How to construct parametrized families of free boundaries near nondegenerate solutions

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.

[1]  Existence and stability of solutions for a fourth order overdetermined problem , .

[2]  Jimmy Lamboley,et al.  Structure of shape derivatives around irregular domains and applications , 2006 .

[3]  A. Henrot,et al.  Shape Variation and Optimization: A Geometrical Analysis , 2018 .

[4]  D. Kinderlehrer,et al.  Regularity in free boundary problems , 1977 .

[5]  A. Tromba.  A general approach to Morse theory , 1977 .

[6]  Lorenzo Cavallina.  The simultaneous asymmetric perturbation method for overdetermined free boundary problems , 2022, Nonlinear Analysis.

[7]  H. Shahgholian,et al.  Convexity of free boundaries with Bernoulli type boundary condition , 1997 .

[8]  L. Caffarelli,et al.  Existence and regularity for a minimum problem with free boundary. , 1981 .

[9]  A. Enciso,et al.  Solutions to the overdetermined boundary problem for semilinear equations with position-dependent nonlinearities , 2017, Advances in Mathematics.

[10]  A. Henrot,et al.  An overdetermined problem with non-constant boundary condition , 2013, 1302.4620.

[11]  S. Smale.  Morse theory and a non-linear generalization of the Dirichlet problem , 1964 .

[12]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[13]  James Serrin,et al.  A symmetry problem in potential theory , 1971 .

[14]  S. Bauer.  On the Segal conjecture for compact Lie groups. , 1989 .

[15]  R. Palais.  The Morse lemma for Banach spaces , 1969 .

[16]  L. Cavallina,et al.  On a two-phase Serrin-type problem and its numerical computation , 2018, ESAIM: Control, Optimisation and Calculus of Variations.

[17]  M. Flucher,et al.  Bernoulli's free-boundary problem, qualitative theory and numerical approximation. , 1997 .

[18]  L. Cavallina,et al.  Symmetry breaking solutions for a two-phase overdetermined problem of Serrin-type , 2020, 2001.10212.

[19]  Nikola Kamburov,et al.  Nontrivial solutions to Serrin's problem in annular domains , 2019, Annales de l'Institut Henri Poincaré C, Analyse non linéaire.

[20]  Stability analysis of the two-phase torsional rigidity near a radial configuration , 2018, Applicable Analysis.

[21]  Arian Novruzi,et al.  Structure of shape derivatives , 2002 .