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

TL;DR

This paper proves a boundary unique continuation property for divergence form elliptic equations on quasiconvex domains, establishing sign stability of solutions and confirming Lin's conjecture.

Contribution

It demonstrates the existence of a countable covering where solutions maintain a sign and verifies Lin's conjecture in quasiconvex domains.

Findings

Existence of a countable collection of balls where solutions have a consistent sign.

The set difference on the boundary has Minkowski dimension less than d-1-ε.

Confirmation of Lin's conjecture for quasiconvex domains.

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.