I will present new regularity results for the stationary Hamilton–Jacobi equation $H(x,\nabla u(x))=0$ in the external domain ${\mathbb R}^{ n} \setminus K$
with $u=0$ on the compact set $K$. This problem arises in various contexts, including front propagation and optimal control.
For Hamiltonians that are nondegenerate and convex in the second variable, I will show that all sublevel sets of the unique nonnegative viscosity solution are John domains for all times. This regularity is sharp, as explicit counterexamples demonstrate. Furthermore, if $K$ is itself a John domain, one can establish a uniform lower bound on the John constant for all sublevels. This is joint work with Elisa Davoli (TU Wien).