Robin Harmonic Measure

This thesis investigates two probability measures associated with solutions to the Laplace equation on bounded Lipschitz domains. First, utilising a solvability estimate for the Dirichlet problem, we prove that the (Dirichlet) harmonic measure ω is absolutely continuous with respect to the surface measure σ and that its density satisfies the Reverse Hölder inequality of order 2 for strictly star-shaped domains in R2. Following this framework, we establish a solvability estimate for the Robin problem. However, this proves to be unnecessary. Instead, with a direct analytical approach, we achieve a stronger regularity result. Specifically, we prove in Rn that the density of the Robin harmonic measure possesses both a uniform upper bound and a strict, positive lower bound independent of the solvability framework. We present numerical findings in two dimensions using the FEM, analysing the Robin harmonic density and its asymptotic behaviour as α → 0,∞. Crucially, we utilise these numerical results to directly compare the two measures, visually showcasing the stronger regularity of the Robin harmonic density on non-convex corner Lipschitz domains where the Dirichlet harmonic density typically exhibits singular behaviour.