Unique continuation at the boundary for divergence form elliptic equations on quasiconvex domains

Abstract

Let $\hat{I}\text{c◯}\subset {\mathbb{R}}^d$ 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 $\hat{I}\text{c◯}$, and $u$ vanishes on $\hat{I}\pounds =\partial \hat{I}\text{c◯}\cap B$ 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 $B_i\cap \hat{I}\text{c◯}$ maintains a consistent sign. Furthermore, for any compact subset $K$ of $\hat{I}\pounds$, the set difference $K\setminus {\bigcup }_i{B}_i$ is shown to possess a Minkowski dimension that is strictly less than $d-1-\hat{I}\mu$. As a consequence, we prove Lin's conjecture in quasiconvex domains.

Correct many typos and add reference. arXiv admin note: text overlap with arXiv:2303.02046, arXiv:2201.12307 by other authors