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

TL;DR

This paper completes the classification of blow-up limits for minimizers in the vectorial Bernoulli free boundary problem and analyzes the asymptotic behavior of solutions under measure constraints.

Contribution

It provides a complete classification of blow-up limits and studies the asymptotics of minimizers with measure constraints in a bounded domain.

Findings

Classification of all blow-up limits achieved.

Asymptotic behavior of minimizers characterized as measure constraint varies.

Insights into singular solutions of the vectorial Bernoulli problem.

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.