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

Jules Candau-Tilh

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.

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

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.

Paper Structure

This paper contains 12 sections, 7 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Application to three perturbed isoperimetric problems
  4. Notation and organisation of the article
  5. Existence of generalized minimizers
  6. Density estimates for perturbed isoperimetric problems
  7. Density estimates for $\rho$-minimizers of the perimeter
  8. Generalized minimizers as $\rho$-minimizers of the perimeter
  9. Application to perturbed isoperimetric problems
  10. An anisotropic liquid drop model
  11. A prescribed nonlocal curvature problem
  12. A model of a nonlocal perimeter interacting with Dirichlet eigenvalues

Key Result

Proposition \oldthetheorem

Assume that $\mathcal{E}$ satisfies (S1). Then $infclass=infgen$.

Theorems & Definitions (24)

  • Proposition \oldthetheorem
  • Theorem \oldthetheorem
  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • ...and 14 more