 0$, or $\delta \to 0$.

**3.1. Compact embedding.** The proof of our embedding theorems is based on the following lemma.

**Lemma 3.6.**

(i) Assume that

$$ W^{1,q}(\Omega_m; \nu_0, \nu_1) \to L^{q_0}(\Omega_m; \nu_2) \text{ is compact for every } m, \quad (3.4) $$

$$ \lim_{m \to \infty} \sup_{\|u\|_{1,q,\Omega,\nu_0,\nu_1} \le 1} \|u\|_{q_0,\Omega^m,\nu_2} = 0. \quad (3.5) $$

Then $W^{1,q}(\Omega; \nu_0, \nu_1) \to L^{q_0}(\Omega; \nu_2)$ is compact. On the other hand, if $W^{1,q}(\Omega; \nu_0, \nu_1) \to L^{q_0}(\Omega; \nu_2)$ is compact, then (3.5) holds.
(ii) If

$$W^{1,q}(\Omega_m; \mathbf{V}_0, \mathbf{V}_1) \to L^{q_0}(\Omega_m; \mathbf{V}_2) \quad \text{is continuous for every } m, \qquad (3.6)$$

$$\lim_{m \to \infty} \sup_{\|u\|_{1,q,\Omega,\mathbf{V}_0,\mathbf{V}_1} \le 1} \|u\|_{q_0,\Omega^m,\mathbf{V}_2} < \infty. \qquad (3.7)$$

Then $W^{1,q}(\Omega; \mathbf{V}_0, \mathbf{V}_1) \to L^{q_0}(\Omega; \mathbf{V}_2)$ is continuous. If $W^{1,q}(\Omega; \mathbf{V}_0, \mathbf{V}_1) \to L^{q_0}(\Omega; \mathbf{V}_2)$ is continuous, then (3.7) holds.

*Proof.* (i) Notice that (3.5) is equivalent to the statement that to every $\epsilon > 0$ there exists an $m_0$, such that

$$\|u\|_{q_0,\Omega,\mathbf{V}_2}^{q_0} \le \epsilon \|u\|_{1,q,\Omega,\mathbf{V}_0,\mathbf{V}_1}^{q_0} + \|u\|_{q_0,\Omega_{m_0},\mathbf{V}_2}^{q_0} \quad \forall u \in W^{1,q}(\Omega; \mathbf{V}_0, \mathbf{V}_1). \qquad (3.8)$$

Now let $u_j$ be a bounded sequence in $W^{1,q}(\Omega; \mathbf{V}_0, \mathbf{V}_1)$, $\|u_j\|_{1,q,\Omega,\mathbf{V}_0,\mathbf{V}_1} \le C_1$ and let $\epsilon > 0$ be given. From (3.8) and (3.4) it follows that there is a subsequence which is a Cauchy sequence in $L^{q_0}(\Omega_{m_0}; \mathbf{V}_2)$. Therefore, we can find $m_\epsilon$ such that for every $i, j \ge m_\epsilon$

$$\|u_j - u_i\|_{q_0,\Omega,\mathbf{V}_2}^{q_0} \le \epsilon \|u_j - u_i\|_{1,q,\Omega,\mathbf{V}_0,\mathbf{V}_1}^{q_0} + \|u_j - u_i\|_{q_0,\Omega_{m_0},\mathbf{V}_2}^{q_0} \\ \le (2^{q_0} C_1^{q_0} + 1)\epsilon.$$

Consequently, $u_j$ contains a Cauchy sequence in $L^{q_0}(\Omega; \mathbf{V}_2)$. Hence $W^{1,q}(\Omega; \mathbf{V}_0, \mathbf{V}_1) \to L^{q_0}(\Omega; \mathbf{V}_2)$ is compact.

Now, let $W^{1,q}(\Omega; \mathbf{V}_0, \mathbf{V}_1) \to L^{q_0}(\Omega; \mathbf{V}_2)$ be compact and assume that (3.8) does not hold. Then there exist an $\epsilon > 0$ and a sequence $(u_j)$ in $W^{1,q}(\Omega; \mathbf{V}_0, \mathbf{V}_1)$, such that

$$\|u_j\|_{q_0,\Omega,\mathbf{V}_2}^{q_0} > \epsilon \|u_j\|_{1,q,\Omega,\mathbf{V}_0,\mathbf{V}_1}^{q_0} + \|u_j\|_{q_0,\Omega_j,\mathbf{V}_2}^{q_0}.$$

Write $\tilde{u}_j = u_j/\|u_j\|_{1,q,\Omega,\mathbf{V}_0,\mathbf{V}_1}^{q_0}$, hence

$$\|\tilde{u}_j\|_{q_0,\Omega,\mathbf{V}_2}^{q_0} > \epsilon + \|\tilde{u}_j\|_{q_0,\Omega_j,\mathbf{V}_2}^{q_0}. \qquad (3.9)$$

Since $\tilde{u}_j$ is bounded in $W^{1,q}(\Omega; \mathbf{V}_0, \mathbf{V}_1)$, there is a subsequence converging to $\tilde{u}$ in $L^{q_0}(\Omega; \mathbf{V}_2)$. Now taking the limit in (3.9), we get

$$\|\tilde{u}\|_{q_0,\Omega,\mathbf{V}_2}^{q_0} \ge \epsilon + \|\tilde{u}\|_{q_0,\Omega,\mathbf{V}_2}^{q_0},$$

which is a contradiction.

(ii) The second part of Lemma 3.6 can be proved in a similar way to (i). □

**Proof of Proposition 1.1.** We have

$$\|v\|_{q_0,B(0,1)} \le C_1 \|v\|_{1,q,B(0,1)}, \quad \forall v \in W^{1,q}(B(0,1)),$$

where $B(0,1) \subset \mathbb{R}^n$ and $C_1 = C_1(q, q_0, n, B(0,1)) > 0$.

Let $u \in W^{1,q}(\Omega; \mathbf{V}_0, \mathbf{V}_1)$. By Proposition 3.2,

$$\|\bar{u}\|_{1,q,M,\mathbf{V}_0,\mathbf{V}_1} \le C_2 \|u\|_{1,q,\Omega,\mathbf{V}_0,\mathbf{V}_1}, \qquad (3.10)$$

where $\bar{u} = \mathcal{E}u \in W^{1,q}(M; \mathbf{V}_0, \mathbf{V}_1)$ and $C_2 = C_2(n, q, \mathbf{R}_2, \mathbf{K}_q) > 0$.

Let $\psi_{k,i}(0) \in D^m, m \ge m_*$.

$$\left( \int_{\hat{U}_{k,i}} |\bar{u}|^{q_0} dv_g \right)^{\frac{1}{q_0}} = \left( \int_{B(0,1)} |\bar{u} \circ \psi_{k,i}|^{q_0} \sqrt{\det[g_{ab}]} dx \right)^{\frac{1}{q_0}} \\ \le C_1 \|G_{k,i}\|^{\frac{1}{q_0}} \hat{r}_{k,i}^{\frac{n}{q_0}} \left( \int_{B(0,1)} |\bar{u} \circ \psi_{k,i}(\hat{r}_{k,i}y)|^q dy \right)^{\frac{1}{q}} \\ \le C_1 \|G_{k,i}\|^{\frac{1}{q_0}} \|G_{k,i}^{-1}\|^{\frac{n}{q}} \hat{r}_{k,i}^{\frac{n}{q_0}} \left( \hat{r}_{k,i}^{-n} \int_{\hat{U}_{k,i}} |\bar{u}|^q dv_g + \hat{r}_{k,i}^{-n+q} \|d\psi_{k,i}\| \int_{\hat{U}_{k,i}} |\nabla \bar{u}|^q dv_g \right)^{\frac{1}{q}}.$$
From $(W_2)$ and $(W_1)$, we get

$$
\begin{aligned}
\int_{\hat{U}_{k,i}} |\bar{u}|^{q_0} d\mathbf{V}_2 &\leq \left[ b_2^{\frac{1}{q_0}}(\psi_{k,i}(0)) C_1 \|G_{k,i}\|^{\frac{1}{q_0}} \|G_{k,i}^{-1}\|^{\frac{1}{q}} \hat{r}_{k,i}^{\frac{n}{q}-\frac{n}{q}+1} \right]^{q_0} \\
&\quad \cdot \left( \hat{r}_{k,i}^{-q} \int_{\hat{U}_{k,i}} |\bar{u}|^q d\mathbf{V}_g + \|d\psi_{k,i}\| \int_{\hat{U}_{k,i}} |\nabla \bar{u}|^q d\mathbf{V}_g \right)^{\frac{q_0}{q}} \\
&\leq C_1^{q_0} \left[ \frac{b_2^{1/q_0}(\psi_{k,i}(0))}{b_1^{1/q}(\psi_{k,i}(0))} \|G_{k,i}\|^{\frac{1}{q_0}} \|G_{k,i}^{-1}\|^{\frac{1}{q}} \hat{r}_{k,i}^{\frac{n}{q_0}-\frac{n}{q}+1} \right]^{q_0} \\
&\quad \cdot \left( \hat{r}_{k,i}^{-q} \|d\psi_{k,i}\|_{B(0,\hat{r}_{k,i})}^{-1} \int_{\hat{U}_{k,i}} |\bar{u}|^q d\mathbf{V}_1 + \int_{\hat{U}_{k,i}} |\nabla \bar{u}|^q d\mathbf{V}_1 \right)^{\frac{q_0}{q}} \\
&\leq C_1^{q_0} (B_{q,q_0}^m)^{q_0} \left( K_q \int_{\hat{U}_{k,i}} |\bar{u}|^q d\mathbf{V}_0 + \int_{\hat{U}_{k,i}} |\nabla \bar{u}|^q d\mathbf{V}_1 \right)^{\frac{q_0}{q}}.
\end{aligned}
$$

From $(U_3)$,

$$
\begin{aligned}
\|\bar{u}\|_{q_0, D^m, \mathbf{V}_2}^{q_0} &\le C_1^{q_0} (B_{q,q_0}^m)^{q_0} \left( \sum_{k,i} K_q \int_{\hat{U}_{k,i}} |\bar{u}|^q d\mathbf{V}_0 + \int_{\hat{U}_{k,i}} |\nabla \bar{u}|^q d\mathbf{V}_1 \right)^{\frac{q_0}{q}} \\
&\le C_1^{q_0} (B_{q,q_0}^m)^{q_0} [\mathbf{R}_1(\mathbf{K}_q + 1)]^{\frac{q_0}{q}} \|\bar{u}\|_{1,q,M,\mathbf{V}_0,\mathbf{V}_1}^{q_0}.
\end{aligned}
$$

By (3.10)

$$
\|u\|_{q_0, \Omega^m, \mathbf{V}_2} \le C_2 C_1 \mathcal{B}_{q,q_0}^m [\mathbf{R}_1(\mathbf{K}_q + 1)]^{\frac{1}{q}} \|u\|_{1,q, \Omega, \mathbf{V}_0, \mathbf{V}_1}. \quad (3.11)
$$

Recall that $V_i$ and $W$ are bounded from above and from below by positive constants on each compact subset of $M$. Therefore the proof of the proposition follow from Lemma 3.6 and (3.11). $\square$

Using Lemma 2.3 and the argument from the proof of Proposition 1.1 (as in the proof of [46, Proposition 3.6]), we have the

**Corollary 3.7.** Assume that $(U_1) - (U_4)$, $(W_1)$ and $(W_2)$ are verified. Suppose $1 \le q < n$ and $nq/(n-q) \ge q_0 \ge q$.

(i) If $\lim_{m \to \infty} B_{q,q_0}^m < \infty$, then $W^{1,q}(T\Omega; V_0, V_1) \to L^{q_0}(T\Omega; V_2)$ is continuous.

(ii) If $\lim_{m \to \infty} B_{q,q_0}^m = 0$, then $W^{1,q}(T\Omega; V_0, V_1) \to L^{q_0}(T\Omega; V_2)$ is compact.

**3.2. Compact traces.** Proceeding similarly to Lemma 3.6, we have the

**Lemma 3.8.**

(i) Assume that

$$
W^{1,q}(\Omega_m; \mathbf{V}_0, \mathbf{V}_1) \to L^{q_1}(\Gamma_m; W) \text{ is compact for every } m, \quad (3.12)
$$

$$
\lim_{m \to \infty} \sup_{\|u\|_{1,q,\Omega,\mathbf{V}_0,\mathbf{V}_1} \le 1} \|u\|_{q_1, \Gamma^m, W} = 0. \quad (3.13)
$$

Then $W^{1,q}(\Omega; V_0, V_1) \to L^{q_1}(\Gamma; W)$ is compact. On the other hand, if the trace operator is compact, then (3.13) holds.

(ii) If

$$
W^{1,q}(\Omega_m; \mathbf{V}_0, \mathbf{V}_1) \to L^{q_1}(\Gamma_m; W) \quad \text{is continuous for every } m, \quad (3.14)
$$

$$
\lim_{m \to \infty} \sup_{\|u\|_{1,q,\Omega,\mathbf{V}_0,\mathbf{V}_1} \le 1} \|u\|_{q_1,\Gamma^m,W} < \infty. \quad (3.15)
$$

Then $W^{1,q}(\Omega; V_0, V_1) \to L^{q_1}(\Gamma; W)$ is continuous. If the trace operator is continuous, then (3.15) holds.
**Proof of Proposition 1.2.** We have

$$ \|v\|_{q_1, B(0,1) \cap \{x_n=0\}} \le c_1 \|v\|_{1,q, B(0,1)}, \quad \forall v \in W^{1,q}(B(0,1)), $$

where $B(0,1) \subset \mathbb{R}^n$ and $c_1 = c_1(q, q_1, n, B(0,1)) > 0$.

Let $u \in W^{1,q}(\Omega; \mathbf{V}_0, \mathbf{V}_1)$. By Proposition 3.2,

$$ \|\bar{u}\|_{1,q,M,\mathbf{V}_0,\mathbf{V}_1} \le c_2 \|u\|_{1,q,\Omega,\mathbf{V}_0,\mathbf{V}_1}, \quad (3.16) $$

where $\bar{u} = \mathcal{E}u \in W^{1,q}(M; \mathbf{V}_0, \mathbf{V}_1)$ and $c_2 = c_2(n, q, \mathbf{R}_2, \mathbf{K}_q) > 0$.

