With Nicolas Forcadel et Régis Monneau, we recently uploaded on HAL and arxiv two new preprints about Hamilton-Jacobi equations.
The first one is interested in the understanding of viscosity solutions for HJ equations posed on a domain. This theory allows one to easily construct weak solutions for boundary value problems. These weak viscosity solutions can be studied thanks to the relaxation operator introduced by J. Guerand. In this new work, a link with classical Godunov fluxes is exhibited. As an application, the classical Neumann and Dirichlet problems are discussed.
The second one is interested in uniqueness of viscosity solutions. We elaborate on a method introduced by P.-L. Lions and P. Souganidis to prove new comparison principles for HJ equations posed on a domain. We follow them by performing a blow analysis and by reducing to a 1D problem but we depart from their method by blowing up the sub-solution and the super-solution simultaneously.