Expansion of harmonic functions near the boundary of Dini domains

TL;DR

This paper investigates the local behavior of harmonic functions near boundary points in Dini domains, establishing unique tangent functions and convergence rates, which enhances understanding of boundary regularity in harmonic analysis.

Contribution

It introduces a novel expansion of harmonic functions near boundary points in Dini domains, including uniqueness of tangent functions and convergence estimates.

Findings

Harmonic functions can be uniquely expanded into homogeneous polynomials near boundary points.

Existence of unique tangent functions at boundary points in Dini domains.

Convergence rates to tangent functions are quantitatively estimated.

Abstract

Let $u$ be a harmonic function in a $C^1$-Dini domain, such that $u$ vanishes on an open set of the boundary. We show that near every point in the open set, $u$ can be written uniquely as the sum of a non-trivial homogeneous harmonic polynomial and an error term of higher degree (depending on the Dini parameter). In particular, this implies that $u$ has a unique tangent function at every such point, and that the convergence rate to the tangent function can be estimated. We also study the relationship of tangent functions at nearby points in a special case.