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

Let Ω ⊂ Rᵈ be a quasiconvex Lipschitz domain and $A(x)$ be a d × d uniformly elliptic, symmetric matrix with Lipschitz coefficients. Assume a nontrivial u solves -∇ · (A(x) ∇ u) = 0 in Ω, and u vanishes on Σ = ∂ Ω ∩ B for some ball B. The main contribution of this paper is to demonstrate the existence of a countable collection of open balls (Bᵢ)ᵢ such that the restriction of u to Bᵢ ∩ Ω maintains a consistent sign. Furthermore, for any compact subset K of Σ, the set difference K ᵢ Bᵢ 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.