Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.13021 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929249087127552 |
|---|---|
| author | Righi, Robert Shen, Zhongwei |
| author_facet | Righi, Robert Shen, Zhongwei |
| contents | In this paper we establish $W^{1,p}$ estimates for solutions $u_\varepsilon$ to Laplace's equation with the Dirichlet condition in a bounded and perforated, not necessarily periodically, $C^1$ domain $Ω_{\varepsilon, η}$ in $\mathbb{R}^d$. The bounding constants depend explicitly on two small parameters $\varepsilon$ and $η$, where $\varepsilon$ represents the scale of the minimal distance between holes, and $η$ denotes the ratio between the size of the holes and $\varepsilon$. The proof relies on a large-scale $L^p$ estimate for $\nabla u_\varepsilon$, whose proof is divided into two parts. In the first part, we show that as $\varepsilon, η$ approach zero, harmonic functions in $Ω_{\varepsilon, η}$ may be approximated by solutions of an intermediate problem for a Schrödinger operator in $Ω$. In the second part, a real-variable method is employed to establish the large-scale $L^p$ estimate for $\nabla u_\varepsilon$ by using the approximation at scales above $\varepsilon$. The results are sharp except in the case $d\ge 3$ and $p=d$ or $d^\prime$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_13021 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dirichlet Problems in Perforated Domains Righi, Robert Shen, Zhongwei Analysis of PDEs 35Q35, 35B27, 76D07 In this paper we establish $W^{1,p}$ estimates for solutions $u_\varepsilon$ to Laplace's equation with the Dirichlet condition in a bounded and perforated, not necessarily periodically, $C^1$ domain $Ω_{\varepsilon, η}$ in $\mathbb{R}^d$. The bounding constants depend explicitly on two small parameters $\varepsilon$ and $η$, where $\varepsilon$ represents the scale of the minimal distance between holes, and $η$ denotes the ratio between the size of the holes and $\varepsilon$. The proof relies on a large-scale $L^p$ estimate for $\nabla u_\varepsilon$, whose proof is divided into two parts. In the first part, we show that as $\varepsilon, η$ approach zero, harmonic functions in $Ω_{\varepsilon, η}$ may be approximated by solutions of an intermediate problem for a Schrödinger operator in $Ω$. In the second part, a real-variable method is employed to establish the large-scale $L^p$ estimate for $\nabla u_\varepsilon$ by using the approximation at scales above $\varepsilon$. The results are sharp except in the case $d\ge 3$ and $p=d$ or $d^\prime$. |
| title | Dirichlet Problems in Perforated Domains |
| topic | Analysis of PDEs 35Q35, 35B27, 76D07 |
| url | https://arxiv.org/abs/2402.13021 |