Let $\psi_{1,j}(0) \in \Gamma^m, m \ge m_*$. Then,

$$ \begin{aligned} \left( \int_{\hat{U}_{1,j} \cap \Gamma} |u|^{q_1} d\sigma_g \right)^{\frac{1}{q_1}} &= \left( \int_{B(0, \hat{r}_{1,j}) \cap \{x_n=0\}} |u \circ \psi_{1,j}|^{q_1} \sqrt{\det[g_{\Gamma,ab}]} d\sigma \right)^{\frac{1}{q_1}} \\ &\le c_1 \|G_{\Gamma,1,j}\|^{\frac{1}{q_1}} \hat{r}_{1,j}^{\frac{n-1}{q_1}} \left( \int_{B(0,1)} |\bar{u} \circ \psi_{1,j}(\hat{r}_{1,j}y)|^q dy \right)^{\frac{1}{q}} \\ &\le c_1 \|G_{\Gamma,1,j}\|^{\frac{1}{q_1}} \|G_{1,j}^{-1}\|^{\frac{1}{q}} \hat{r}_{1,j}^{\frac{n-1+n}{q}} \left( \hat{r}_{1,j}^{-n} \int_{\hat{U}_{1,j}} |\bar{u}|^q dv_g + \hat{r}_{1,j}^{-n+q} \|d\psi_{1,j}\| \int_{\hat{U}_{1,j}} |\nabla \bar{u}|^q dv_g \right)^{\frac{1}{q}}, \end{aligned} $$

Proceeding similarly the proof of Proposition 1.1, from ($W_2$) and ($W_3$), we get

$$ \int_{\Gamma} |u|^{q_1} d\mathbf{W} \le c_1^{q_1} (\mathcal{B}_{\Gamma, q, q_0}^m)^{q_1} \left( K_q \int_{\hat{U}_{1,j}} |\bar{u}|^q d\mathbf{V}_0 + \int_{\hat{U}_{1,j}} |\nabla \bar{u}|^q d\mathbf{V}_1 \right)^{\frac{q_1}{q}}. $$

From ($U_3$),

$$ \begin{aligned} \|u\|_{q_1, \Gamma^m, \mathbf{W}}^{q_1} &\le c_1^{q_1} (\mathcal{B}_{\Gamma, q, q_0}^m)^{q_1} \left( \sum_j K_q \int_{\hat{U}_{1,j}} |\bar{u}|^q d\mathbf{V}_0 + \int_{\hat{U}_{1,j}} |\nabla \bar{u}|^q d\mathbf{V}_1 \right)^{\frac{q_1}{q}} \\ &\le c_1^{q_1} (\mathcal{B}_{\Gamma, q, q_0}^m)^{q_1} [\mathbf{R}_1(\mathbf{K}_q + 1)]^{\frac{q_1}{q}} \|\bar{u}\|_{1,q,M,\mathbf{V}_0,\mathbf{V}_1}^{q_1}. \end{aligned} $$

By (3.10)

$$ \|u\|_{q_1, \Gamma^m, \mathbf{W}} \le c_2 c_1 \mathcal{B}_{\Gamma, q, q_0}^m [\mathbf{R}_1(\mathbf{K}_q + 1)]^{\frac{1}{q}} \|u\|_{1,q, \Omega, \mathbf{V}_0, \mathbf{V}_1}. \quad (3.17) $$

Recall that $\mathbf{V}_i$ and $\mathbf{W}$ are bounded from above and from below by positive constants on each compact subset of $M$. Therefore the proof of the proposition follow from Lemma 3.8 and (3.17). $\square$

Following the proof of Proposition 1.2 and using Lemma 3.2 (as in the proof of [46, Proposition 3.6]), we have the

**Corollary 3.9.** Assume that $(U_1) - (U_4)$, $(\mathbf{W}_1)$ and $(\mathbf{W}_3)$ are verified. Suppose $1 \le q < n$ and $(n-1)q/(n-q) \ge q_1 \ge q$.

(i) If $\lim_{m \to \infty} \mathcal{B}_{\Gamma, q, q_0}^m < \infty$, then there exists a continuous trace operator $W^{1,q}(T\Omega; \mathbf{V}_0, \mathbf{V}_1) \to L^{q_1}(T\Omega_{\Gamma}; \mathbf{W})$.

(ii) If $\lim_{m \to \infty} \mathcal{B}_{\Gamma, q, q_0}^m = 0$, then the trace operator $W^{1,q}(T\Omega; \mathbf{V}_0, \mathbf{V}_1) \to L^{q_1}(T\Omega_{\Gamma}; \mathbf{W})$ is compact.

## 4. PRINCIPAL EIGENVALUES FOR INDEFINITE WEIGHT

Before proving Theorem 1.5, we will prove the preliminary results Proposition 4.3 - 4.5.

### 4.1. Elliptic estimates. Similarly to Lemma 3.1 we have the

**Lemma 4.1.** Suppose $0 < r < R \le \hat{r}_{1,j}$ and $j \in \mathbb{N}$ fixed. There exist a smooth function $\zeta : \hat{U}_{1,j} \to [0, 1]$ such that $\zeta = 1$ in $\psi_{1,j}(B(0,r))$, $\text{supp} \zeta \subset \psi_{1,j}(B(0,R))$, and $|\nabla \zeta|_g \le c_1(n) \|d(\psi_{1,j}^{-1})\|(R-r)^{-1}$ in $\hat{U}_{1,j}$.
For $h \in \mathbb{R}$ and $t \in (0, \hat{r}_{1,j})$. We define,

$$
\begin{align*}
\mathcal{Q}[u, v] &:= \int_{\Omega} g(\nabla u, \nabla v) \mathrm{d}V_1 + \int_{\Omega} uv \mathrm{d}V_0, \\
\Psi(h, t) &:= \sqrt{\int_{\mathcal{U}(h,t)} u_h^2 \mathrm{d}V_2 + \int_{\mathcal{U}_{\Gamma}(h,t)} u_h^2 \mathrm{d}W},
\end{align*}
$$

where $u, v \in W^{1,2}(\Omega; V_0, V_1)$,

$$
\begin{align*}
u_h &:= (u - h)^+, \\
\mathcal{U}(h,t) &:= \{q \in \psi_{1,j}(B(0,t) \cap \{x_n > 0\}) \mid u(q) > h\}, \\
\mathcal{U}_{\Gamma}(h,t) &:= \{q \in \psi_{1,j}(B(0,t) \cap \{x_n = 0\}) \mid u(q) > h\}.
\end{align*}
$$

The proof of the next lemma will be given in the appendix.

**Lemma 4.2.** Suppose $\mathcal{R}_{1,j}^2(p)V_1(p) \le K_2V_2(p)$ for a.e. $p \in \hat{U}_{1,j}$. Assume that $c_2, f \in L^{q_2}(\hat{U}_{1,j} \cap \Omega; V_2)$, and $c_3, f_1 \in L^{q_3}(\hat{U}_{1,j} \cap \Gamma; W)$ for some $q_2, q_3 > \max\{2_v/(2_v-2), 2_w/(2_w-2)\}$. Suppose

$$
\mathcal{Q}[u, v_h] + \int_{\hat{U}_{1,j} \cap \Omega} c_2 u v_h \, dV_2 + \int_{\hat{U}_{1,j} \cap \Gamma} c_3 u v_h \, dW \le \int_{\hat{U}_{1,j} \cap \Omega} f v_h \, dV_2 + \int_{\hat{U}_{1,j} \cap \Gamma} f_1 v_h \, dW, \quad (4.1)
$$

for all $h \ge 0$, where $v_h = u_h \zeta^2$ and $\zeta$ is given in Lemma 4.1.

There exist $\epsilon = \epsilon(q_2, q_3, 2v, 2w) > 0$ and $c_2 = c_2(n, C_{2v}, C_{2w}, 2v, 2w, q_2, q_3) > 0$ such that if

$$
h_2 > h_1 \\
\ge c_2 \max \left\{ \|c_2\|_{q_2, \hat{U}_{1,j} \cap \Omega, v_2}^{q_2 2v / [q_2(2v-2)-2v]}, \|c_3\|_{q_3, \Gamma \cap \hat{U}_{1,j}, w}^{q_3 2w / [q_3(2w-2)-2w]} \right\}, \quad 1 \max\{\|u^+\|_{2, \hat{U}_{1,j} \cap \Omega, v_2}, \|u^+\|_{2, \hat{U}_{1,j} \cap \Gamma, w}\},
$$

then

$$
\begin{align}
& \Psi(h_2, r) \nonumber \\
& \le c_3 \left[ \frac{\hat{r}_{1,j} - r_{1,j}}{(R-r)(h_2-h_1)^\epsilon} + \frac{\|f\|_{q_2, \hat{U}_{1,j} \cap \Omega, v_2} + \|f_1\|_{q_3, \hat{U}_{1,j} \cap \Gamma, w} + h_2}{(h_2-h_1)^{1+\epsilon}} \right] \Psi^{1+\epsilon}(h_1, R), \tag{4.2}
\end{align}
$$

where $c_3 = c_3(n, C_{2v}, C_{2w}, 2v, 2w, q_2, q_3, K_2) > 0$.

**Proposition 4.3.** Suppose $\mathcal{R}_{1,j}^2(p)V_1(p) \le K_2V_2(p)$ for a.e. $p \in \hat{U}_{1,j}$. Assume that $c_2, f \in L^{q_2}(\hat{U}_{1,j} \cap \Omega; V_2)$, and $c_3, f_1 \in L^{q_3}(\hat{U}_{1,j} \cap \Gamma; W)$ for some $q_2, q_3 > \max\{2_v/(2_v-2), 2_w/(2_w-2)\}$. Suppose

$$
\mathcal{Q}[u, v_h] + \int_{\Omega} c_2 u v_h \, dV_2 + \int_{\Gamma} c_3 u v_h \, dW \le \int_{\Omega} f v_h \, dV_2 + \int_{\Gamma} f_1 v_h \, dW,
$$

for all $h \ge 0$, $v_h = u_h \zeta^2$ and $\zeta \in C_0^\infty(\hat{U}_{1,j})$. Then

$$
\begin{align*}
\text{ess sup}_{U_{1,j} \cap \Omega} u^+ + \text{ess sup}_{U_{1,j} \cap \Gamma} u^+ &\le C \left( 1 + \max \left\{ \|c_2\|_{q_2, \hat{U}_{1,j} \cap \Omega, v_2}^{q_2 2v / [q_2(2v-2)-2v]}, \|c_3\|_{q_3, \hat{U}_{1,j} \cap \Gamma, w}^{q_3 2w / [q_3(2w-2)-2w]} \right\} \right) \\
& \quad \cdot \left( \|u^+\|_{q_2, \hat{U}_{1,j} \cap \Omega, v_2} + \|u^+\|_{q_2, \hat{U}_{1,j} \cap \Gamma, w} \right) \\
& \quad + C \left( \|f\|_{q_2, \hat{U}_{1,j} \cap \Omega, v_2} + \|f_1\|_{q_3, \hat{U}_{1,j} \cap \Gamma, w} \right),
\end{align*}
$$

where $C = C(n, C_{2v}, C_{2w}, 2v, 2w, q_2, q_3, K_2) > 0$.

*Proof.* We carry out the iteration. Define for $i \in \mathbb{N} \cup \{0\}$,

$$
h_i := h_0 + h \left(1 - \frac{1}{2^i}\right) \le h_0 + h \quad \text{and} \quad R_i := r_{1,j} + \frac{1}{2^i}(\hat{r}_{1,j} - r_{1,j}).
$$

where

$$
h_0 = c_2 \max \left\{ \|c_2\|_{q_2, \hat{U}_{1,j} \cap \Omega, v_2}^{q_2 2v / [q_2(2v-2)-2v]}, \|c_3\|_{q_3, \Gamma \cap \hat{U}_{1,j}, w}^{q_3 2w / [q_3(2w-2)-2w]} \right\}, \quad 1 \max\{\|u^+\|_{2, \hat{U}_{1,j} \cap \Omega, v_2}, \|u^+\|_{2, \hat{U}_{1,j} \cap \Gamma, w}\}
$$

$c_2$ is given Lemma 4.2, and $h > 0$ will be determined later.
We have

$$h_i - h_{i-1} = \frac{h}{2^i} \quad \text{and} \quad R_{i-1} - R_i = \frac{\hat{r}_{1,j} - r_{1,j}}{2^i}.$$

Hence, from (4.2),

$$ \begin{aligned} \Psi(h_i, R_i) & \le \mathcal{C}_3 \left[ 2^i + \frac{2^i (\|f\|_{q_2, \hat{U}_{1,j} \cap \Omega, v_2} + \|f_1\|_{q_3, \hat{U}_{1,j} \cap \Gamma, w} + h_0 + h)}{h} \right] \frac{2^{\epsilon i}}{h^{\epsilon}} \Psi^{1+\epsilon}(h_{i-1}, R_{i-1}) \\ & \le 2\mathcal{C}_3 (\|f\|_{q_2, \hat{U}_{1,j} \cap \Omega, v_2} + \|f_1\|_{q_3, \hat{U}_{1,j} \cap \Gamma, w} + h_0 + h) \frac{2^{(1+\epsilon)i}}{h^{1+\epsilon}} \Psi^{1+\epsilon}(h_{i-1}, R_{i-1}), \end{aligned} \quad (4.3) $$

Next we prove inductively for any $i \in \mathbb{N} \cup \{0\}$,

$$ \Psi(h_i, R_i) \le \frac{\Psi(h_0, R_0)}{\gamma^i} \quad \text{for some } \gamma > 1, \quad (4.4) $$

if $h$ is sufficiently large. It is true for $i = 0$. Suppose it is true for $i - 1$. We have

$$ \Psi^{1+\epsilon}(h_{i-1}, R_{i-1}) \le \left( \frac{\Psi(h_0, R_0)}{\gamma^{i-1}} \right)^{1+\epsilon} = \frac{\Psi^{\epsilon}(h_0, R_0)}{\gamma^{i\epsilon-(1+\epsilon)}} \frac{\Psi(h_0, R_0)}{\gamma^i}. \quad (4.5) $$

Then, by (4.3) and (4.5), we obtain

$$ \Psi(h_i, R_i) \le 2\mathcal{C}_3 \gamma^{1+\epsilon} \frac{\|f\|_{q_2, \hat{U}_{1,j} \cap \Omega, v_2} + \|f_1\|_{q_3, \hat{U}_{1,j} \cap \Gamma, w} + h_0 + h}{h^{1+\epsilon}} \Psi^{\epsilon}(h_0, R_0) \frac{2^{i(1+\epsilon)}}{\gamma^{i\epsilon}} \frac{\Psi(h_0, R_0)}{\gamma^i}. $$

Choose $\gamma$ first such that $\gamma^\epsilon = 2^{1+\epsilon}$. Note $\gamma > 1$. Next, we need

$$ 2\mathcal{C}_3 \gamma^{1+\epsilon} \left( \frac{\Psi(h_0, R_0)}{h} \right)^{\epsilon} \frac{\|f\|_{q_2, \hat{U}_{1,j} \cap \Omega, v_2} + \|f_1\|_{q_3, \hat{U}_{1,j} \cap \Gamma, w} + h_0 + h}{h} \le 1. $$

Therefore, we choose

$$ h = C(\|f\|_{q_2, \hat{U}_{1,j} \cap \Omega, v_2} + \|f_1\|_{q_3, \hat{U}_{1,j} \cap \Gamma, w} + h_0 + \Psi(h_0, R_0)), $$

for $C = C(n, C_{2v}, C_{2w}, 2v, 2w, q_2, q_3, K_2) > 0$ large. Which prove (4.4).

