Asymptotic expansions for harmonic functions at conical boundary points

TL;DR

This paper investigates the asymptotic behavior of harmonic functions near conical boundary points, establishing conditions for the existence of precise expansions and demonstrating the necessity of certain hypotheses through counterexamples.

Contribution

It provides new theorems characterizing the asymptotics of harmonic functions at boundary points with tangent cones, under minimal regularity assumptions.

Findings

Doubling index has a finite limit or diverges at boundary points.

Finite order of vanishing implies a specific homogeneous harmonic expansion.

Counterexample shows some hypotheses are essential for the expansion to hold.

Abstract

We prove three theorems about the asymptotic behavior of solutions $u$ to the homogeneous Dirichlet problem for the Laplace equation at boundary points with tangent cones. First, under very mild hypotheses, we show that the doubling index of $u$ either has a unique finite limit, or goes to infinity; in other words, there is a well-defined order of vanishing. Second, under more quantitative hypotheses, we prove that if the order of vanishing of $u$ is finite at a boundary point $0$, then locally $u(x) = |x|^m \psi(x/|x|) + o(|x|^m)$, where $|x|^m \psi(x/|x|)$ is a homogeneous harmonic function on the tangent cone. Finally, we construct a convex domain in three dimensions where such an expansion fails at a boundary point, showing that some quantitative hypotheses are necessary in general. The assumptions in all of the results only involve regularity at a single point, and in particular are much weaker than what is necessary for unique continuation, monotonicity of Almgren's frequency, Carleman estimates, or other related techniques.