Accepted Paper

Inserted: 29 dec 2023
Last Updated: 23 mar 2026

Journal: Journal of Differential Equations
Year: 2026

Abstract:

Let u be the unique nonnegative viscosity solution of the Hamilton-Jakobi equation $H(x,\nabla u)=0$ in the external domain $R^n\setminus K$ with $u = 0$ on $K$. Under general conditions on $H$, we prove that all sublevels of $u$ are John domains. Moreover, if $K$ itself is a John domain, we provide a uniform lower bound on the John constant of all sublevels. We exhibit counterexamples showing that John regularity is sharp in this setting.

Keywords: regularity, viscosity solution, John domains, Hamilton-Jakobi equation