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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.05044 |
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| _version_ | 1866909208866193408 |
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| author | Cai, Yingying |
| author_facet | Cai, Yingying |
| contents | Let $Ω\subset \mathbb{R}^d$ be a quasiconvex Lipschitz domain and $A(x)$ be a $d \times d$ uniformly elliptic, symmetric matrix with Lipschitz coefficients. Assume a nontrivial $u$ solves $-\nabla \cdot (A(x) \nabla u) = 0$ in $Ω$, and $u$ vanishes on $Σ= \partial Ω\cap B$ for some ball $B$. The main contribution of this paper is to demonstrate the existence of a countable collection of open balls $(B_i)_i$ such that the restriction of $u$ to $B_i \cap Ω$ maintains a consistent sign. Furthermore, for any compact subset $K$ of $Σ$, the set difference $K \setminus \bigcup_i B_i$ is shown to possess a Minkowski dimension that is strictly less than $d - 1 - ε$. As a consequence, we prove Lin's conjecture in quasiconvex domains. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_05044 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Unique continuation at the boundary for divergence form elliptic equations on quasiconvex domains Cai, Yingying Analysis of PDEs Let $Ω\subset \mathbb{R}^d$ be a quasiconvex Lipschitz domain and $A(x)$ be a $d \times d$ uniformly elliptic, symmetric matrix with Lipschitz coefficients. Assume a nontrivial $u$ solves $-\nabla \cdot (A(x) \nabla u) = 0$ in $Ω$, and $u$ vanishes on $Σ= \partial Ω\cap B$ for some ball $B$. The main contribution of this paper is to demonstrate the existence of a countable collection of open balls $(B_i)_i$ such that the restriction of $u$ to $B_i \cap Ω$ maintains a consistent sign. Furthermore, for any compact subset $K$ of $Σ$, the set difference $K \setminus \bigcup_i B_i$ is shown to possess a Minkowski dimension that is strictly less than $d - 1 - ε$. As a consequence, we prove Lin's conjecture in quasiconvex domains. |
| title | Unique continuation at the boundary for divergence form elliptic equations on quasiconvex domains |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2405.05044 |