A fractional Cahn-Hilliard system

Abstract

In this talk, we introduce a fractional variant of the Cahn-Hilliard system. We first focus on the model case of the fractional Laplacian with homogeneous Dirichlet boundary conditions, and briefly show how to prove the existence and uniqueness of a weak solution. The proof relies on the variational method known as “minimizing-movements scheme”, which fits naturally with the gradient-flow structure of the equation. The interest of the proposed method lies in its extreme generality and flexibility. In particular, relying on the variational structure of the equation, it can be applied to show existence of a weak solution also for a more general class of integro-differential operators, not necessarily linear or symmetric. These include, e.g., fractional versions of the p-Laplacian. Moreover, by adapting the argument to the case of regional fractional operators, we can prove the existence of solutions also in the interesting case of fractional Neumann boundary conditions. These aspects will be the object of the second part of the talk. This is a joint work with E. Davoli and C. Gavioli.