Taking $i \to \infty$ in (4.4), we conclude

$$ \Psi(h_0 + h, r_{1,j}) = 0. $$

Hence, we have

$$ \begin{aligned} \operatorname*{ess\,sup}_{U_{1,j} \cap \Omega} u^+ + \operatorname*{ess\,sup}_{U_{1,j} \cap \Gamma} u^+ & \le 2(C+1)(\|f\|_{q_2, \hat{U}_{1,j} \cap \Omega, v_2} + \|f_1\|_{q_3, \hat{U}_{1,j} \cap \Gamma, w} + h_0 + \Psi(h_0, R_0)) \\ & \le 2(C+1)(\|f\|_{q_2, \hat{U}_{1,j} \cap \Omega, v_2} + \|f_1\|_{q_3, \hat{U}_{1,j} \cap \Gamma, w} \\ & \phantom{\le 2(C+1)(} + \mathcal{C}_2 \max \left\{ \|\mathcal{C}_2\|_{q_2, \hat{U}_{1,j}, v_2}^{q_2 2v / [q_2(2v-2)-2v]}, \|\mathcal{C}_3\|_{q_3, \Gamma \cap \hat{U}_{1,j}, w}^{q_3 2w / [q_3(2w-2)-2w]} \right\}, 1) \\ & \phantom{\le 2(C+1)(} \cdot \max\{\|u^+\|_{2, \hat{U}_{1,j} \cap \Omega, v_2}, \|u^+\|_{2, \hat{U}_{1,j} \cap \Gamma, w}\} \\ & \phantom{\le 2(C+1)(} + \|u^+\|_{2, \hat{U}_{1,j} \cap \Omega, v_2} + \|u^+\|_{2, \hat{U}_{1,j} \cap \Gamma, w}). \end{aligned} $$

This finishes the proof.

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

Similarly, we can show that:

**Proposition 4.4.** Suppose $\mathcal{R}_{1,j}^2(p)V_1(p) \le K_2V_2(p)$ for a.e. $p \in \hat{U}_{1,j}$. Assume that $c_2, f \in L^{q_2}(\hat{U}_{1,j} \cap \Omega; V_2)$ for some $q_2 > 2v/(2v-2)$. Suppose

$$ \mathcal{Q}[u, v_h] + \int_{\Omega} c_2 u v_h dV_2 \le \int_{\Omega} f v_h dV_2, $$
for all $h \ge 0$, $v_h = u_h \zeta^2$ and $\zeta \in C_0^\infty(\hat{U}_{1,j})$. Then

$$ \underset{U_{1,j} \cap \Omega}{\operatorname{ess\,sup}} u^+ \le C \left( 1 + \|c_2\|_{q_2, \hat{U}_{1,j} \cap \Omega, \mathbf{v}_2}^{q_2 2\nu / [q_2(2\nu-2)-2\nu]} \right) \|u^+\|_{2, \hat{U}_{1,j} \cap \Omega, \mathbf{v}_2} + C \|f\|_{q_2, \hat{U}_{1,j} \cap \Omega, \mathbf{v}_2} $$

where $C = C(n, C_{2\nu}, 2\nu, q_2, K_2) > 0$.

**Proposition 4.5.** Suppose $\mathbb{R}_{0,i}^2(p)\mathbf{V}_1(p) \le K_2\mathbf{V}_2(p)$ for a.e. $p \in \hat{U}_{0,i}$. Assume that $c_2, f \in L^{q_2}(\hat{U}_{0,i}; \mathbf{V}_2)$, for some $q_2 > 2\nu/(2\nu-2)$. Suppose

$$ \mathcal{Q}[u, v_h] + \int_{\Omega} c_2 u v_h d\mathbf{V}_2 \le \int_{\Omega} f v_h d\mathbf{V}_2, $$

for all $h \ge 0$, $v_h = u_h \zeta^2$ and $\zeta \in C_0^\infty(\hat{U}_{0,i})$. Then

$$ \underset{U_{0,i}}{\operatorname{ess\,sup}} u^+ \le C \left( 1 + \|c_2\|_{q_2, \hat{U}_{0,i}, \mathbf{v}_2}^{q_2 2\nu / [q_2(2\nu-2)-2\nu]} \right) \|u^+\|_{2, \hat{U}_{0,i}, \mathbf{v}_2} + C \|f\|_{q_2, \hat{U}_{0,i}, \mathbf{v}_2} $$

where $C = C(n, C_{2\nu}, 2\nu, q_2, K_2) > 0$.

**4.2. Proof of Theorem 1.5.** Steep 1. For all $u \in L^2(\Omega; \mathbf{V}_2)$, there exists a unique solution $A_0 u \in W^{1,2}(\Omega; \mathbf{V}_0, \mathbf{V}_1) \subset L^{2\nu}(\Omega; \mathbf{V}_2)$ such that:

$$ (\tau u, v)_{\mathbf{V}_2} = \mathcal{Q}[A_0 u, v] \quad \forall v \in W^{1,2}(\Omega; \mathbf{V}_0, \mathbf{V}_1), $$

where $(\cdot, \cdot)_{\mathbf{V}_2}$ denotes the inner product associated with $\|\cdot\|_{2,\Omega,\mathbf{V}_2}$. We have the operator $A : W^{1,2}(\Omega; \mathbf{V}_0, \mathbf{V}_1) \to W^{1,2}(\Omega; \mathbf{V}_0, \mathbf{V}_1)$ defined by $Au := A_0 u$ is symmetric.

**Steep 2.** A is compact.

Indeed, suppose first that $\mathrm{supp} \tau \subset \Omega_m$ for some $m \in \mathbb{N}$. Then

$$ \|Au_i - Au_j\|_{1,2,\Omega,\mathbf{V}_0,\mathbf{V}_1}^2 \le \|\tau\|_{L^\infty(\Omega_m)} \|u_i - u_j\|_{2\nu/(2\nu-1),\Omega_m,\mathbf{V}_2} \|Au_i - Au_j\|_{2\nu,\Omega_m,\mathbf{V}_2}. $$

From (1.4),

$$ \|Au_i - Au_j\|_{1,2,\Omega,\mathbf{V}_0,\mathbf{V}_1} \le C_{2\nu} \|\tau\|_{L^\infty(\Omega)} \|u_i - u_j\|_{2\nu/(2\nu-1),\Omega_m,\mathbf{V}_2}. $$

Hence, A is compact. This is because $\mathbf{V}_0$ and $\mathbf{V}_1$ are bounded from above and below by positive constants on each compact subset of M. Consequently, $W^{1,2}(\Omega_m; \mathbf{V}_0, \mathbf{V}_1) \to L^{2\nu/(2\nu-1)}(\Omega_m; \mathbf{V}_2)$ is compact.

In general, set

$$ \tau_m(p) := \begin{cases} \tau(p) & \text{if } p \in \Omega_m, \\ 0 & \text{if } p \in \Omega \setminus \Omega_m, \end{cases} $$

and let $A_m : W^{1,2}(\Omega; \mathbf{V}_0, \mathbf{V}_1) \to W^{1,2}(\Omega; \mathbf{V}_0, \mathbf{V}_1)$, defined by $(\tau_m u, v)_{\mathbf{V}_2} = \mathcal{Q}[A_m u, v]$, for $u, v \in W^{1,2}(\Omega; \mathbf{V}_0, \mathbf{V}_1)$. Then

$$ \|A_m u - Au\|_{1,2,\Omega,\mathbf{V}_0,\mathbf{V}_1}^2 \le \|\tau_m - \tau\|_{2\nu/(2\nu-2),\Omega,\mathbf{V}_2} \|u\|_{2\nu,\Omega,\mathbf{V}_2} \|A_m u - Au\|_{2\nu,\Omega,\mathbf{V}_2}. $$

From (1.4),

$$ \|A_m u - Au\|_{1,2,\Omega,\mathbf{V}_0,\mathbf{V}_1} \le C \|\tau_m - \tau\|_{2\nu/(2\nu-2),\Omega,\mathbf{V}_2} \|u\|_{2\nu,\Omega,\mathbf{V}_2}. $$

We conclude A is compact.

Therefore there exist infinitely many eigenvalues $\cdots \le \lambda_2^- \le \lambda_1^- < 0 < \lambda_1^+ \le \lambda_2^+ \le \cdots$ of

$$ \begin{cases} -\mathrm{div}(\mathbf{V}_1 \nabla u) + \mathbf{V}_0 u = \lambda \mathbf{V}_2 \tau u & \text{in } \Omega, \\ \mathbf{V}_1 \frac{\partial u}{\partial \nu} = 0 & \text{on } \Gamma, \end{cases} $$

Also, $u \in W_{\mathrm{loc}}^{2,2}(\bar{\Omega})$ and as consequence of Propositions 4.4 and 4.5, we have $u \in L_{\mathrm{loc}}^{\infty}(\bar{\Omega})$.

**Steep 3.** If $(W_4)$ holds, we show that $u \in L^\infty(\Omega)$. Moreover, if $u \in W_0^{1,2}(\bar{\Omega}; \mathbf{V}_0, \mathbf{V}_1)$, then

$$ \lim_{m \to \infty} \operatorname{ess\,sup}_{\Omega^m} |u| = 0. \qquad (4.6) $$
Let $q_2 > 2\nu/(2\nu - 2)$. From Propositions 4.4 and 4.5, we have there is a constant $C_1 = C_1(n, C_{2\nu}, 2\nu, q_2, K_2) > 0$ such that

$$ \underset{U_{1,j} \cap \Omega}{\operatorname{ess \, sup}} |u| \le C_1 \left( 1 + \|\lambda \tau\|_{q_2, \hat{U}_{1,j} \cap \Omega, \nu_2}^{q_2 2\nu / [q_2(2\nu-2)-2\nu]} \right) \|u\|_{2, \hat{U}_{1,j} \cap \Omega, \nu_2}, $$

$$ \underset{U_{0,i}}{\operatorname{ess \, sup}} |u| \le C_1 \left( 1 + \|\lambda \tau\|_{q_2, \hat{U}_{0,i}, \nu_2}^{q_2 2\nu / [q_2(2\nu-2)-2\nu]} \right) \|u\|_{2, \hat{U}_{0,i}, \nu_2}, $$

for all $i, j \in \mathbb{N}$.

Let $\epsilon > 0$. Since $u \in W_0^{1,2}(\bar{\Omega}; \nu_0, \nu_1)$, we have there exist $\zeta \in C_0^\infty(M)$ such that $\|\zeta - u\|_{1,2,\Omega,\nu_0,\nu_1} < \epsilon$. Let $m_0 \in \mathbb{N}$ such that $\operatorname{supp} \zeta \subset D_{m_0}$. If $\hat{U}_{1,j} \cap \Omega, \hat{U}_{0,i} \subset \Omega \setminus D_{m_0}$, from (W4), (1.4) and (1.5), we have

$$ \underset{U_{1,j} \cap \Omega}{\operatorname{ess \, sup}} |u| \le C_1 \left[ 1 + \left( \|\lambda \tau\|_{L^\infty(\Omega)}^{q_2} K_3 \right)^{2\nu/[q_2(2\nu-2)-2\nu]} \right] K_3^{\frac{2\nu-2}{2\nu}} \|u\|_{2\nu, \hat{U}_{1,j} \cap \Omega, \nu_2} $$

$$ \le C_1 \left[ 1 + \left( \|\lambda \tau\|_{L^\infty(\Omega)}^{q_2} K_3 \right)^{2\nu/[q_2(2\nu-2)-2\nu]} \right] $$

$$ \cdot K_3^{\frac{2\nu-2}{2\nu}} \left( \|u - \zeta\|_{2\nu, \hat{U}_{1,j} \cap \Omega, \nu_2} + \|\zeta\|_{2\nu, \hat{U}_{1,j} \cap \Omega, \nu_2} \right) $$

$$ \le C_1 \left[ 1 + \left( \|\lambda \tau\|_{L^\infty(\Omega)}^{q_2} K_3 \right)^{2\nu/[q_2(2\nu-2)-2\nu]} \right] C_{2\nu} K_3^{\frac{2\nu-2}{2\nu}} \epsilon $$

and

$$ \underset{U_{0,i}}{\operatorname{ess \, sup}} |u| \le C_1 \left[ 1 + \left( \|\lambda \tau\|_{L^\infty(\Omega)}^{q_2} K_3 \right)^{2\nu/[q_2(2\nu-2)-2\nu]} \right] C_{2\nu} K_3^{\frac{2\nu-2}{2\nu}} \epsilon. $$

Therefore

$$ \lim_{m \to \infty} \operatorname{ess \, sup}_{\Omega^m} |u| = 0. $$

With this, we conclude the proof of Theorem 1.5.

