# (Non)local and (non)linear free boundary problems

## TL;DR

This paper reviews recent advances in free boundary problems that incorporate nonlocal and nonlinear energy functionals, highlighting their complex behaviors and potential for modeling diverse physical regimes.

## Contribution

It introduces new free boundary models with nonlocal and nonlinear energies, exploring their structural properties and stability issues across scales.

## Key findings

- Nonlocal parameter interpolates between volume and perimeter functionals.
- Nonlinear superpositions model different energy regimes.
- Scale invariance may be broken, affecting minimizer stability.

## Abstract

We discuss some recent developments in the theory of free boundary problems, as obtained in a series of papers in collaboration with L. Caffarelli, A. Karakhanyan and O. Savin. The main feature of these new free boundary problems is that they deeply take into account nonlinear energy superpositions and possibly nonlocal functionals. The nonlocal parameter interpolates between volume and perimeter functionals, and so it can be seen as a fractional counterpart of classical free boundary problems, in which the bulk energy presents nonlocal aspects. The nonlinear term in the energy superposition takes into account the possibility of modeling different regimes in terms of different energy levels and provides a lack of scale invariance, which in turn may cause a structural instability of minimizers that may vary from one scale to another.