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

Abstract Let Ω ⊂ Rd be a quasiconvex Lipschitz domain and A(x) be a d × d uniformly eliiptic, symmetric matrix with Lipschitz coefficients. Assume 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\ Ui 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.