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

Let 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, and u vanishes on = 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 maintains a consistent sign. Furthermore, for any compact subset K of, the set difference K ᵢ Bᵢ is shown to possess a Minkowski dimension that is strictly less than d - 1 -. As a consequence, we prove Lin's conjecture in quasiconvex domains.