A class of unstable free boundary problems

TL;DR

This paper investigates a novel class of unstable free boundary problems involving a nonlinear combination of elastic energy and surface tension, providing explicit examples, boundary conditions, and regularity results, marking a first in nonlinear perimeter studies.

Contribution

It introduces the first analysis of free boundary problems with a nonlinear perimeter functional, revealing instability and deriving new boundary and regularity results.

Findings

Explicit example of instability in nonlinear free boundary problems

Derivation of free boundary conditions emphasizing domain effects

New regularity results for minimal solutions in nonlinear setting

Abstract

We consider the free boundary problem arising from an energy functional which is the sum of a Dirichlet energy and a nonlinear function of either the classical or the fractional perimeter. The main difference with the existing literature is that the total energy is here a nonlinear superposition of the either local or nonlocal surface tension effect with the elastic energy. In sharp contrast with the linear case, the problem considered in this paper is unstable, namely a minimizer in a given domain is not necessarily a minimizer in a smaller domain. We provide an explicit example for this instability. We also give a free boundary condition, which emphasizes the role played by the domain in the geometry of the free boundary. In addition, we provide density estimates for the free boundary and regularity results for the minimal solution. As far as we know, this is the first case in which a nonlinear function of the perimeter is studied in this type of problems. Also, the results obtained in this nonlinear setting are new even in the case of the local perimeter, and indeed the instability feature is not a consequence of the possibly nonlocality of the problem, but it is due to the nonlinear character of the energy functional.