Description
I will present the framework of periodic homogenisation of nonlocal Hamilton-Jacobi equations, associated with Levy-Itô integro-differential operators. A typical equation is the fractional diffusion coupled with a transport term, where the diffusion is only weakly elliptical. Homogenization is established in two steps: (i) the resolution of a cellular problem - where Lipshitz regularity of the corrector plays a key role and (ii) the convergence of the oscillating solutions towards an averaged profile - where comparison principles are involved. I shall discuss recent results on the regularity of solutions and comparison principles for nonlocal equations, and the difficulties we face when compared with local PDEs. The talk is based on recent results obtained in collaboration with D. Ghilli, E. Topp and O.Ley, E. Topp, T. Minh Le.
