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

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 Hölder 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.

arXiv comment19 pages