4.7 Article

Nondegeneracy implies the existence of parametrized families of free boundaries

期刊

JOURNAL OF DIFFERENTIAL EQUATIONS
卷 383, 期 -, 页码 1-23

出版社

ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jde.2023.11.011

关键词

Overdetermined problem; Free boundary problem; Nondegenerate critical points; Shape derivatives; Implicit function theorem; Serrin's overdetermined problem

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In this paper, the concept of variational free boundary problem is introduced, and a unified functional-analytical framework is provided for constructing families of solutions. The notion of nondegeneracy of a critical point is extended to this setting.
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. (c) 2023 The Author. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons .org /licenses /by -nc -nd /4 .0/).

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