Refined Rellich boundary inequalities for the derivatives of a harmonic function

TL;DR

This paper refines Rellich boundary inequalities for harmonic functions, extending their applicability to non-Lipschitz domains and providing sharp $L^p$ estimates in two dimensions using complex analysis techniques.

Contribution

It introduces refined inequalities that remain valid in non-Lipschitz domains and derives sharp $L^p$ estimates for harmonic functions in two dimensions.

Findings

Refined Rellich inequalities valid for non-Lipschitz domains.

Sharp $L^p$ estimates for harmonic functions in 2D.

Use of Riemann mapping and interpolation for estimates.

Abstract

The classical Rellich inequalities imply that the $L^2$-norms of the normal and tangential derivatives of a harmonic function are equivalent. In this note, we prove several refined inequalities, which make sense even if the domain is not Lipschitz. For two-dimensional domains, we obtain a sharp $L^p$-estimate for $1<p\leq 2$ by using a Riemann mapping and interpolation argument.