A concentration-compactness principle for perturbed isoperimetric problems with general assumptions

TL;DR

This paper extends the concentration-compactness principle to perturbed isoperimetric problems, establishing existence and density estimates for generalized minimizers under broad conditions, including classical and nonlocal functionals.

Contribution

It introduces a generalized minimizer framework for perturbed isoperimetric problems with broad applicability and mild assumptions, covering classical, anisotropic, and fractional perimeters.

Findings

Existence of generalized minimizers under mild assumptions

Density estimates for these minimizers

Applicability to a wide class of functionals including nonlocal terms

Abstract

Derived from the concentration-compactness principle, the concept of generalized minimizer can be used to define generalized solutions of variational problems which may have components ``infinitely'' distant from each other. In this article and under mild assumptions we establish existence and density estimates of generalized minimizers of perturbed isoperimetric problems. Our hypotheses encapsulate a wide class of functionals including the classical, anisotropic and fractional perimeter. The perturbation term may for instance take the form of a potential, a translation invariant kernel or a nonlocal term involving the Wasserstein distance.