Existence of the gradient estimate constant C2 on bounded Lipschitz domains

Establish the existence of a constant C2 such that, for every bounded Lipschitz domain D ⊂ ℝ^N and every bounded real-valued harmonic function U on D that satisfies the uniform α-Hölder condition along lines parallel to a fixed axis (for example, |U(x', xN) − U(x', yN)| ≤ C |xN − yN|^α for all x' ∈ ℝ^{N−1} and admissible xN, yN within D), the gradient bound |∇U(x)| ≤ C2 · d(x, ∂D)^{α−1} holds for all x ∈ D. This addresses the key difficulty in extending the main theorem from unbounded epigraph domains to bounded Lipschitz domains.

Background

The paper proves that for unbounded Lipschitz epigraph domains E_Ψ, if a bounded harmonic function U satisfies an α-Hölder condition along vertical lines, then U enjoys global α-Hölder continuity. The proof constructs three constants (C1, C2, C3) controlling, respectively, the normal derivative, the full gradient via a distance-to-boundary estimate, and the global Hölder modulus.

In the remark following the proof, the author discusses extending these results to more general Lipschitz domains, including bounded ones. While the existence of C1 seems feasible in a broad setting, the argument used to obtain C2 relies on gradient behavior in unbounded domains (specifically, decay at infinity). For bounded domains, the author explicitly notes that overcoming this obstacle is not clear, identifying the existence of the gradient estimate constant C2 as the main challenge for such extensions.