Regularity of sublevels for Hamilton-Jacobi solutions

Ulisse Stefanelli (U of Vienna)

Feb 28. 2025, 11:00 — 11:50

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).

 

Further Information
Venue:
ESI Boltzmann Lecture Hall
Associated Event:
Degenerate and Singular PDEs (Workshop)
Organizer(s):
Verena Bögelein (PLUS, Salzburg)
Ugo Gianazza (U of Pavia)
Juha Kinnunen (Aalto U)
Naian Liao (PLUS, Salzburg)