**Proposition 4.6.** *Assume the hypotheses of Corollary 1.3 hold. Suppose $\nu_1 \in C^1(M)$, $\tau \in L^{2\nu/(2\nu-2)}(\Omega; \nu_2) \cap L^\infty(\Omega)$, and $\{\tau > 0\}$ and $\{\tau < 0\}$ do not have empty interiors. Assume that $\hat{r}_{k,i} - r_{k,i} < 1$ and $g^{ab}(y)x_a x_b \ge \theta|x|^2$ for all $x \in \mathbb{R}^n$ and $y \in B(0, \hat{r}_{k,i})$, for some $\theta > 0$. If $u$ is a weak solution of (1.2), we have the estimate*

$$ \int_{U_{k,i}} |\nabla^2 u|^2 d\nu_3 \le C(n) \left[ (C_1 + C_2) \int_{\hat{U}_{k,i}} u^2 d\nu_0 + C_3 \int_{\hat{U}_{k,i}} |\nabla u|^2 d\nu_1 \right], $$

where

$$ C_1 = \left[ \frac{\|\lambda \tau \nu_2\|_{L^\infty(\hat{U}_{k,i})} + \|\nu_0\|_{L^\infty(\hat{U}_{k,i})}}{\hat{r}_{k,i} - r_{k,i}} \right]^2 \frac{\|\nu_3\|_{L^\infty(\hat{U}_{k,i})}}{\nu_0 (\inf_{\hat{U}_{k,i}} \nu_1)^2} \frac{R_4^2 R_6^{3/2}}{\theta^2}, $$

$$ C_2 = \frac{R_4^2 R_5^2 R_6^{1/2}}{\theta} \frac{\|\nu_3\|_{L^\infty(\hat{U}_{k,i})}}{\inf_{\hat{U}_{k,i}} \nu_1}, $$

$$ C_3 = \left[ 1 + \frac{\|\sqrt{\det[g^{ab}]}\|_{L^\infty(\hat{U}_{k,i})} \max\{\mathbf{R}_7, \mathbf{R}_8\}}{\theta \inf_{\hat{U}_{k,i}} \mathbf{V}_1 (\hat{r}_{k,i} - r_{k,i})} \right]^2 \frac{R_4^2 R_6^{1/2} \|\nu_3\|_{L^\infty(\hat{U}_{k,i})}}{\theta (\inf_{\hat{U}_{k,i}} \mathbf{V}_1) \inf_{\hat{U}_{k,i}} \mathbf{V}_0}, $$

$R_4 := \sup_{a,b} \|g^{ab}\|_{L^\infty(\hat{U}_{k,i})}$, $R_5 = \sup_{a,b,c} \|\Gamma_{ab}^c\|_{L^\infty(\hat{U}_{k,i})}$ ($\Gamma_{ab}^c$ are the Christoffel symbols), $R_6 := (\sup_{\hat{U}_{k,i}} \det[g_{ab}]) (\sup_{\hat{U}_{k,i}} \det[g^{ab}])$, $R_7 := \sup_{a,b} \|\nu_1 \sqrt{\det[g_{cd}]} g^{ab}\|_{L^\infty(\hat{U}_{k,i})}$, and $R_8 := \sup_{a,b} \|\nabla_{\delta_{\mathbb{R}^n}} (\nu_1 \sqrt{\det[g_{cd}]} g^{ab})\|_{L^\infty(\hat{U}_{k,i})}$.

*Proof.* Using a chart $\psi_{1,j} : B(0, \hat{r}_{1,j}) \to \hat{U}_{1,j}$, by (2.1), we have

$$ |\nabla^2 u|^2 = g^{ab} g^{cd} \left( u_{x_a x_c} u_{x_b x_d} - \Gamma_{ac}^m u_{x_m} u_{x_b x_d} - \Gamma_{bd}^m u_{x_m} u_{x_a x_c} + \Gamma_{ac}^m \Gamma_{bd}^{\ell} u_{x_m} u_{x_\ell} \right), \quad (4.7) $$

where, for simplicity, we are writing $u(x) = u \circ \psi_{1,j}(x)$ for $x \in B(0, \hat{r}_{1,j}) \cap \{x_n > 0\}$.
For every function $f: B(0, \hat{r}_{1,j}) \cap \{x_n \ge 0\} \to \mathbb{R}$, we define $\bar{f}(x) := f(x_1, \dots, x_{n-1}, |x_n|)$ for $x \in B(0, \hat{r}_{1,j})$. Hence

$$ \int_{B(0, \hat{r}_{1,j})} \overline{P^{cd}} \bar{u}_{x_c} v_{x_d} dx + \int_{B(0, \hat{r}_{1,j})} \bar{Q} \bar{u} v dx = 0, \quad \forall v \in H^1(B(0, \hat{r}_{1,j})), $$

where $P^{cd} = V_1 \sqrt{\det[g_{ab}]} g^{cd}$ and $Q = \sqrt{\det[g_{ab}]}(V_0 - \lambda \tau V_2)$.

Following the argument in [27, Interior $H^2$-regularity], we have the estimate

$$ \int_{B(0, r_{1,j})} |\bar{u}_{x_a x_b}|^2 dx \le c(n) \left( c_1 \int_{B(0, \hat{r}_{1,j})} \bar{u}^2 dx + c_2 \int_{B(0, \hat{r}_{1,j})} |\nabla \bar{u}|^2 dx \right), \quad (4.8) $$

where

$$ c_1 = \left[ \frac{\|\sqrt{\det[g^{ab}]}\|_{L^\infty(B(0, \hat{r}_{1,j}))} (\|\tau V_2 \sqrt{\det[g_{ab}]}\|_{L^\infty(B(0, \hat{r}_{1,j}))} + \|V_0 \sqrt{\det[g_{ab}]}\|_{L^\infty(B(0, \hat{r}_{1,j}))})}{\theta(\hat{r}_{1,j} - r_{1,j}) \inf_{B(0, \hat{r}_{1,j})} V_1} \right]^2, $$ $$ c_2 = \left[ 1 + \frac{\|\sqrt{\det[g^{ab}]}\|_{L^\infty(B(0, \hat{r}_{1,j}))} \max\{c_3, c_4\}}{\theta(\hat{r}_{1,j} - r_{1,j}) \inf_{B(0, \hat{r}_{1,j})} V_1} \right]^2, $$

$c_3 = \sup_{c,d} \|V_1 \sqrt{\det[g^{ab}]} g^{cd}\|_{L^\infty(B(0, \hat{r}_{1,j}))}$, and $c_4 = \sup_{c,d} \|\nabla(V_1 \sqrt{\det[g^{ab}]} g^{cd})\|_{L^\infty(B(0, \hat{r}_{1,j}))}$.

Employing (4.7) and (4.8), we conclude the proof of Theorem 1.5 4.6. $\square$

The following remark shows possible conditions that limit $R_i$, $i = 4, \dots, 8$, along $M$.

**Remark 4.7.** See [51, Theorem 2.5] for more details. See also [52]. Let $(M_2^n, g)$, $n \ge 2$, be a Riemannian manifold with boundary $\partial M_2$. To given $c > 0$, $k \in \mathbb{N}$, and dimension $n$, there exist $R_1, R_2, R_3 > 0$ and $c_1 > 0$ such that the following holds:

(a) If $p \in \partial M_2$, $0 < r_1 \le R_1$, $0 < r_2 \le R_2$, and $\kappa_p: B(0, r_1) \times [0, r_2) \to M_2$ is a normal boundary chart, and if $|\nabla^i R| \le c$ and $|\bar{\nabla}^i l| \le c$ for $i = 0, \dots, k$ on the image of $\kappa_p$, then in these coordinates we get

$$ |D^\alpha g_{ij}| \le c_1 \quad \text{and} \quad |D^\alpha g^{ij}| \le c_1 \quad \text{whenever} \quad |\alpha| \le k. $$

Here, $R$ is the curvature and $l$ the second fundamental form tensor. $\nabla$ is the Levi-Civita connection of $M_2$, and $\bar{\nabla}$ is the one of $\partial M_2$.

(b) If, on the other hand,

$$ |D^\alpha g_{ij}| \le c \quad \text{and} \quad |D^\alpha g^{ij}| \le c \quad \text{for} \quad |\alpha| \le k + 2, $$

then, on the image of $\kappa_p$,

$$ |\nabla^i R| \le c_1 \quad \text{and} \quad |\bar{\nabla}^i l| \le c_1 \quad \text{for} \quad i = 0, \dots, k. $$

### 5. EIGENVALUE PROBLEM WITH $L^1$-DATA

We begin with the following auxiliary result. Then we will proceed to prove Theorem 1.8.

**Lemma 5.1.** Assume $n < q$. Then there exists a constant $C = C(n, q, K_3) > 0$, such that

$$ \|u\|_{L^\infty(\hat{U}_{k,i} \cap \Omega)} \le C[\|d\psi_{k,i}\|(\hat{r}_{k,i} - r_{k,i}) + 1]\|u\|_{1,q, \hat{U}_{k,i} \cap \Omega, V_0, V_1} \quad \forall u \in W^{1,q}(\Omega; V_0, V_1). \quad (5.1) $$

*Proof.* Let $u \in W^{1,q}(\Omega; V_0, V_1)$. Using Proposition 3.2, we write $\bar{u} = \mathcal{E}u$.

**Step 1.** For $x \in B(0, r_{k,i})$, we have the estimate:

$$ \begin{aligned} & \int_{B(x, \hat{r}_{k,i} - |x|)} |\bar{u} \circ \psi_{k,i}(y) - \bar{u} \circ \psi_{k,i}(x)| dy \\ & \le C_1(n) \|d\psi_{k,i}\| \int_{B(x, \hat{r}_{k,i} - |x|)} \frac{|\nabla \bar{u}| \circ \psi_{k,i}(y)}{|y - x|^{n-1}} dy. \end{aligned} \quad (5.2) $$
The proof follows the same argument of [27, Morrey's inequality].

**Steep 2.** We apply inequality (5.2) as follows:

$$|\bar{u} \circ \psi_{k,i}(x)| \le \int_{B(x,\hat{r}_{k,i}-|x|)} |\bar{u} \circ \psi_{k,i}(x) - \bar{u} \circ \psi_{k,i}(y)| dy + \int_{B(x,\hat{r}_{k,i}-|x|)} |\bar{u} \circ \psi_{k,i}(y)| dy$$

$$ \begin{aligned} & \le C_1 \|d\psi_{k,i}\| \int_{B(x,\hat{r}_{k,i}-|x|)} \frac{|\nabla \bar{u}| \circ \psi_{k,i}(y)}{|x-y|^{n-1}} dy \\ & \qquad + C_2(n)(\hat{r}_{k,i} - |x|)^{-n} \int_{B(x,\hat{r}_{k,i}-|x|)} |\bar{u} \circ \psi_{k,i}(y)| dy \\ & \le C_1 \|d\psi_{k,i}\| \left( \int_{B(x,\hat{r}_{k,i}-|x|)} (|\nabla \bar{u}| \circ \psi_{k,i}(y))^q dy \right)^{\frac{1}{q}} \left( \int_{B(x,\hat{r}_{k,i}-|x|)} |x-y|^{-\frac{(n-1)q}{q-1}} dy \right)^{\frac{q-1}{q}} \\ & \qquad + C_3(n)(\hat{r}_{k,i} - |x|)^{-\frac{n}{q}} \left( \int_{B(x,\hat{r}_{k,i}-|x|)} |\bar{u} \circ \psi_{k,i}(y)|^q dy \right)^{\frac{1}{q}} \\ & \le C_4(n,q) \|d\psi_{k,i}\| (\hat{r}_{k,i} - |x|)^{\frac{q-n}{q}} \left( \int_{B(x,\hat{r}_{k,i}-|x|)} (|\nabla \bar{u}| \circ \psi_{k,i}(y))^q dy \right)^{\frac{1}{q}} \\ & \qquad + C_3(\hat{r}_{k,i} - |x|)^{-\frac{n}{q}} \left( \int_{B(x,\hat{r}_{k,i}-|x|)} |\bar{u} \circ \psi_{k,i}(y)|^q dy \right)^{\frac{1}{q}} \\ & \le C_4 \|d\psi_{k,i}\| \|G_{k,i}^{-1}\|^{\frac{1}{q}} (\hat{r}_{k,i} - |x|)^{\frac{q-n}{q}} \left( \int_{\hat{U}_{k,i}} |\nabla \bar{u}|^q d\nu_g \right)^{\frac{1}{q}} \\ & \qquad + C_3 \|G_{k,i}^{-1}\|^{\frac{1}{q}} (\hat{r}_{k,i} - |x|)^{-\frac{n}{q}} \left( \int_{\hat{U}_{k,i}} |\bar{u}|^q d\nu_g \right)^{\frac{1}{q}} \end{aligned} $$

From (W₂) and (W₅),

$$ \begin{aligned} & |\bar{u} \circ \psi_{k,i}(x)| \\ & \le C_4 \|d\psi_{k,i}\|^{1-\frac{1}{q}} b_1^{-\frac{1}{q}} (\psi_{k,i}(0)) \|G_{k,i}^{-1}\|^{\frac{1}{q}} (\hat{r}_{k,i} - r_{k,i})^{\frac{q-n}{q}} \left( \int_{\hat{U}_{k,i}} |\nabla \bar{u}|^q d\nu_1 \right)^{\frac{1}{q}} \\ & \qquad + C_3 \|G_{k,i}^{-1}\|^{\frac{1}{q}} (\hat{r}_{k,i} - r_{k,i})^{-\frac{n}{q}} \left( \int_{\hat{U}_{k,i}} |\bar{u}|^q d\nu_g \right)^{\frac{1}{q}} \\ & \le C_4 K_3^{\frac{1}{q}} \|d\psi_{k,i}\| (\hat{r}_{k,i} - r_{k,i}) \left( \int_{\hat{U}_{k,i}} |\nabla \bar{u}|^q d\nu_1 \right)^{\frac{1}{q}} + C_3 K_3^{\frac{1}{q}} \left( \int_{\hat{U}_{k,i}} |\bar{u}|^q d\nu_0 \right)^{\frac{1}{q}}. \end{aligned} $$

This conclude the proof of Lemma 5.1. □

**5.1. Proof of Theorem 1.8 (i).** Steep 1. We first prove that

$$ (u_j) \text{ is bounded in } L^1(\Omega; \nu_0). \qquad (5.3) $$
Indeed, if not, one can suppose (passing eventually to a subsequence) that $\|u_j\|_{1,\Omega,\nu_0} \to \infty$. Set $v_j = \frac{u_j}{\|u_j\|_{1,\Omega,\nu_0}}$. Then, $v_j \in W^{1,2}(\Omega; \nu_0, \nu_1)$, $\|v_j\|_{1,\Omega,\nu_0} = 1$, and

$$ egin{cases} -\operatorname{div}(\nu_1 \nabla v_j) + \nu_0 v_j = \lambda \nu_2 \tau v_j + \nu_2 \frac{f_{0,j}}{\|u_j\|_{1,\Omega,\nu_0}} & \text{in } \Omega, \\ \nu_1 \frac{\partial v_j}{\partial \nu} = \nu_1 \frac{f_{1,j}}{\|u_j\|_{1,\Omega,\nu_0}} & \text{on } \Gamma, \end{cases} \quad (5.4) $$

