Weighted Sobolev Spaces and an Eigenvalue Problem for an Elliptic Equation with $ L^1 $ Data
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| Formato: | Preprint |
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2024
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| _version_ | 1866908066585247744 |
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| author | Apaza, Juan Pablo Alcon |
| author_facet | Apaza, Juan Pablo Alcon |
| contents | The aim of this work is to study the continuity and compactness of the operators $W^{1, q}(Ω; \mathtt {V}_0, \mathtt {V}_1 ) \rightarrow L^{q_0} (Ω; \mathtt {V}_2)$ and $W^{1, q} (Ω; \mathtt {V}_0, \mathtt {V}_1 ) \rightarrow L^{q_1}(\partial Ω; \mathtt {W})$ in weighted Sobolev spaces. To study additional properties of these Sobolev spaces, we will also study the equation: $$ \left\{\begin{aligned} -\operatorname{div}\left(\mathtt {V}_1 \nabla u\right)+\mathtt {V}_0 u & =λ\mathtt {V}_2 τu+\mathtt {V}_2 f_0 & & \text { in } Ω, \\ \mathtt {V}_1 \frac{\partial u}{\partial ν} & =\mathtt {W}_1 f_1 & & \text { on } \partial Ω, \end{aligned}\right. $$ where $Ω$ is an open subset of a Riemannian manifold, $λ$ is a real number, $f_0 \in L^1 (Ω; \mathtt {V}_0), f_1 \in L^1(\partial Ω; \mathtt {W})$, $τ$ 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 $L^2 (Ω; \mathtt {V}_0)$ and $L^2(\partial Ω; \mathtt {W})$. For the case where $f_0=0$ and $f_1=0$, we are also interested in studying the limit ess $\sup _{Ω\backslash Ω_m}|u| \rightarrow 0$, where $Ω_m$ is a sequence of open sets such that $Ω_m \subset Ω_{m+1}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_00162 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weighted Sobolev Spaces and an Eigenvalue Problem for an Elliptic Equation with $ L^1 $ Data Apaza, Juan Pablo Alcon Analysis of PDEs 35J25, 46E35, 93B60, 35R05 The aim of this work is to study the continuity and compactness of the operators $W^{1, q}(Ω; \mathtt {V}_0, \mathtt {V}_1 ) \rightarrow L^{q_0} (Ω; \mathtt {V}_2)$ and $W^{1, q} (Ω; \mathtt {V}_0, \mathtt {V}_1 ) \rightarrow L^{q_1}(\partial Ω; \mathtt {W})$ in weighted Sobolev spaces. To study additional properties of these Sobolev spaces, we will also study the equation: $$ \left\{\begin{aligned} -\operatorname{div}\left(\mathtt {V}_1 \nabla u\right)+\mathtt {V}_0 u & =λ\mathtt {V}_2 τu+\mathtt {V}_2 f_0 & & \text { in } Ω, \\ \mathtt {V}_1 \frac{\partial u}{\partial ν} & =\mathtt {W}_1 f_1 & & \text { on } \partial Ω, \end{aligned}\right. $$ where $Ω$ is an open subset of a Riemannian manifold, $λ$ is a real number, $f_0 \in L^1 (Ω; \mathtt {V}_0), f_1 \in L^1(\partial Ω; \mathtt {W})$, $τ$ 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 $L^2 (Ω; \mathtt {V}_0)$ and $L^2(\partial Ω; \mathtt {W})$. For the case where $f_0=0$ and $f_1=0$, we are also interested in studying the limit ess $\sup _{Ω\backslash Ω_m}|u| \rightarrow 0$, where $Ω_m$ is a sequence of open sets such that $Ω_m \subset Ω_{m+1}$. |
| title | Weighted Sobolev Spaces and an Eigenvalue Problem for an Elliptic Equation with $ L^1 $ Data |
| topic | Analysis of PDEs 35J25, 46E35, 93B60, 35R05 |
| url | https://arxiv.org/abs/2410.00162 |