Lin’s conjecture on boundary unique continuation in Lipschitz domains

Prove that for any bounded harmonic function u in a Lipschitz domain Ω ⊂ ℝ^d, if u vanishes on a relatively open subset U of the boundary ∂Ω and the gradient ∇u vanishes on a subset of U of positive surface measure, then u is identically zero in Ω.

Background

Boundary unique continuation concerns whether vanishing boundary data imply triviality of solutions inside the domain. A classical conjecture of L. Bers was disproved by Bourgain–Wolff for C^1 domains, prompting refined questions. In this context, Lin formulated a stronger boundary unique continuation conjecture involving both vanishing of the function and its gradient on a boundary set of positive measure.

The conjecture has been proved in several settings: C^{1,1}, C^{1,α}, C^{1,Dini}, C^1 or Lipschitz with small constant, and convex domains. The present paper establishes the conjecture for quasiconvex Lipschitz domains. However, despite these advances, the conjecture in full generality, especially for arbitrary Lipschitz domains in dimension d ≥ 3, remains open.