Metric formulae for nonconvex Hamilton--Jacobi equations and applications

Abstract

We consider the Hamilton-Jacobi equation $H(x,Du)=0$ in $\mathbb R^n$, with $H$ not enjoying any convexity properties in the second variable. Our aim is to establish existence and nonexistence theorems for viscosity solutions of associated Dirichlet problems, find representation formulae and prove comparison principles. Our analysis is based on the introduction of a metric intrinsically related to the $0$--sublevels of the Hamiltonian, given by an inf-sup game theoretic formula. We also study the case where the equation is critical; i.e., $H(x,Du)= - \varepsilon$ does not admit any viscosity subsolution, for $\varepsilon >0$.