The Cahn-Hilliard equation has been extensively studied over the past decades as it plays a fun- damental role in several applications, ranging from materials science and physics to biology and engineering. Meanwhile, it has recently been discovered that the behavior of certain phenomena can be described in a more physically accurate way by nonlocal operators.

A more realistic version of the standard Cahn–Hilliard equation can thus be derived from a nonlocal free energy, which leads to the substitution of the Laplacian with its nonlocal version, the so-called fractional Laplacian. In the local setting, the system is usually complemented with homogeneous Neumann conditions, since they are consistent with the physical nature of the problem. Their nonlocal counterparts, together with existence and uniqueness of appropriate weak solutions to the corresponding systems, have been established in recent works.

The first objective of our project is to investigate the asymptotic behavior of solutions to these nonlocal Cahn-Hilliard equations, in order to obtain rigorous nonlocal-to-local convergence re- sults. Under appropriate assumptions on the potential and the initial data, we expect the sequence of weak solutions to the nonlocal problems to converge to the weak solution of the corresponding local equation.

Building on these results, the second part of the project is concerned with extending the analysis to more general nonlocal free energies and the associated integro-differential operators. By introducing a suitable notion of weak solution, we aim to establish analogous convergence results in these broader settings.

These questions are of relevant interest since one of the main advantages of the nonlocal setting is that regularity properties of the solutions are often easier to obtain. Consequently, having nonlocal-to-local convergence results would enable the approximation of solutions to local systems by means of their nonlocal counterparts. In particular, we plan to exploit these asymptotics in the context of optimal control problems arising in tumor growth modeling, in order to derive first-order optimality conditions.