On viscosity solutions of certain Hamilton-Jacobi equations: Regularity results and generalized Sard's Theorems

Ludovic Rifford (1)

1 JAD - Laboratoire Jean Alexandre Dieudonné

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 $C^{1,1}_{loc}$ 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 .

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Ludovic Rifford. On viscosity solutions of certain Hamilton-Jacobi equations: Regularity results and generalized Sard's Theorems. Communications in Partial Differential Equations, 2007, 33 (3), pp.517-559. ⟨hal-00769099⟩