**Claim 5.2.** The sequence $(v_j)$ is bounded in $W^{1,q}(\Omega; \nu_0, \nu_1)$ for every $1 \le q < n/(n-1)$.

*Proof.* We also remark that if $1 \le q < n/(n-1)$, then its conjugate exponent $q' > n$. Hence

$$ egin{align*} \|v_j\|_{1,q,\Omega,\nu_0,\nu_1} &= \sup_{T \in (W^{1,q}(\Omega;\nu_0,\nu_1))^* \atop \|T\| \le 1} \langle T, v_j \rangle \\ &= \sup_{\|w\|_{1,q',\Omega,\nu_0,\nu_1} \le 1} \int_{\Omega} g(\nabla v_j, \nabla w) d\nu_1 + \int_{\Omega} v_j w d\nu_0 \\ &= \sup_{\|w\|_{1,q',\Omega,\nu_0,\nu_1} \le 1} \lambda \int_{\Omega} \tau v_j w d\nu_2 + \int_{\Omega} \frac{f_{0,j}}{\|u_j\|_{1,\Omega,\nu_0}} w d\nu_2 + \int_{\Gamma} \frac{f_{1,j}}{\|u_j\|_{1,\Gamma,\omega}} w d\nu_1. \end{align*} $$

From (5.1), (W5), and (U3),

$$ C_1 \|v_j\|_{1,q,\Omega,\nu_0,\nu_1} \le |\lambda| \|\tau\|_{L^\infty(\Omega)} \int_{\Omega} |v_j| d\nu_0 + \int_{\Omega} \frac{f_{0,j}}{\|u_j\|_{1,\Omega,\nu_0}} d\nu_0 + \int_{\Gamma} \frac{f_{1,j}}{\|u_j\|_{1,\Omega,\nu_0}} d\nu. $$

where $C_1 = C_1(n, q, K_3, R_1) > 0$. Therefore $(v_j)$ is bounded. $\square$

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

Let $\hat{v}_j \in W^{1,2}(\Omega; \nu_0, \nu_1)$ be a weak solution of the problem

$$ egin{cases} -\operatorname{div}(\nu_1 \nabla \hat{v}_j) + \nu_0 \hat{v}_j = \lambda \nu_2 \tau \hat{v}_j & \text{in } \Omega, \\ \nu_1 \frac{\partial \hat{v}_j}{\partial \nu} = 0 & \text{on } \Gamma, \end{cases} \quad (5.5) $$

We will now split the proof of (5.3) into two cases.

*Case n = 3.* From the hypotheses of Lemma 1.8, we have that there is $\sigma \in (1, 3/2)$ such that $\sigma_\nu \ge 2$. It follows by Claim 5.2 and (1.4) that $(v_j)$ is bounded in $L^{\sigma_\nu}(\Omega; \nu_2)$.

If $\sigma_\nu > 2$. From (5.5), (1.4), and Hölder inequality,

$$ egin{align} \|\hat{v}_j\|_{1,2,\Omega,\nu_0,\nu_1}^2 &\le |\lambda| \|\tau\|_{\sigma_\nu/(\sigma_\nu-2),\Omega,\nu_2} \|\hat{v}_j\|_{\sigma_\nu,\Omega,\nu_2} \|v_j\|_{\sigma_\nu,\Omega,\nu_2} \tag{5.6} \\ &\le C_{\sigma_\nu} |\lambda| \|\tau\|_{\sigma_\nu/(\sigma_\nu-2),\Omega,\nu_2} \|\hat{v}_j\|_{1,\sigma,\Omega,\nu_0,\nu_1} \|v_j\|_{\sigma_\nu,\Omega,\nu_2} \nonumber \\ &\le C_2 \|\hat{v}_j\|_{1,2,\Omega,\nu_0,\nu_1} \|v_j\|_{\sigma_\nu,\Omega,\nu_2}, \nonumber \end{align} $$

where $C_2 = C_2(C_{\sigma_\nu}, |\lambda|, \|\tau\|_{\sigma_\nu/(\sigma_\nu-2),\Omega,\nu_2}, \nu_0(\Omega), \nu_1(\Omega), \sigma) > 0$.

If $\sigma_\nu = 2$. Then

$$ egin{align} \|\hat{v}_j\|_{1,2,\Omega,\nu_0,\nu_1}^2 &\le |\lambda| \|\tau\|_{L^\infty(\Omega)} \|\hat{v}_j\|_{\sigma_\nu,\Omega,\nu_2} \|v_j\|_{\sigma_\nu,\Omega,\nu_2} \tag{5.7} \\ &\le C_3 \|\hat{v}_j\|_{1,2,\Omega,\nu_0,\nu_1} \|v_j\|_{\sigma_\nu,\Omega,\nu_2}, \nonumber \end{align} $$

where $C_3 = C_3(C_{\sigma_\nu}, |\lambda|, \|\tau\|_{L^\infty(\Omega)}, \nu_0(\Omega), \nu_1(\Omega), \sigma) > 0$.

From (5.6) and (5.7), we conclude that $\hat{v}_j$ is bounded in $W^{1,2}(\Omega; \nu_0, \nu_1)$.

On the other hand,

$$ egin{cases} -\operatorname{div}[\nu_1 \nabla (v_j - \hat{v}_j)] + \nu_0 (v_j - \hat{v}_j) = \nu_2 \frac{f_{0,j}}{\|u_j\|_{1,\Omega,\nu_0}} & \text{in } \Omega, \\ \nu_1 \frac{\partial (v_j - \hat{v}_j)}{\partial \nu} = \nu_1 \frac{f_{1,j}}{\|u_j\|_{1,\Omega,\nu_0}} & \text{on } \Gamma, \end{cases} \quad (5.8) $$
It follows by an argument similar to that applied in the proof of Claim 5.2 that $(v_j - \hat{v}_j)$ is bounded in $W^{1,q}(\Omega; \mathbf{V}_0, \mathbf{V}_1)$ for every $1 \le q < n/(n-1) = 3/2$. So, by convexity (and since $W^{1,q}(\Omega; \mathbf{V}_0, \mathbf{V}_1) \subset W^{1,1}(\Omega; \mathbf{V}_0, \mathbf{V}_1)$ if $q > 1$), it converges weakly (up to a subsequence) in $W^{1,q}(\Omega; \mathbf{V}_0, \mathbf{V}_1)$ for all $1 \le q < n/(n-1)$.

Passing eventually again to a subsequence, one may suppose that

$$v_j \to v \quad \text{in} \quad W^{1,q}(\Omega; \mathbf{V}_0, \mathbf{V}_1). \qquad (5.9)$$

Therefore, by (5.8),

$$\hat{v}_j \to v \quad \text{in} \quad W^{1,q}(\Omega; \mathbf{V}_0, \mathbf{V}_1). \qquad (5.10)$$

Using (5.6) and (5.7), this (weak) convergence is also valid in $W^{1,2}(\Omega; \mathbf{V}_0, \mathbf{V}_1)$, and furthermore, $v \in W^{1,2}(\Omega; \mathbf{V}_0, \mathbf{V}_1)$. Now, by (5.9) and (5.10), it follows that one can pass to the limit in (5.5) as $j \to \infty$. Hence

$$\left\{ \begin{array}{ll} -\operatorname{div}(\mathbf{V}_1 \nabla v) + \mathbf{V}_0 v = \lambda \mathbf{V}_2 \tau v & \text{in } \Omega, \\ \mathbf{V}_1 \frac{\partial v}{\partial \nu} = 0 & \text{on } \Gamma, \end{array} \right. \qquad (5.11)$$

Then, since $\lambda$ is not an eigenvalue, $v=0$. We have obtained that $v_j \to 0$ (up to a subsequence) weakly in $W^{1,q_2}(\Omega; \mathbf{V}_0, \mathbf{V}_1)$, where $q_2 \in (1, n/(n-1))$ is given in $(H_1)$,

We have $\mathbf{V}_0$ and $\mathbf{V}_1$ bounded from above and below by positive constants on each open set $\Omega_m$. Also, $W^{1,q_2}(\Omega_m) \to L^{q_2}(\Omega_m)$ is compact. Then we can assume that $v_j \to 0$ a.e. in $\Omega$. Then, using $(H_1)$, by the Vitali convergence theorem, we conclude $v_j \to 0$ in $L^1(\Omega; \mathbf{V}_0)$, which is a contradiction, since $\|v_j\|_{1,\Omega,\mathbf{V}_0} = 1$. This shows that our assertion (5.3) is true.

Case $n \ge 4$. Set $\hat{v}_j^{(1)} = \hat{v}_j$ and, for each integer $k \ge 2$, let $\hat{v}_j^{(k)}$ be the unique solution of the problem

$$\left\{ \begin{array}{ll} -\operatorname{div}(\mathbf{V}_1 \nabla \hat{v}_j^{(k)}) + \mathbf{V}_0 \hat{v}_j^{(k)} = \lambda \mathbf{V}_2 \tau \hat{v}_j^{(k-1)} & \text{in } \Omega, \\ \mathbf{V}_1 \frac{\partial \hat{v}_j^{(k)}}{\partial \nu} = 0 & \text{on } \Gamma. \end{array} \right. $$

Let $1 < \sigma < n/(n-1)$, where $\sigma$ is given in $(H_1)$. We denote $\sigma^{(0)\star} := \sigma$, and for each integer $1 \le i < n/\sigma$,

$$\sigma^{(i)\star} := \frac{(i\sigma)\mathbf{v}_i}{i}.$$

**Claim 5.3.** The sequence $(\hat{v}_j^{(1)})$ is bounded in $W^{1,\sigma^{(1)\star}}(\Omega; \mathbf{V}_0, \mathbf{V}_1)$.

*Proof.* We have

$$\begin{align*} \|\hat{v}_j^{(1)}\|_{1,\sigma^{(1)\star},\Omega,\mathbf{V}_1,\mathbf{V}_0} & = \sup\left\{ \int_{\Omega} g(\nabla \hat{v}_j^{(1)}, \nabla w) d\mathbf{V}_1 + \int_{\Omega} \hat{v}_j^{(1)} w d\mathbf{V}_0 \mid \|w\|_{1,(\sigma^{(1)\star})',\Omega,\mathbf{V}_0,\mathbf{V}_1} \le 1 \right\} \\ & = \sup\left\{ \int_{\Omega} \lambda \tau \hat{v}_j^{(1)} w d\mathbf{V}_2 \mid \|w\|_{1,(\sigma^{(1)\star})',\Omega,\mathbf{V}_0,\mathbf{V}_1} \le 1 \right\} \\ & \le \lambda \|\tau\|_{L^{\infty}(\Omega)} \sup\{ \|\hat{v}_j^{(1)}\|_{\sigma^{(0)\star},\Omega,\mathbf{V}_2} \|w\|_{(\sigma^{(0)\star})',\Omega,\mathbf{V}_2} \mid \|w\|_{1,(\sigma^{(1)\star})',\Omega,\mathbf{V}_0,\mathbf{V}_1} \le 1 \}. \end{align*}$$

Using Claim 5.2, $W^{1,(\sigma^{(1)\star})'}(\Omega; \mathbf{V}_0, \mathbf{V}_1) \to L^{((\sigma^{(1)\star})')_v}(\Omega; \mathbf{V}_2)$, and $(\sigma^{(0)\star})' \le ((\sigma^{(1)\star})')_v$ (see $(H_1)$), we conclude the proof of the claim. $\square$

By the same arguments as those used in the proof of Claim 5.3, one can show that $(\hat{v}_j^{(k)})$ is bounded in $W^{1,\sigma^{(k)\star}}(\Omega; \mathbf{V}_0, \mathbf{V}_1)$ with $\sigma^{(k)\star} \ge 2$. Thus, by applying the idea used in the case $n=3$, we conclude the proof of (5.3).

**Steep 2.** Since $(u_j) \subset W^{1,2}(\Omega; \mathbf{V}_0, \mathbf{V}_1)$ is bounded in $L^1(\Omega; \mathbf{V}_0)$, applying the arguments done in the proof of Claim 5.2 and we find that $(u_j)$ is bounded in each space $W^{1,q}(\Omega; \mathbf{V}_0, \mathbf{V}_1)$, $1 \le q < n/(n-1)$. So, there exists $u \in \cap_{1 \le q < n/(n-1)} W^{1,q}(\Omega; \mathbf{V}_0, \mathbf{V}_1)$ such that, up to a subsequence, $u_n \rightharpoonup u$ weakly in
$W^{1,q}(\Omega; \mathbf{V}_0, \mathbf{V}_1)$ for all $1 \le q < n/(n-1)$, which means that $u$ is a solution by approximation of the problem (1.1).

**5.2. Proof of Theorem 1.8 (ii).** Let us first assume that (1.1) has a solution $u$. Choose $v \in E_\lambda$ an arbitrary eigenfunction of (1.2) associated to the eigenvalue $\lambda$. We find

$$ \int_{\Omega} g(\nabla u, \nabla v) d\mathbf{V}_1 + \int_{\Omega} uv d\mathbf{V}_0 = \lambda \int_{\Omega} \tau uv d\mathbf{V}_2 + \int_{\Omega} f_0 v d\mathbf{V}_2 + \int_{\Gamma} f_1 v d\mathbf{W}_1. $$

Then

$$ \int_{\Omega} f_0 v d\mathbf{V}_2 + \int_{\Omega} f_1 v d\mathbf{W}_1 = 0. \qquad (5.12) $$

Conversely, assume that $f_0 \in L^1(\Omega; \mathbf{V}_0)$, $f_1 \in L^1(\Gamma; \mathbf{W})$, and (5.12) holds for every $v \in E_\lambda = \{u \in W^{1,2}(\Omega; \mathbf{V}_0, \mathbf{V}_1) | u$ solves (1.2)\}.$

**Claim 5.4.** There exists a sequences $(\tilde{f}_{0,j}) \subset L^2(\Omega; \mathbf{V}_0)$ and $(\tilde{f}_{1,j}) \subset L^2(\Gamma; \mathbf{W})$ such that

$$ \int_{\Omega} \tilde{f}_{0,j} v d\mathbf{V}_2 + \int_{\Omega} f_1 v d\mathbf{W}_1 = 0 \quad \forall v \in E_{\lambda}, \qquad (5.13) $$

$$ \tilde{f}_{0,j} \to f_0 \text{ in } L^1(\Omega; \mathbf{V}_0) \quad \text{and} \quad \tilde{f}_{1,j} \to f_1 \text{ in } L^1(\Gamma; \mathbf{W}). \qquad (5.14) $$

*Proof.* On $L^2(\Omega; \mathbf{V}_2) \times L^2(\Gamma; \mathbf{W}_1)$ we define the norm

$$ \|(u, z)\|_{\mathbf{V}_2, \mathbf{W}_1} := \int_{\Omega} u^2 d\mathbf{V}_2 + \int_{\Gamma} z^2 d\mathbf{W}_1 $$

and by $\langle \cdot, \cdot \rangle_{\mathbf{V}_2, \mathbf{W}_1}$ it is associated scalar product.

