Continuous harmonic functions on a ball that are not in $H^s$ for $s>1/2$

TL;DR

This paper constructs harmonic functions on a ball that are continuous up to the boundary but do not belong to high-order Sobolev spaces, demonstrating the optimality of regularity results in certain boundary value problems.

Contribution

It provides explicit examples of harmonic functions with limited Sobolev regularity in any dimension, extending classical two-dimensional examples using spherical harmonics.

Findings

Existence of harmonic functions not in $H^s$ for large $s$

Examples valid in all dimensions $n \\ge 2$

Implication for optimality of regularity results in boundary problems

Abstract

We show that there are harmonic functions on a ball ${\mathbb{B}_n}$ of $\mathbb{R}^n$, $n\ge 2$, that are continuous up to the boundary (and even H\"older continuous) but not in the Sobolev space $H^s(\mathbb{B}_n)$ for any $s$ sufficiently big. The idea for the construction of these functions is inspired by the two-dimensional example of a harmonic continuous function with infinite energy presented by Hadamard in 1906. To obtain examples in any dimension $n\ge 2$ we exploit certain series of spherical harmonics. As an application, we verify that the regularity of the solutions that was proven for a class of boundary value problems with nonlinear transmission conditions is, in a sense, optimal.