On Viscosity Solutions of Certain Hamilton–Jacobi Equations: Regularity Results and Generalized Sard's Theorems

Abstract

Under usual assumptions on the Hamiltonian, we prove that any viscosity solution of the corresponding Hamilton–Jacobi equation on the manifold M is locally semiconcave and outside the closure of its singular set (which is nowhere dense in M). Moreover, we prove that, under additional assumptions and in low dimension, any viscosity solution of that Hamilton–Jacobi equation satisfies a generalized Sard theorem. In consequence, almost every level set of such a function is a locally Lipschitz hypersurface in M.