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On the blow-up of the vectorial Bernoulli free boundary problem

Giovanni Siclari, Bozhidar Velichkov

TL;DR

This work completes the blow-up classification for minimizers of the vectorial Bernoulli free boundary problem by linking blow-up behavior to a capacitary variational problem $\Lambda^*(A)$ and proving that, for linear data, the energy threshold can surpass the naive bound set by $\|A\|^2$ when $\operatorname{rk}(A)>1$. It develops existence and penalization schemes for measure-constrained vectorial free boundaries, derives a shape-variation formula, and shows how penalization recovers the constrained problem for small penalties, enabling a robust asymptotic analysis as the measure constraint approaches the domain’s full measure. The linear-case analysis reduces the computation of $\Lambda^*(A)$ to a lower-dimensional capacitary problem, proves that $\Lambda^*(A)$ depends only on the blow-up point (and is upper semicontinuous), and establishes boundedness of the contact set for blow-ups. The asymptotic results as $m\to|D|$ yield convergence of minimizers to the harmonic extension of the boundary data with an explicit energy limit tied to a capacitary minimization, while the radiality lemma provides a powerful tool to compare energies across symmetric competitors. Together, these results advance the understanding of singular homogeneous global solutions and the full blow-up classification in the vectorial setting, with implications for higher-regularity theory of vectorial free boundaries.

Abstract

In this paper, we complete the classification of the blow-up limits of minimizers of the vectorial Bernoulli free boundary problem. Furthermore, we study the vectorial Bernoulli free boundary problem in a bounded box $D$, with a constraint $m$ on the measure of the positivity set, and the asymptotic of minimizers as the measure constraint $m$ tends to $|D|$. Such a study with a linear datum on the fixed boundary is the main ingredient for the characterization of the singular homogeneous global solutions of the vectorial problem and, thus, for the classification of the blow-up limits.

On the blow-up of the vectorial Bernoulli free boundary problem

TL;DR

This work completes the blow-up classification for minimizers of the vectorial Bernoulli free boundary problem by linking blow-up behavior to a capacitary variational problem and proving that, for linear data, the energy threshold can surpass the naive bound set by when . It develops existence and penalization schemes for measure-constrained vectorial free boundaries, derives a shape-variation formula, and shows how penalization recovers the constrained problem for small penalties, enabling a robust asymptotic analysis as the measure constraint approaches the domain’s full measure. The linear-case analysis reduces the computation of to a lower-dimensional capacitary problem, proves that depends only on the blow-up point (and is upper semicontinuous), and establishes boundedness of the contact set for blow-ups. The asymptotic results as yield convergence of minimizers to the harmonic extension of the boundary data with an explicit energy limit tied to a capacitary minimization, while the radiality lemma provides a powerful tool to compare energies across symmetric competitors. Together, these results advance the understanding of singular homogeneous global solutions and the full blow-up classification in the vectorial setting, with implications for higher-regularity theory of vectorial free boundaries.

Abstract

In this paper, we complete the classification of the blow-up limits of minimizers of the vectorial Bernoulli free boundary problem. Furthermore, we study the vectorial Bernoulli free boundary problem in a bounded box , with a constraint on the measure of the positivity set, and the asymptotic of minimizers as the measure constraint tends to . Such a study with a linear datum on the fixed boundary is the main ingredient for the characterization of the singular homogeneous global solutions of the vectorial problem and, thus, for the classification of the blow-up limits.
Paper Structure (16 sections, 26 theorems, 248 equations)

This paper contains 16 sections, 26 theorems, 248 equations.

Table of Contents

  1. Introduction
  2. Existence and properties of minimizers
  3. An explicit bound for $\Lambda^*({\rm{Id}_d})$
  4. Strategy of the proof and plan of the paper
  5. Existence and penalization for a measure constrained vectorial free boundary problem
  6. Existence
  7. A penalized functional
  8. Shape variation of the free boundary and consequences
  9. Shape Variation
  10. Equivalence with the constrained problem
  11. Asymptotics with respect to the measure constraint
  12. The linear case
  13. Computation of $\Lambda^*(A)$
  14. $\Lambda^*(A)$ for blows-up of minimizers of the vectorial Bernoulli problem
  15. Boundedness of the contact set
  16. ...and 1 more sections

Key Result

Theorem 1.4

Let $A$ be a $k \times d$ matrix with ${\rm{rk}}(A)=n$ and $1\le n\le d$. Let $Q \in \mathbb{R}^{d,d}$ be an orthogonal matrix such that $A= Q$, for some matrix $A_1 \in \mathbb{R}^{k,n}$ of rank $n$. Then, the following holds:

Theorems & Definitions (60)

  • Definition 1.1: Vectorial minimizers in $D$
  • Definition 1.2: Blow-ups
  • Definition 1.3: Global vectorial minimizers
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Theorem 1.9
  • Lemma 1.10: Radiality reduction lemma
  • ...and 50 more