How to construct parametrized families of free boundaries near nondegenerate solutions

TL;DR

This paper develops a unified analytical framework for constructing families of solutions to variational free boundary problems by extending nondegeneracy concepts, demonstrated through applications to the two-phase Serrin's overdetermined problem.

Contribution

It introduces the notion of variational free boundary problems and extends nondegeneracy, enabling the construction of solution families near nondegenerate solutions.

Findings

Unified framework for variational free boundary problems

Construction of solution families near nondegenerate solutions

Application to two-phase Serrin's overdetermined problem

Abstract

In this paper, we introduce the notion of variational free boundary problem. Namely, we say that a free boundary problem is variational if its solutions can be characterized as the critical points of some shape functional. Moreover, we extend the notion of nondegeneracy of a critical point to this setting. As a result, we provide a unified functional-analytical framework that allows us to construct families of solutions to variational free boundary problems whenever the shape functional is nondegenerate at some given solution. As a clarifying example, we apply this machinery to construct families of nontrivial solutions to the two-phase Serrin's overdetermined problem in both the degenerate and nondegenerate case.