Calculus of Variations and Geometric Measure Theory

E. Davoli - U. Stefanelli

Level sets of Eikonal functions are John regular

created by davoli on 29 Dec 2023


Submitted Paper

Inserted: 29 dec 2023
Last Updated: 29 dec 2023

Year: 2023


Let u be the unique viscosity solution of $\alpha(x)
\nabla u
=1$ in the external domain $R^n\setminus K$ with $u = 0$ on $K$. In case $\alpha$ is continuous, bounded, and uniformly positive and $K$ is a bounded John domain, we prove that all superlevels of $u$ are John domains, too. Moreover, we give counterexamples showing that John regularity is sharp in this setting.

Keywords: regularity, viscosity solution, Eikonal equation, John domains