Continuous Harmonic Functions on a Ball that are not in $$H^s$$ for $$s>1/2$$
Abstract
Abstract We show that there are harmonic functions on a ball $${\mathbb {B}_n}$$ B n of $$\mathbb {R}^n$$ R n , $$n\ge 2$$ n ≥ 2 , that are continuous, and even Hölder continuous, up to the boundary but not in the Sobolev space $$H^s(\mathbb {B}_n)$$ H s ( B n ) for s bigger than a certain sharp bound. The idea for the construction 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 $$n\ge 2$$ n ≥ 2 we exploit certain series of spherical harmonics.
Fields of Study & Subjects
Advanced Mathematical Modeling in Engineering
Stability and Controllability of Differential Equations
Numerical methods in inverse problems
Publication details
- Journal / source
- Journal of Geometric Analysis
- Publisher
- Springer Science and Business Media LLC
- Volume
- 36
- Issue
- 6
- Publication date
- 11 May 2026
- Language
- EN