Set $\hat{E}_\lambda := \{(u, z) \in L^2(\Omega; \mathbf{V}_2) \times L^2(\Gamma; \mathbf{W}_1) | u \in E_\lambda \text{ and } z = u|_\Gamma\}$. Let $\{e_1, \dots, e_{n_0}\}$ be an orthonormal basis in the finite dimensional space $E_\lambda$. By Theorem 1.5 we have $e_i \in L^\infty(\Omega)$, $i = 1, \dots, n_0$.

We choose sequences $(f_{0,j}) \subset L^2(\Omega; \mathbf{V}_0)$ and $(f_{1,j}) \subset L^2(\Gamma; \mathbf{W})$ such that $f_{0,j} = 0$ in $\Omega^j$, $f_{1,j} = 0$ in $\Gamma^j$, $f_{0,j} \to f_0$ in $L^1(\Omega; \mathbf{V}_0)$, and $f_{1,j} \to f_0$ in $L^1(\Gamma; \mathbf{W})$.

Define

$$ \tilde{f}_{0,j} := f_{0,j} - \sum_{i=1}^{n_0} \left( \int_{\Omega} f_{0,j} e_i d\mathbf{V}_2 \right) e_i \quad \text{and} \quad \tilde{f}_{1,j} := f_{1,j} - \sum_{i=1}^{n_0} \left( \int_{\Gamma} f_{1,j} e_i d\mathbf{W}_1 \right) e_i. $$

Hence (5.13) is verified. Also, since $e_i \in L^\infty(\Omega)$, then (5.14) holds. $\square$

Therefore, by Proposition B.1(ii), problem (1.1) has a solution $u_j$ for corresponding data $\tilde{f}_{0,j}$ and $\tilde{f}_{1,j}$, such that $u_j = 0$ in $\Omega^j$. As in the proof of (i), the sequence $(u_j)$ converges to a solution $u$ of problem (1.1).

# 6. BEHAVIOR OF THE FIRST EIGENVALUE

**6.1. Proof of Theorem 1.10 (i).** Steep 1. We prove that:

$$ \mathcal{Q}_N[\phi, \phi] - \lambda_1(N) \ge C_\nu(\Gamma) \frac{\lambda_2(N) - \lambda_1(N)}{\lambda_2(N) + \lambda_1(N)}. \qquad (6.1) $$

Let $\psi = \kappa_1 - \phi$. We have,

$$ C_\nu(\Gamma) \le \int_N \psi^2 d\mathbf{V}_0 + \int_N |\nabla \psi|^2 d\mathbf{V}_1. $$

Hence,

$$ C_{\nu}(\Gamma) \le \lambda_{1}(N) + \mathcal{Q}_{N}[\phi, \phi] - 2\lambda_{1}(N) \int_{N} \tau \kappa_{1} \phi d\mathbf{V}_{2}, \qquad (6.2) $$

where $\mathcal{Q}_N[\phi, \phi] := \int_N \phi^2 d\mathbf{V}_0 + \int_N |\nabla \phi|^2 d\mathbf{V}_1$.
Using,

$$ \phi = \sum_i \frac{Q_N[\phi, \kappa_i]}{Q_N[\kappa_i, \kappa_i]} \kappa_i \quad \text{and} \quad \phi = \sum_i (\tau\phi, \kappa_i)_{N,v_2} \kappa_i, $$

where $(\tau\phi, \kappa_i)_{N,v_2} := \int_N \tau\phi\kappa_i dV_2$, we have

$$ Q_N[\phi, \phi] = \sum_i \lambda_i(N)b_i^2 \quad \text{and} \quad (\tau\phi, \phi)_{N,v_2} = \sum_i b_i^2, $$

where $b_i := \int_N \tau\phi\kappa_i dV_2$.

Consequently,

$$ Q_N[\phi, \phi] \geq \lambda_1(N)b_1^2 + (1 - b_1^2)\lambda_2(N), $$

This implies,

$$ \begin{aligned} Q_N[\phi, \phi] - \lambda_1(N) &\geq (1 - b_1^2)(\lambda_2(N) - \lambda_1(N)) \\ &\geq (1 - b_1)(1 + b_1)(\lambda_2(N) - \lambda_1(N)). \end{aligned} \quad (6.3) $$

By (6.2) and (6.3), we obtain (6.1)

Steep 2. Now, we proceed to prove:

$$ \begin{aligned} & \lambda_1(N) - |\lambda_1^-(M)| \\ &\geq \operatorname{Cap}^-(M\setminus N) \frac{\lambda_1^+(M) - |\lambda_1^-(M)|}{\lambda_1^+(M) + |\lambda_1^-(M)|} \quad \text{if} \quad |\lambda_1^-(M)| \leq \lambda_1^+(M) \leq |\lambda_2^-(M)|. \end{aligned} \quad (6.4) $$

Let $\psi^{\pm} = \phi_1^{\pm} \mp \operatorname{sign}(b_1^{\pm})\kappa_1$, where $b_i^{\pm} := \int_M \tau\phi_i^{\pm} \kappa_1 dV_2$, $i \in \mathbb{N}$. We have,

$$ \operatorname{Cap}^{\pm}(M\setminus N) \leq \int_M |\psi^{\pm}|^2 dV_0 + \int_M |\nabla\psi^{\pm}|^2 dV_1. $$

Hence,

$$ \operatorname{Cap}^{\pm}(M\setminus N) \leq |\lambda_1^{\pm}(M)| + \lambda_1(N) - 2 \left| \int_M \tau\phi_1^{\pm} \kappa_1 dV_2 \right|. \quad (6.5) $$

Employing

$$ \kappa_1 = \sum_i \frac{Q_M[\kappa_1, \phi_i^-]}{Q_M[\phi_i^-, \phi_i^-]} \phi_i^- + \sum_j \frac{Q_M[\kappa_1, \phi_j^+]}{Q_M[\phi_j^+, \phi_j^+]} \phi_j^+. $$

We have

$$ Q_M[\kappa_1, \kappa_1] = \sum_i |\lambda_i^-(M)|(b_i^-)^2 + \sum_j |\lambda_j^+(M)|(b_j^+)^2 \quad (6.6) $$

and $\sum_i (b_i^-)^2 + \sum_j (b_j^+)^2 < \infty$.

Since we can assume that $\lambda_1(N) = \lambda_k^+(M)$ and $\kappa_1 = \phi_k^+$ for some $k$, we have $\sum_i (b_i^-)^2 + \sum_j (b_j^+)^2 \geq 1$ and

$$ Q_M[\kappa_1, \kappa_1] \geq |\lambda_1^-(M)|(b_1^-)^2 + [1 - (b_1^-)^2]\lambda_1^+(M). $$

This implies,

$$ \begin{aligned} Q_M[\kappa_1, \kappa_1] - |\lambda_1^-(M)| &\geq [1 - (b_1^-)^2](\lambda_1^+(M) - |\lambda_1^-(M)|) \\ &\geq (1 - |b_1^-|)(1 + |b_1^-|)(\lambda_1^+(M) - |\lambda_1^-(M)|) \end{aligned} \quad (6.7) $$

By (6.5) and (6.7),

$$ \begin{aligned} & \lambda_1(N) (|\lambda_1^-(M)| + \lambda_1^+(M)) - |\lambda_1^-(M)| (|\lambda_1^-(M)| + \lambda_1^+(M)) \\ &\geq \operatorname{Cap}^-(M\setminus N) (\lambda_1^+(M) - |\lambda_1^-(M)|) \quad \text{if} \quad |\lambda_1^-(M)| \leq \lambda_1^+(M) \leq |\lambda_2^-(M)|. \end{aligned} $$

Which proves (6.4). Similarly we can show the other inequalities related to (1.8).
## 6.2. Proof of Theorem 1.10 (ii). From (1.9), we have

$$ 
\begin{aligned}
\mu_1 &:= (1 - \ell) - (\lambda_1(N))^{-1} C_\nu(\Gamma) - 2(\lambda_1(N))^{-1/2} (C_\nu(\Gamma))^{1/2} > 0, \\
\mu_2 &:= 1 - \frac{\mathrm{Cap}^\pm(M \setminus N)}{\min\{|\lambda_1^-(M)|, \lambda_1^+(M)\}} - 2|\lambda_1^\pm(M)|^{-1/2} (\mathrm{Cap}^\pm(M \setminus N))^{1/2} > 0.
\end{aligned} 
\quad (6.8) $$

*Steep 1. One has the inequality*

$$ \lambda_1(\Omega) - \lambda_1(N) \le \mu_1^{-1} \left[ \lambda_1(N)\ell + 2C_\nu(\Gamma) + 4(\lambda_1(N))^{1/2}(C_\nu(\Gamma))^{1/2} \right]. \quad (6.9) $$

Indeed. Let $u_i \in W^{1,2}(N; \mathbf{V}_0, \mathbf{V}_1)$ such that $0 < Q_N[u_i, u_i] \le C_\nu(\Gamma) + 1/i$, $i \in \mathbb{N}$. Denote $\psi_i := u_i - \kappa_1$. Then

$$ \int_N \tau \psi_i^2 d\mathbf{V}_2 = \int_N \tau \kappa_1^2 d\mathbf{V}_2 + \int_N \tau u_i^2 d\mathbf{V}_2 - 2 \int_N \tau \kappa_1 u_i d\mathbf{V}_2. $$

Hence,

$$ \left| \int_N \tau \psi_i^2 d\mathbf{V}_2 - 1 \right| \le (\lambda_1(N))^{-1} (C_\nu(\Gamma) + 1/i) + 2(\lambda_1(N))^{-1/2} (C_\nu(\Gamma) + 1/i)^{1/2}, \quad (6.10) $$

since $\lambda_1(N) \le Q_N[u_i, u_i]/(\tau u_i, u_i)_{N, \mathbf{V}_2}$.

Also, we have

$$ Q_N[\psi_i, \psi_i] = Q_N[\kappa_1, \kappa_1] - 2Q_N[\kappa_1, u_i] + Q_N[u_i, u_i]. $$

This yields

$$ Q_\Omega[\psi_i, \psi_i] \le \lambda_1(N) + 2(\lambda_1(N))^{1/2}(C_\nu(\Gamma) + 1/i)^{1/2} + (C_\nu(\Gamma) + 1/i) \quad (6.11) $$

From (6.8), (6.10), (6.11), and $\lambda_1(\Omega) \le Q_\Omega[\psi_i, \psi_i]/(\tau \psi_i, \psi_i)_{\Omega, \mathbf{V}_2}$, we conclude (6.9).

*Steep 2. The following inequality holds:*

$$ \begin{aligned} \lambda_1(N) - |\lambda_1^\pm(M)| & \le \mu_2^{-1} \left[ \left( 1 + \frac{|\lambda_1^\pm|}{\min\{|\lambda_1^\mp|, \lambda_1^+\}} \right) \mathrm{Cap}^\pm(M \setminus N) + 4|\lambda_1^\pm(M)|^{1/2} (\mathrm{Cap}^\pm(M \setminus N))^{1/2} \right]. \end{aligned} \quad (6.12) $$

Let $u_i^\pm \in W^{1,2}(M; \mathbf{V}_0, \mathbf{V}_1)$ such that $u_i^\pm - \phi_1^\pm = 0$ in $M \setminus N$, and $0 < Q_M[u_i^\pm, u_i^\pm] \le \mathrm{Cap}^\pm(M \setminus N) + 1/i$, $i \in \mathbb{N}$.

From [23, Lemma 1.1 and 1.2], for all $u \in W^{1,2}(M; \mathbf{V}_0, \mathbf{V}_1)$ with $Q_M[u, u] = 1$:

$$ \frac{1}{\lambda_1^-(M)} \le \int_M \tau u^2 d\mathbf{V}_2 \le \frac{1}{\lambda_1^+(M)}. \quad (6.13) $$

Denote $\psi_i^\pm := u_i^\pm - \phi_1^\pm$. Then

$$ \int_N \tau (\psi_i^\pm)^2 d\mathbf{V}_2 = \int_M \tau (\psi_i^\pm)^2 d\mathbf{V}_2 = \mathrm{sign}(\lambda_1^\pm(M)) 1 + \int_M \tau (u_i^\pm)^2 d\mathbf{V}_2 - 2 \int_M \tau \phi_1^\pm u_i^\pm d\mathbf{V}_2. \quad (6.14) $$

From (6.13) and $|\lambda_1^\pm(M)(\tau \phi_1^\pm, u_i^\pm)_{M, \mathbf{V}_2}| = |Q_M[\phi_1^\pm, u_i^\pm]| \le \sqrt{Q_M[\phi_1^\pm, \phi_1^\pm]} \sqrt{Q_M[u_i^\pm, u_i^\pm]}$, we have

$$ \left| \int_N \tau (\psi_i^\pm)^2 d\mathbf{V}_2 - \mathrm{sign}(\lambda_1^\pm(M)) 1 \right| \le \max\{|\lambda_1^-(M)|^{-1}, |\lambda_1^+(M)|^{-1}\} (\mathrm{Cap}^\pm(M \setminus N) + 1/i) + 2|\lambda_1^\pm(M)|^{-1/2} (\mathrm{Cap}^\pm(M \setminus N) + 1/i)^{1/2}. \quad (6.15) $$

Using

$$ Q_N[\psi_i^\pm, \psi_i^\pm] = Q_M[\psi_i^\pm, \psi_i^\pm] = Q_M[\phi_1^\pm, \phi_1^\pm] - 2Q_M[\phi_1^\pm, u_i^\pm] + Q_M[u_i^\pm, u_i^\pm], $$
we obtain

$$
\begin{align}
\mathcal{Q}_N[\psi_i^{\pm}, \psi_i^{\pm}] &\le |\lambda_1^{\pm}(M)| + 2|\lambda_1^{\pm}(M)|^{1/2} (\mathrm{Cap}^{\pm}(M\setminus N) + 1/i)^{1/2} + (\mathrm{Cap}^{\pm}(M\setminus N) + 1/i). \tag{6.16} \\
\text{From (6.8), (6.15), (6.16), and } \lambda_1(N) &\le \mathcal{Q}_N[\psi_i, \psi_i]/(\tau\psi_i, \psi_i)_{N,\mathcal{V}_2}, \text{ we conclude (6.12).}
\end{align}
$$

APPENDIX A. PROOF OF LEMMA 4.2

Similarly to Lemma 3.1 we have the

**Lemma A.1.** Suppose $0 < r < R \le \hat{r}_{1,j}$ and $j \in \mathbb{N}$ fixed. There exist a smooth function $\zeta : \hat{U}_{1,j} \to [0, 1]$ such that $\zeta = 1$ in $\psi_{1,j}(B(0, r))$, $\text{supp} \zeta \subset \psi_{1,j}(B(0, R))$, and $|\nabla \zeta|_g \le C_1(n) \|d(\psi_{1,j}^{-1})\|(R-r)^{-1}$ in $\hat{U}_{1,j}$.

