Transversally Lipschitz Harmonic Functions are Lipschitz

TL;DR

This paper proves that harmonic functions in a smooth bounded domain are Lipschitz continuous if they are Lipschitz along a family of transversal curves, extending classical regularity results using a generalized Lipschitz space.

Contribution

It establishes that transversally Lipschitz harmonic functions are globally Lipschitz, generalizing previous regularity results with a new approach involving majorants.

Findings

Harmonic functions Lipschitz along transversal curves are Lipschitz in the domain.

Introduces a generalized Lipschitz space using majorants.

Extends classical regularity theorems for harmonic functions.

Abstract

Let \Omega\subset\mathbb{R}^n be a bounded domain with C^\infty boundary. We show that a harmonic function in \Omega that is Lipschitz along a family of curves transversal to b\Omega is Lipschitz in \Omega. The space of Lipschitz functions we consider is defined using the notion of a majorant which is a certain generalization of the power functions t^\alpha, 0<\alpha<1.