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

Let $\mathrm{\Omega }\subset {\mathbb{R}}^d$ 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 $-\mathrm{\nabla }\cdot (A(x)\mathrm{\nabla }u)=0$ in $\mathrm{\Omega }$, and $u$ vanishes on $\mathrm{\Sigma }=\mathrm{\partial }\mathrm{\Omega }\cap B$ 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 $B_i\cap \mathrm{\Omega }$ maintains a consistent sign. Furthermore, for any compact subset $K$ of $\mathrm{\Sigma }$, 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.