We will follow the proof in [30, Theorem 4.1]; see also [5, Lemma A.2].

**Claim 1.** We have

$$
\int_{\hat{U}_{1,j} \cap \Omega} |\nabla(u_{h_2}\zeta)|^2 dV_1 \\
\le C_1(n) \left[ \frac{(\hat{r}_{1,j} - r_{1,j})^2}{(R-r)^2} \mathbf{K}_2 \int_{\mathcal{U}(h_2,R)} u_{h_2}^2 dV_2 + \int_{\hat{U}_{1,j} \cap \Omega} g(\nabla u, \nabla v_{h_2}) dV_1 \right].
$$

*Proof.* We have

$$
\begin{align*}
\int_{\hat{U}_{1,j} \cap \Omega} g(\nabla u, \nabla v_{h_2}) \, dV_1 &= \int_{\hat{U}_{1,j} \cap \Omega} (\zeta^2 g(\nabla u, \nabla u_{h_2}) + 2u_{h_2} \zeta g(\nabla u, \nabla \zeta)) \, dV_1 \\
&\geq \int_{\hat{U}_{1,j} \cap \Omega} \zeta^2 |\nabla u_{h_2}|^2 \, dV_1 - \int_{\hat{U}_{1,j} \cap \Omega} 2|\zeta| |\nabla u_{h_2}| |\nabla \zeta| |u_{h_2}| \, dV_1 \\
&\geq \frac{1}{2} \int_{\hat{U}_{1,j} \cap \Omega} \zeta^2 |\nabla u_{h_2}|^2 \, dV_1 - 2 \int_{\hat{U}_{1,j} \cap \Omega} |\nabla \zeta|^2 u_{h_2}^2 \, dV_1.
\end{align*}
$$

From Lemma A.1 and (W₄), we conclude the claim.

□

The inequality (4.1) and the Claim 1 implies

$$
\min\{1, C_1\} \mathcal{Q}[u_{h_2}\zeta, u_{h_2}\zeta]
$$

$$
\le \int_{\hat{U}_{1,j} \cap \Omega} |\nabla(u_{h_2}\zeta)|^2 dV_1 + C_1 \int_{\hat{U}_{1,j} \cap \Omega} uv_{h_2} dV_0 \le C_1 \left[ \frac{(\hat{r}_{1,j} - r_{1,j})^2}{(R-r)^2} \mathbf{K}_2 \int_{\mathcal{U}(h_2,R)} u_{h_2}^2 dV_0 \right. \\
\left. - \int_{\hat{U}_{1,j} \cap \Omega} c_2 uv_{h_2} dV_2 - \int_{\hat{U}_{1,j} \cap \Gamma} c_3 uv_{h_2} dW + \int_{\hat{U}_{1,j} \cap \Omega} fv_{h_2} dV_2 + \int_{\hat{U}_{1,j} \cap \Gamma} f_1 v_{h_2} dW \right]. \quad \text{(A.1)}
$$

Next, we will estimate the terms on the right-hand side of (A.1).

**Claim 2.** The following estimates are valid:

(i) One has

$$
\begin{align*}
& - \int_{\hat{U}_{1,j} \cap \Gamma} c_3 u (u_{h_2} \zeta^2) dW \\
&\le 2 C_{2\mathbf{W}}^2 \|c_3\|_{q_3, \hat{U}_{1,j} \cap \Gamma, \mathbf{W}} \mathbf{W}(\{u_{h_2} \zeta \ne 0\} \cap \hat{U}_{1,j} \cap \Gamma)^{\frac{2\mathbf{W}-2}{2\mathbf{W}} - \frac{1}{q_3}} \mathcal{Q}[u_{h_2} \zeta, u_{h_2} \zeta] \\
&\quad + h_2^2 \|c_3\|_{q_3, \hat{U}_{1,j} \cap \Gamma, \mathbf{W}} \mathbf{W}(\{u_{h_2} \zeta \ne 0\} \cap \hat{U}_{1,j} \cap \Gamma)^{1-\frac{1}{q_3}}.
\end{align*}
$$

(ii) For all $\delta > 0$,

$$
\int_{\hat{U}_{1,j} \cap \Gamma} f_1 u_{h_2} \zeta^2 dW_g \\
\le C_{2\mathbf{W}} \left( \delta^{-1} \mathbf{W}(\{u_{h_2} \zeta \ne 0\} \cap \hat{U}_{1,j} \cap \Gamma)^{\frac{2(2\mathbf{W}-1)}{2\mathbf{W}} - \frac{2}{q_3}} \|f_1\|_{q_3, \hat{U}_{1,j} \cap \Gamma, \mathbf{W}}^2 + \delta \mathcal{Q}[u_{h_2} \zeta, u_{h_2} \zeta] \right).
$$
(iii) One has

$$
\begin{align*}
& - \int_{\hat{U}_{1,j} \cap \Omega} c_2 u(u_{h2} \zeta^2) dV_2 \\
& \le 2 C_{2v}^2 \|c_2\|_{q_2, \hat{U}_{1,j} \cap \Omega, v_2} V_2(\{u_{h2} \zeta \ne 0\})^{\frac{2\gamma-2}{2v} - \frac{1}{q_2}} Q[u_{h2} \zeta, u_{h2} \zeta] \\
& \phantom{\le} + h_2^2 \|c_2\|_{q_2, \hat{U}_{1,j} \cap \Omega, v_2} V_2(\{u_{h2} \zeta \ne 0\})^{1-\frac{1}{q_2}}.
\end{align*}
$$

(iv) For all $\delta > 0$,

$$
\int_{\hat{U}_{1,j} \cap \Omega} f u_{h2} \zeta^2 dV_2 \\
\le C_{2v} \left( \delta^{-1} V_2(\{u_{h2} \zeta \ne 0\})^{\frac{2(2\gamma-1)}{2v} - \frac{2}{q_2}} \|f\|_{q_2, \hat{U}_{1,j} \cap \Omega, v_2}^2 + \delta Q[u_{h2} \zeta, u_{h2} \zeta] \right).
$$

Proof. We will start with the following inequalities, which will be used later. Hölder’s inequality and
(1.5) imply

$$
\begin{align}
\int_{\hat{U}_{1,j} \cap \Gamma} (u_{h2}\zeta)^2 dW &\le W(\{u_{h2}\zeta \ne 0\} \cap \hat{U}_{1,j} \cap \Gamma)^{1-\frac{2}{2w}} \left( \int_{\hat{U}_{1,j} \cap \Gamma} |u_{h2}\zeta|^{2w} dW \right)^{\frac{2}{2w}} \tag{A.2} \\
&\le C_{2w}^2 W(\{u_{h2}\zeta \ne 0\} \cap \hat{U}_{1,j} \cap \Gamma)^{1-\frac{2}{2w}} Q[u_{h2}\zeta, u_{h2}\zeta]. \nonumber
\end{align}
$$

On the other hand, by (1.4),

$$
\begin{align}
\int_{\hat{U}_{1,j} \cap \Omega} (u_{h2}\zeta)^2 dV_2 &\le V_2(\{u_{h2}\zeta \ne 0\})^{1-\frac{2}{2\gamma}} \left( \int_{\hat{U}_{1,j} \cap \Omega} |u_{h2}\zeta|^{2\gamma} dV_2 \right)^{\frac{2}{2\gamma}} \tag{A.3} \\
&\le C_{2\gamma}^2 V_2(\{u_{h2}\zeta \ne 0\})^{1-\frac{2}{2\gamma}} Q[u_{h2}\zeta, u_{h2}\zeta]. \notag
\end{align}
$$

(i) Since $2/2w + 1/q_3 < 1$,

$$
\begin{align*}
- \int_{\hat{U}_{1,j} \cap \Gamma} c_3 u (u_{h2} \zeta^2) dW &= - \int_{\{u_{h2} \zeta \ne 0\} \cap \hat{U}_{1,j} \cap \Gamma} c_3 (u_{h2}^2 + h_2 u_{h2}) \zeta^2 dW \\
&\le 2 \int_{\{u_{h2} \zeta \ne 0\} \cap \hat{U}_{1,j} \cap \Gamma} |c_3| u_{h2}^2 \zeta^2 dW + h_2^2 \int_{\{u_{h2} \zeta \ne 0\} \cap \hat{U}_{1,j} \cap \Gamma} |c_3| \zeta^2 dW \\
&\le 2 \left( \int_{\hat{U}_{1,j} \cap \Gamma} |c_3|^{q_3} dW \right)^{\frac{1}{q_3}} \left( \int_{\{u_{h2} \zeta \ne 0\} \cap \hat{U}_{1,j} \cap \Gamma} |u_{h2} \zeta|^{2w} dW \right)^{\frac{2}{2w}} \\
&\quad \cdot W(\{u_{h2} \zeta \ne 0\} \cap \hat{U}_{1,j} \cap \Gamma)^{1-\frac{2}{2w}-\frac{1}{q_3}} \\
&\quad + h_2^2 \|c_3\|_{q_3, \hat{U}_{1,j} \cap \Gamma, w} W(\{u_{h2} \zeta \ne 0\} \cap \hat{U}_{1,j} \cap \Gamma)^{1-\frac{1}{q_3}} \\
&\le 2 C_{2w}^2 \|c_3\|_{q_3, \hat{U}_{1,j} \cap \Gamma, w} W(\{u_{h2} \zeta \ne 0\} \cap \hat{U}_{1,j} \cap \Gamma)^{\frac{2w-2}{2w}-\frac{1}{q_3}} Q[u_{h2} \zeta, u_{h2} \zeta] \\
&\quad + h_2^2 \|c_3\|_{q_3, \hat{U}_{1,j} \cap \Gamma, w} W(\{u_{h2} \zeta \ne 0\} \cap \hat{U}_{1,j} \cap \Gamma)^{1-\frac{1}{q_3}},
\end{align*}
$$

by (A.2). This conclude the proof of (i).
(ii) Since $1/q_3 + 1/2_w < 1$ and $0 \le \zeta \le 1$,

$$ \int_{\hat{U}_{1,j} \cap \Gamma} f_1 u_{h_2} \zeta^2 dW \\ \le \left( \int_{\hat{U}_{1,j} \cap \Gamma} |f_1|^{q_3} dW \right)^{\frac{1}{q_3}} \left( \int_{\hat{U}_{1,j} \cap \Gamma} |u_{h_2} \zeta|^{2w} dW \right)^{\frac{1}{2w}} W(\{u_{h_2} \zeta \neq 0\} \cap \hat{U}_{1,j} \cap \Gamma)^{1-\frac{1}{q_3}-\frac{1}{2w}}. $$

By (A.2),

$$ \int_{\hat{U}_{1,j} \cap \Gamma} |f_1| u_{h_2} \zeta^2 dW \\ \le C_{2w} \|f_1\|_{q_3, \hat{U}_{1,j} \cap \Gamma, w} \sqrt{Q[u_{h_2} \zeta, u_{h_2} \zeta]} W(\{u_{h_2} \zeta \neq 0\} \cap \hat{U}_{1,j} \cap \Gamma)^{\frac{2w-1}{2w} - \frac{1}{q_3}} \\ \le C_{2w} \delta^{-1} W(\{u_{h_2} \zeta \neq 0\} \cap \hat{U}_{1,j} \cap \Gamma)^{\frac{2(2w-1)}{2w} - \frac{2}{q_3}} \|f_1\|_{q_3, \hat{U}_{1,j} \cap \Gamma, w}^2 + C_{2w} \delta Q[u_{h_2} \zeta, u_{h_2} \zeta]. $$

This proves (ii). The proofs of (iii) and (iv) follow the same lines as in (i) and (ii). $\square$

Let us observe that $1 - 1/q_2 \le 2(2v - 1)/2v - 2/q_2$, $1 - 1/q_3 \le 2(2w - 1)/2w - 2/q_3$, $\{u_{h_2} \zeta \ne 0\} \subset U(h_2, R)$, $\{u_{h_2} \zeta \ne 0\} \cap \hat{U}_{1,j} \cap \Gamma \subset U_\Gamma(h_2, R)$, $V_2(U(h_2, R)) \le h_2^{-1} \int_{U(h_2, R)} u^+ dV_2$, and $W(U_\Gamma(h_2, R)) \le h_2^{-1} \int_{U_\Gamma(h_2, R)} u^+ dW$.

Hence, by (A.1) and Claim 2, there exists a constant $N = N(n, C_{2v}, C_{2w}, 2v, 2w, q_2, q_3) > 0$ such that if

$$ h_2 \ge N \max \left\{ \|c_2\|_{q_2, \hat{U}_{1,j} \cap \Omega, v_2}^{q_2 2v / [q_2(2v-2)-2v]}, \|c_3\|_{q_3, \Gamma \cap \hat{U}_{1,j}, w}^{q_3 2w / [q_3(2w-2)-2w]}, 1 \right\} \max \{ \|u^+\|_{2, \hat{U}_{1,j} \cap \Omega, v_2}, \|u^+\|_{2, \hat{U}_{1,j} \cap \Gamma, w} \}, $$

then

$$ V_2(U(h_2, R)) < 1, \quad W(U_\Gamma(h_2, R)) < 1 \qquad (A.4) $$

and

$$ C_{2v} Q[u_{h_2} \zeta, u_{h_2} \zeta] \\ \le \frac{(\hat{r}_{1,j} - r_{1,j})^2}{(R-r)^2} K_2 \int_{U(h_2, R)} u_{h_2}^2 dV_2 + (\|f\|_{q_2, \hat{U}_{1,j} \cap \Omega, v_2}^2 + h_2^2) V_2(\{u_{h_2} \zeta \neq 0\})^{1-\frac{1}{q_2}} \quad (A.5) \\ + (\|f_1\|_{q_3, \hat{U}_{1,j} \cap \Gamma, w}^2 + h_2^2) W(\{u_{h_2} \zeta \neq 0\} \cap \hat{U}_{1,j} \cap \Gamma)^{1-\frac{1}{q_3}}, $$

where $C_2 = C_2(n, C_{2v}, C_{2w}) > 0$.

