References & Citations
Mathematics > Analysis of PDEs
Title: Unique continuation at the boundary for divergence form elliptic equations on quasiconvex domains
(Submitted on 8 May 2024 (v1), last revised 23 May 2024 (this version, v3))
Abstract: Let $\Omega \subset \mathbb{R}^d$ 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 $\Omega$, and $u$ vanishes on $\Sigma = \partial \Omega \cap B$ 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 $B_i \cap \Omega$ maintains a consistent sign. Furthermore, for any compact subset $K$ of $\Sigma$, the set difference $K \setminus \bigcup_i B_i$ is shown to possess a Minkowski dimension that is strictly less than $d - 1 - \epsilon$. As a consequence, we prove Lin's conjecture in quasiconvex domains.
Submission history
From: Cai Yingying [view email][v1] Wed, 8 May 2024 13:29:18 GMT (25kb)
[v2] Thu, 9 May 2024 00:47:43 GMT (25kb)
[v3] Thu, 23 May 2024 13:53:25 GMT (25kb)
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