On viscosity solutions of certain Hamilton-Jacobi equations: Regularity results and generalized Sard's Theorems Ludovic Rifford ? May 15, 2007 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 C1,1loc 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 . 1 Introduction Let M be a smooth manifold without boundary. We denote by TM (resp. T ?M) the tangent bundle of M , (x, v) a point in TM , and pi : TM ? M the canonical projection. Similarly, we denote by T ?M the cotangent bundle of M , (x, p) a point in T ?M , and pi? : T ?M ? M the canonical projection. We will assume that the manifold M is equipped with a complete Riemannian metric g. For every v ? TxM , we set ?v? := √ gx(v, v). And we denote by ? · ? the dual norm on T ?M . Let H : T ?M ? R be an Hamiltonian of class Ck (with k ≥ 2) which satisfies the three following conditions: (H1) (Uniform superlinearity) For every K ≥ 0, there is
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