Sobolev regularity for the first order Hamilton-Jacobi equation
  We provide Sobolev estimates for solutions of first order Hamilton-Jacobi equations with Hamiltonians which are superlinear in the gradient variable. We also show that the solutions are differentiable almost everywhere. The proof relies on an inverse H\"older inequality. Applications to mean field games are discussed. 
Pierre Cardaliaguet, Alessio Porretta and Daniela Tonon
math.AP