Free Boundary Problems with long range interaction

In this talk we consider a class of variational shape optimisation problems for densities that repel each other at a certain distance. Typical examples are given by the Dirichlet functional and the Rayleigh functional minimised in the class of functions that attain some H^1 boundary conditions, subject to the constraint that the supports of different densities are at a certain fixed distance. We show a connection with solutions to variational elliptic systems with nonlocal competing interactions, investigate the optimal regularity of the solutions (and prove uniform estimates with respect to the distance parameter), prove a free-boundary condition and derive some preliminary results characterising the free boundary. The talk is based in the following works:

* N. Soave, H. Tavares, S. Terracini, A. Zilio, Variational problems with long-range interaction, Arch. Rational Mech. Anal. 228 (2018), 743–772.

* N. Soave, H. Tavares and A. Zilio, Free boundary problems with long-range interactions: uniform Lipschitz estimates in the radius. Mathematische Annalen (2022), https://doi.org/10.1007/s00208-022-02406-8