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

TL;DR

This paper investigates weighted Sobolev spaces, operator properties, and eigenvalue problems for elliptic equations with $L^1$ data on Riemannian manifolds, extending existence results to less regular data.

Contribution

It introduces new analysis of weighted Sobolev spaces and eigenvalue problems for elliptic equations with $L^1$ data, including existence and limit behavior results.

Findings

Continuity and compactness of certain operators in weighted Sobolev spaces established.

Existence results for elliptic equations with $L^1$ data extended to Riemannian manifolds.

Limit behavior of solutions with zero data analyzed.

Abstract

The aim of this work is to study the continuity and compactness of the operators $W^{1, q}(\Omega ; \mathtt {V}_0, \mathtt {V}_1 ) \rightarrow L^{q_0} (\Omega ; \mathtt {V}_2)$ and $W^{1, q} (\Omega ; \mathtt {V}_0, \mathtt {V}_1 ) \rightarrow L^{q_1}(\partial \Omega ; \mathtt {W})$ in weighted Sobolev spaces. To study additional properties of these Sobolev spaces, we will also study the equation: $$ \left\{\begin{aligned} -\operatorname{div}\left(\mathtt {V}_1 \nabla u\right)+\mathtt {V}_0 u & =\lambda \mathtt {V}_2 \tau u+\mathtt {V}_2 f_0 & & \text { in } \Omega, \\ \mathtt {V}_1 \frac{\partial u}{\partial \nu} & =\mathtt {W}_1 f_1 & & \text { on } \partial \Omega, \end{aligned}\right. $$ where $\Omega$ is an open subset of a Riemannian manifold, $\lambda$ is a real number, $f_0 \in L^1 (\Omega ; \mathtt {V}_0), f_1 \in L^1(\partial \Omega ; \mathtt {W})$, $\tau$ is a function that changes sign, and $\mathtt {V}_i, \mathtt {W}, \mathtt {W}_1$ are weight functions satisfying suitable conditions. We aim to obtain existence results similar to those for the case where the data are given in $L^2 (\Omega ; \mathtt {V}_0)$ and $L^2(\partial \Omega ; \mathtt {W})$. For the case where $f_0=0$ and $f_1=0$, we are also interested in studying the limit ess $\sup _{\Omega \backslash \Omega_m}|u| \rightarrow 0$, where $\Omega_m$ is a sequence of open sets such that $\Omega_m \subset \Omega_{m+1}$.