Existence of $C^{1,1}$ critical sub-solutions of the Hamilton-Jacobi   equation on compact manifolds
  We prove the existence of a critical subsolution of the Hamilton-Jacobi equation which is differentiable with Lipschitz differential, improving a previous result of Fathi and Siconolfi. This text is a consolidated version with a minor mistake corrected from the published version. 
Patrick Bernard (CEREMADE)