Gavioli, C., Davoli, E., & Lombardini, L. (2023, November 28). A fractional Cahn-Hilliard problem [Presentation]. Mathematical Analysis Seminar at the FIM Department of the University of Modena and Reggio Emilia 2023, Modena, Italy.
In this talk, we introduce a fractional variant of the Cahn-Hilliard equation settled in a bounded domain. We first focus on the simpler case of 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. We also investigate the convergence of solutions for this class of nonlocal Cahn-Hilliard problems to their local counterparts, as the order of the fractional Laplacian appearing in the equation is let tend to 1.
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 unique solution also for a more general class of integro-differential operators, not necessarily linear or symmetric. These include, e.g., a fractional version 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.
