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Vectorial Bernoulli Problems and Free Boundary Systems

Giorgio Tortone, Bozhidar Velichkov

TL;DR

This article surveys recent advances in the regularity theory for vectorial Bernoulli free boundary problems and related energy-driven free boundary systems. It synthesizes variational and non-variational frameworks, detailing how blow-up analysis, Weiss-type monotonicity, and epsilon-regularity yield a robust structure: a regular free boundary with $C^{1,\alpha}$-type smoothness and a singular set whose Hausdorff dimension is controlled by critical dimensions, with detailed stratifications in vectorial and spectral settings. The work connects the vectorial Bernoulli problem to optimal domains for Dirichlet eigenvalues and to energy functionals arising in shape optimization, presenting a unified picture of state variables, stability, and regularity across scalar, vectorial, and spectral contexts; it also outlines key open problems, including singular-set refinements, degenerate vs non-degenerate regimes, and extensions to nonlocal operators. The significance lies in clarifying how vector-valued free boundaries behave under variational principles, guiding both theoretical analysis and potential applications in spectral optimization and material design.

Abstract

In this survey we go through some of the recent results about the regularity of vectorial free boundary problems of Bernoulli type and free boundary systems. The aim is to illustrate the general methodologies as well as to outline a selection of notable open questions.

Vectorial Bernoulli Problems and Free Boundary Systems

TL;DR

This article surveys recent advances in the regularity theory for vectorial Bernoulli free boundary problems and related energy-driven free boundary systems. It synthesizes variational and non-variational frameworks, detailing how blow-up analysis, Weiss-type monotonicity, and epsilon-regularity yield a robust structure: a regular free boundary with -type smoothness and a singular set whose Hausdorff dimension is controlled by critical dimensions, with detailed stratifications in vectorial and spectral settings. The work connects the vectorial Bernoulli problem to optimal domains for Dirichlet eigenvalues and to energy functionals arising in shape optimization, presenting a unified picture of state variables, stability, and regularity across scalar, vectorial, and spectral contexts; it also outlines key open problems, including singular-set refinements, degenerate vs non-degenerate regimes, and extensions to nonlocal operators. The significance lies in clarifying how vector-valued free boundaries behave under variational principles, guiding both theoretical analysis and potential applications in spectral optimization and material design.

Abstract

In this survey we go through some of the recent results about the regularity of vectorial free boundary problems of Bernoulli type and free boundary systems. The aim is to illustrate the general methodologies as well as to outline a selection of notable open questions.

Paper Structure

This paper contains 40 sections, 11 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. The one-phase Bernoulli problem
  3. Physical motivation
  4. Variational formulation
  5. Regularity theory for minimizers
  6. General Bernoulli free boundary systems
  7. The vectorial Bernoulli problem
  8. History and motivation
  9. The degenerate case
  10. Non-variational Bernoulli free boundary systems
  11. Structure of the survey
  12. Optimal domains for functionals involving Dirichlet eigenvalues
  13. Qualitative properties of eigenfunctions and optimal domains
  14. Regularity of optimal shapes for $\sum_{i=1}^k\lambda_i(A)$
  15. Free boundary reformulation
  16. ...and 25 more sections

Key Result

Theorem 1.1

Let $u\colon D \to \mathbb{R}$ be a minimizer of $\mathcal{F}$ in $D$. Then $u$ is (locally) Lipschitz continuous in $D$ and the set $\Omega_u := \{u>0\}$ is open. Moreover, the free boundary can be decomposed as the disjoint union of a regular and a singular part where:

Theorems & Definitions (20)

  • Theorem 1.1
  • Definition 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • Remark 2.5
  • Theorem 2.6: MazzoleniTreyVelichkov:AnnIHP
  • Theorem 2.7: KriventsovLin2019:degenerate
  • ...and 10 more