Università degli Studi di Padova

“Ill behaved harmonic functions on a ball”

Giovedì 29 Settembre 2022, ore 14:30 - Aula 2BC30 e Zoom - Roberto Bramati (Ghent University)


In this talk we will show existence of harmonic functions on the unit ball $\mathbb{B}^n$ of $\mathbb{R}^n$ that are continuous or even Hölder continuous up to the boundary, but not in $H^s(\mathbb{B}^n)$ for sufficiently big $s$. The construction of such functions follows ideas of Hadamard and Hardy and relies on the interplay between $L^2$ and $L^{\infty}$ norms of spherical harmonics on the sphere $\mathbb{S}^{n−1}$. If time permits, we will explain how we can use these examples to show optimality of certain regularity results for solutions of a class of boundary value problems with nonlinear transmission conditions.

This is a joint work with Matteo Dalla Riva (Università degli Studi di Palermo) and Brian Luczak (Vanderbilt University).