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

Let Ω⊂Rd be a quasiconvex Lipschitz domain, and let A(x) be a d × d uniformly elliptic, symmetric matrix with Lipschitz coefficients. Assume a nontrivial function 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\⋃iBi is shown to have a Minkowski dimension strictly less than d − 1 − ϵ. As a consequence, we prove Lin’s conjecture in quasiconvex domains.