Weighted Sobolev space theory for Poisson's equation in non-smooth domains
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917148879749120 |
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| author | Seo, Jinsol |
| author_facet | Seo, Jinsol |
| contents | We introduce a general $L_p$-solvability result for the Poisson equation in non-smooth domains $Ω\subset \mathbb{R}^d$, with the zero Dirichlet boundary condition. Our sole assumption on the domain $Ω$ is the Hardy inequality: There exists a constant $N>0$ such that $$ \int_Ω\Big|\frac{f(x)}{d(x,\partialΩ)}\Big|^2\,\mathrm{d} x\leq N\int_Ω|\nabla f|^2 \,\mathrm{d} x\quad\text{for any}\quad f\in C_c^{\infty}(Ω)\,. $$ To describe the boundary behavior of solutions in a general framework, we propose a weight system composed of a superharmonic function and the distance function to the boundary. Additionally, we explore applications across a variety of non-smooth domains, including convex domains, domains with exterior cone condition, totally vanishing exterior Reifenberg domains, and domains $Ω\subset\mathbb{R}^d$ for which the Aikawa dimension of $Ω^c$ is less than $d-2$. Using superharmonic functions tailored to the geometric conditions of the domain, we derive weighted $L_p$-solvability results for various non-smooth domains and specific weight ranges that differ for each domain condition. Furthermore, we provide an application to the Hölder continuity of solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_18865 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weighted Sobolev space theory for Poisson's equation in non-smooth domains Seo, Jinsol Analysis of PDEs We introduce a general $L_p$-solvability result for the Poisson equation in non-smooth domains $Ω\subset \mathbb{R}^d$, with the zero Dirichlet boundary condition. Our sole assumption on the domain $Ω$ is the Hardy inequality: There exists a constant $N>0$ such that $$ \int_Ω\Big|\frac{f(x)}{d(x,\partialΩ)}\Big|^2\,\mathrm{d} x\leq N\int_Ω|\nabla f|^2 \,\mathrm{d} x\quad\text{for any}\quad f\in C_c^{\infty}(Ω)\,. $$ To describe the boundary behavior of solutions in a general framework, we propose a weight system composed of a superharmonic function and the distance function to the boundary. Additionally, we explore applications across a variety of non-smooth domains, including convex domains, domains with exterior cone condition, totally vanishing exterior Reifenberg domains, and domains $Ω\subset\mathbb{R}^d$ for which the Aikawa dimension of $Ω^c$ is less than $d-2$. Using superharmonic functions tailored to the geometric conditions of the domain, we derive weighted $L_p$-solvability results for various non-smooth domains and specific weight ranges that differ for each domain condition. Furthermore, we provide an application to the Hölder continuity of solutions. |
| title | Weighted Sobolev space theory for Poisson's equation in non-smooth domains |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2403.18865 |