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

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.

💡 Research Summary

This paper completes the classification of blow‑up limits for minimizers of the vectorial Bernoulli free‑boundary problem and investigates the problem under a volume constraint. The authors consider the functional

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