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Non-local singular perturbations of non-convex functionals -- recent results

Andrea Braides

TL;DR

The paper surveys singular perturbations of non-convex functionals, emphasizing how fractional and higher-order seminorm perturbations shape Gamma-convergence and interfacial energies in phase transitions and free-discontinuity problems. It develops a unified framework where the Gamma-limit concentrates on interfaces with a density $m_{k+s}$ determined by optimal-profile problems, and it analyzes critical exponents where this density becomes singular. Through BBM and MS limit theories, plus new fractional and local higher-order results, the work connects sharp-interface limits across a range of $k$ and $s$, including higher-dimensional extensions and cases with non-convexity at infinity. The findings illuminate how non-locality and higher-order effects influence the effective surface tension and the structure of minimizers, with implications for phase separation models and variational problems with gradient discontinuities.

Abstract

Singular perturbations have been used to select solutions of (non-convex) variational problems with a multiplicity of minimizers. The prototype of such an approach is the gradient theory of phase transitions by L. Modica, who specialized some earlier Gamma-convergence results by himself and S. Mortola contained in a seminal paper, validating the so-called minimal-interface criterion. I will give an overview of some recent results on perturbations with fractional and higher-order seminorms both in the framework of phase transitions and of free-discontinuity problems, relating these results with the Bourgain-Brezis-Mironescu and Maz'ya-Shaposhnikova limit analysis for fractional Sobolev seminorms, and with the theory of Gamma-expansions.

Non-local singular perturbations of non-convex functionals -- recent results

TL;DR

The paper surveys singular perturbations of non-convex functionals, emphasizing how fractional and higher-order seminorm perturbations shape Gamma-convergence and interfacial energies in phase transitions and free-discontinuity problems. It develops a unified framework where the Gamma-limit concentrates on interfaces with a density determined by optimal-profile problems, and it analyzes critical exponents where this density becomes singular. Through BBM and MS limit theories, plus new fractional and local higher-order results, the work connects sharp-interface limits across a range of and , including higher-dimensional extensions and cases with non-convexity at infinity. The findings illuminate how non-locality and higher-order effects influence the effective surface tension and the structure of minimizers, with implications for phase separation models and variational problems with gradient discontinuities.

Abstract

Singular perturbations have been used to select solutions of (non-convex) variational problems with a multiplicity of minimizers. The prototype of such an approach is the gradient theory of phase transitions by L. Modica, who specialized some earlier Gamma-convergence results by himself and S. Mortola contained in a seminal paper, validating the so-called minimal-interface criterion. I will give an overview of some recent results on perturbations with fractional and higher-order seminorms both in the framework of phase transitions and of free-discontinuity problems, relating these results with the Bourgain-Brezis-Mironescu and Maz'ya-Shaposhnikova limit analysis for fractional Sobolev seminorms, and with the theory of Gamma-expansions.
Paper Structure (13 sections, 13 theorems, 54 equations)

This paper contains 13 sections, 13 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Singular perturbations of double-well energies
  3. Coerciveness in measure
  4. The surface tension
  5. The $\Gamma$-limit
  6. Analysis at critical exponents
  7. Analysis close the exponent $1/2$
  8. Analysis at positive integers from the left
  9. Analysis at positive integers from the right
  10. Some notes on the higher-dimensional case
  11. Singular perturbations of energies with non-convexity at infinity
  12. Perturbations of local type
  13. A result for fractional perturbations

Key Result

Lemma 1

If $F^{k+s}_\varepsilon(u_\varepsilon)\le S<+\infty$ for all $\varepsilon>0$, then there exists $\overline \eta>0$ such that for all $\eta\in (0,\overline\eta)$ there exists a constant $L$ such that for all intervals $J_\varepsilon$ such that $|J_\varepsilon|\ge \varepsilon L$ and $||u_\varepsilon|-

Theorems & Definitions (28)

  • Lemma 1: local interpolation lemma
  • proof
  • Theorem 2: Compactness
  • proof
  • Remark 3
  • Proposition 4: $k+s$-surface tension
  • Lemma 5: A cut-off lemma
  • proof
  • proof : Proof of Proposition \ref{['sut']}
  • Theorem 6: sharp-interface limit
  • ...and 18 more