From (A.3) and (A.5), we have

$$ C_3 \int_{\hat{U}_{1,j} \cap \Omega} (u_{h_2} \zeta)^2 dV_2 \\ \le \frac{(\hat{r}_{1,j} - r_{1,j})^2}{(R-r)^2} V_2(\{u_{h_2} \zeta \neq 0\})^{1-\frac{2}{2v}} \left( \int_{U(h_2, R)} u_{h_2}^2 dV_2 + \int_{U_\Gamma(h_2, R)} u_{h_2}^2 dW \right) \quad (A.6) \\ + (\|f\|_{q_2, \hat{U}_{1,j} \cap \Omega, v_2} + \|f_1\|_{q_3, \hat{U}_{1,j} \cap \Gamma, w} + h_2)^2 \\ \cdot \left( V_2(\{u_{h_2} \zeta \neq 0\})^{1-\frac{2}{2v}} + \frac{1}{q_2} + V_2(\{u_{h_2} \zeta \neq 0\})^{1-\frac{2}{2v}} W(\{u_{h_2} \zeta \neq 0\} \cap \hat{U}_{1,j} \cap \Gamma) \right)^{1-\frac{1}{q_3}}. $$
and, by (A.2),

$$
\begin{align}
& C_4 \int_{\hat{U}_{1,j} \cap \Gamma} (u_{h_2} \zeta)^2 \mathrm{d}W \nonumber \\
& \le \frac{(\hat{r}_{1,j} - r_{1,j})^2}{(R-r)^2} W(\{u_{h_2} \zeta \ne 0\} \cap \hat{U}_{1,j} \cap \Gamma)^{1-\frac{2}{2w}} \left( \int_{\mathcal{U}(h_2,R)} u_{h_2}^2 \mathrm{d}V_2 + \int_{\mathcal{U}_{\Gamma}(h_2,R)} u_{h_2}^2 \mathrm{d}W \right) \tag{A.7} \\
& \quad + (\|f\|_{q_2, \hat{U}_{1,j} \cap \Omega, v_2} + \|f_1\|_{q_3, \hat{U}_{1,j} \cap \Gamma, w} + h_2)^2 \nonumber \\
& \quad \cdot \left( W(\{u_{h_2} \zeta \ne 0\} \cap \hat{U}_{1,j} \cap \Gamma)^{1-\frac{2}{2w}} V_2(\{u_{h_2} \zeta \ne 0\})^{1-\frac{1}{q_2}} \right. \nonumber \\
& \qquad \left. + W(\{u_{h_2} \zeta \ne 0\} \cap \hat{U}_{1,j} \cap \Gamma)^{1-\frac{2}{2w}+1-\frac{1}{q_3}} \right), \nonumber
\end{align}
$$

where $C_i = C_i(n, C_{2v}, C_{2w}, K_2) > 0$, $i = 3, 4$.

On the other hand. Set $\epsilon = \min\{1-1/q_2-2/2w, 1-1/q_3-2/2v, 1-1/q_2-2/2v, 1-1/q_3-2/2w\}$, by Young's inequality,

$$
\begin{align}
& V_2(\{u_{h_2} \zeta \neq 0\})^{1-\frac{2}{2v}} W(\{u_{h_2} \zeta \neq 0\} \cap \hat{U}_{1,j} \cap \Gamma)^{1-\frac{1}{q_3}} \nonumber \\
& \leq \frac{1}{C_5} V_2(\{u_{h_2} \zeta \neq 0\})^{1+\epsilon} + \frac{C_5-1}{C_5} W(\{u_{h_2} \zeta \neq 0\} \cap \hat{U}_{1,j} \cap \Gamma)^{(1-\frac{1}{q_3}) \frac{C_5}{C_5-1}}, \tag{A.8}
\end{align}
$$

$$
\begin{align}
& V_2(\{u_{h_2} \zeta \neq 0\})^{1-\frac{1}{q_2}} W(\{u_{h_2} \zeta \neq 0\} \cap \hat{U}_{1,j} \cap \Gamma)^{1-\frac{2}{2w}} \nonumber \\
& \leq \frac{C_6-1}{C_6} V_2(\{u_{h_2} \zeta \neq 0\})^{(1-\frac{1}{q_2}) \frac{C_6}{C_6-1}} + \frac{1}{C_6} W(\{u_{h_2} \zeta \neq 0\} \cap \hat{U}_{1,j} \cap \Gamma)^{1+\epsilon}, \tag{A.9}
\end{align}
$$

where $C_5(1 - 2/2v) = 1 + \epsilon$ and $C_6(1 - 2/2w) = 1 + \epsilon$. Observe also that

$$
\left(1 - \frac{1}{q_3}\right) \frac{C_5}{C_5 - 1} \geq 1 + \epsilon \quad \text{and} \quad \left(1 - \frac{1}{q_2}\right) \frac{C_6}{C_6 - 1} \geq 1 + \epsilon.
$$

The inequalities (A.4), (A.6) - (A.9) imply

$$
\begin{align}
C_7 \Psi^2(h_2, r) \le \frac{(\hat{r}_{1,j} - r_{1,j})^2}{(R-r)^2} & (V_2(\{u_{h_2} \zeta \ne 0\}) + W(\{u_{h_2} \zeta \ne 0\} \cap \hat{U}_{1,j} \cap \Gamma))^{\epsilon} \Psi(h_2, R)^2 \nonumber \\
& + (\|f\|_{q_2, \hat{U}_{1,j} \cap \Omega, v_2} + \|f_1\|_{q_3, \hat{U}_{1,j} \cap \Gamma, w} + h_2)^2 (V_2(\{u_{h_2} \zeta \ne 0\}) + W(\{u_{h_2} \zeta \ne 0\} \cap \hat{U}_{1,j} \cap \Gamma))^{1+\epsilon}, \tag{A.10}
\end{align}
$$

where $C_7 = C_7(n, C_{2v}, C_{2w}, 2v, 2w, q_2, q_3, K_2) > 0$. Consider (A.10) and the following claim, which is proved as in [30]:

**Claim 3.** If $h_2 > h_1$, then

$$
\begin{align*}
& \int_{\mathcal{U}(h_2,R)} u_{h_2}^2 \mathrm{d}V_2 \le \int_{\mathcal{U}(h_1,R)} u_{h_1}^2 \mathrm{d}V_2, \quad \int_{\mathcal{U}_{\Gamma}(h_2,R)} u_{h_2}^2 \mathrm{d}W \le \int_{\mathcal{U}_{\Gamma}(h_1,R)} u_{h_1}^2 \mathrm{d}W, \\
& V_2(\{u_{h_2} \zeta \ne 0\}) \le \frac{1}{(h_2-h_1)^2} \int_{\mathcal{U}(h_1,R)} u_{h_1}^2 \mathrm{d}V_2, \\
& W(\{u_{h_2} \zeta \ne 0\} \cap \hat{U}_{1,j} \cap \Gamma) \le \frac{1}{(h_2-h_1)^2} \int_{\mathcal{U}_{\Gamma}(h_1,R)} u_{h_1}^2 \mathrm{d}W.
\end{align*}
$$

Therefore,

$$
C_7 \Psi^2(h_2, r) \le \left[ \frac{(\hat{r}_{1,j} - r_{1,j})^2}{(R-r)^2 (h_2 - h_1)^{2\epsilon}} + \frac{(\|f\|_{q_2, \hat{U}_{1,j} \cap \Omega, v_2} + \|f_1\|_{q_3, \hat{U}_{1,j} \cap \Gamma, w} + h_2)^2}{(h_2 - h_1)^{2(1+\epsilon)}} \right] \Psi^{2(1+\epsilon)}(h_1, R).
$$

This proves Lemma 4.2.
APPENDIX B. EXISTENCE OF SOLUTIONS

Recall that $D_m, m \in \mathbb{N}$, denote an increasing sequence of open bounded domains in $M$ such that $M = \cup_m D_m$ and $\bar{D}_m \subset D_{m+1}$ for all $m \in \mathbb{N}$. We write $\Omega_m := D_m \cap \Omega$, $\Omega^m := \Omega \setminus \bar{\Omega}_m$, $\Gamma_m := D_m \cap \Gamma$, and $\Gamma^m := \Gamma \setminus \bar{\Gamma}_m$.

Let $\lambda \in \mathbb{R}$. Let $c \in \mathbb{R}$ such that $\lambda(\tau + c) \ge \text{sign}(\lambda)\lambda\|\tau\|_{L^\infty(\Omega_m)}$ and $\lambda c > 0$. Denote

$$ \left\{ \begin{array}{ll} L_m u := -\text{div}(\mathbf{V}_1 u) + \mathbf{V}_0 u + \mathbf{V}_2 \lambda c & \text{in } \Omega_m \\ B_m u := \mathbf{V}_1 \frac{\partial u}{\partial \nu} & \text{on } \Gamma_m \end{array} \right. \qquad (B.1) $$

The bilinear form associated to (B.1) is

$$ Q_m[u, v] = \int_{\Omega_m} g(\nabla u, \nabla v) d\mathbf{V}_1 + \int_{\Omega_m} uv d\mathbf{V}_0 + \lambda c \int_{\Omega_m} uv d\mathbf{V}_2, $$

where $u, v \in W|_{\Gamma} := \{u \in W^{1,2}(\Omega_m; \mathbf{V}_0, \mathbf{V}_1) \mid u_{\Gamma_m \setminus \Gamma} = 0\}$.

**Proposition B.1.** Suppose $\tau \in L^\infty(\Omega_m)$. Write $\tau_c := \tau + c$.

(i) Let $\lambda \in \mathbb{R}$. Precisely one of the following statements holds: either

$$ \left\{ \begin{array}{ll} \text{for each } f_0 \in L^2(\Omega_m; \mathbf{V}_0) \text{ and } f_1 \in L^2(\Gamma_m; \mathbf{W}) \text{ there exists a unique} \\ \text{weak solution } u \in W_{\Gamma} \text{ of the problem:} \\ \begin{cases} L_m u = \lambda \mathbf{V}_2 \tau_c u + \mathbf{V}_2 f_0 & \text{in } \Omega_m, \\ B_m u = \mathbf{W}_1 f_1 & \text{on } \Gamma_m, \end{cases} \end{array} \right. \qquad (B.2) $$

or else

$$ \left\{ \begin{array}{ll} \text{there exists a weak solution } u \in W_{\Gamma}, \text{ with } u \not\equiv 0, \text{ of} \\ \text{the problem} \\ \begin{cases} L_m u = \lambda \mathbf{V}_2 \tau_c u & \text{in } \Omega_m \\ B_m u = 0 & \text{on } \Gamma_m. \end{cases} \end{array} \right. \qquad (B.3) $$

(ii) The problem (B.2) has a weak solution if and only if

$$ \int_{\Omega_m} f_0 v d\mathbf{V}_2 + \int_{\Gamma_m} f_1 v d\mathbf{W}_1 = 0 \quad \forall v \in E_{\lambda}, $$

where $E_{\lambda} := \{u \in W|_{\Gamma} \mid u \text{ solves (B.3)}\}$.

*Proof.* (i) Steep 1. For each $f_0 \in L^2(\Omega_m; \mathbf{V}_2)$ and $f_1 \in L^2(\Gamma_m; \mathbf{W}_1)$ there exists a unique function $u \in W_{\Gamma}$ solving

$$ Q_m[u, v] = \int_{\Omega_m} f_0 v d\mathbf{V}_2 + \int_{\Gamma_m} f_1 v d\mathbf{W}_1 \quad \forall v \in W_{\Gamma}. \qquad (B.4) $$

Let us write

$$ u = L^{-1}(f_0, f_1) $$

whenever (B.4) holds.

Observe next $u \in W|_{\Gamma}$ is a weak solution of (B.2) if and only if

$$ Q_m[u, v] = \int_{\Omega_m} \lambda \tau_c u v d\mathbf{V}_2 + \int_{\Omega_m} f_0 v d\mathbf{V}_2 + \int_{\Gamma_m} f_1 v d\mathbf{W}_1 \quad \forall v \in W|_{\Gamma}, $$

that is, if and only if

$$ u = L^{-1}(\lambda \tau_c u + f_0, f_1). $$

We rewrite this equality to read

$$ u - K(u) = h $$

for

$$ K(u) := \lambda L^{-1}(\tau_c u, 0) \quad \text{and} \quad h := L^{-1}(f_0, f_1). $$
**Steep 2.** In $L^2(\Omega_m; V_2)$ we consider the norm

$$ \|u\|_{\tau_c} = \left( \int_{\Omega_m} \tau_c u^2 dV_2 \right)^{\frac{1}{2}} . $$

Since in $\Omega_m$ we have the estimate

$$ V_2 \le C_1(\Omega_m, V_0, V_2)V_0. \qquad (B.5) $$

we obtain that $K: L^2(\Omega_m; V_2) \to L^2(\Omega_m; V_2)$ is a bounded, linear, compact operator.

We may consequently apply the Fredholm alternative: either

$$ \left\{ \begin{array}{l} \text{for each } h \in L^2(\Omega_m, V_2) \text{ the equation} \\ u - K(u) = h \\ \text{has a unique solution } u \in L^2(\Omega_m, V_2) \end{array} \right. $$

or else

$$ \left\{ \begin{array}{l} \text{the equation} \\ u - K(u) = 0 \\ \text{has nonzero solutions in } L^2(\Omega_m, V_2). \end{array} \right. $$

Which proves (i).

(ii) We have $K$ is symmetric. Following the argument used in [27, Second Existence Theorem for weak solutions], we conclude the proof. $\square$

**Funding:** This work was supported by Instituto Nacional de Ciência e Tecnologia de Matemática (INCT-Mat), CNPq, Grant 170245/2023-3,

**Data Availability:** No data was used for the research described in the article.

**Declarations**

**Conflict of interest:** The author declare no conflict of interest.

**Declaration of generative AI and AI-assisted technologies in the writing process:** During the preparation of this work, the authors used the AI ChatGPT and the AI Gemini to correct grammar and orthography in the text. After using this tool/service, the authors reviewed and edited the content as needed and takes full responsibility for the content of the publication.

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JUAN A. APAZA, DEPARTAMENTO DE MATEMÁTICA, UNIVERSIDADE FEDERAL DE SÃO CARLOS, 13565-905, SÃO CARLOS-SP, BRAZIL

Email address: juanpabloalconapaza@gmail.com