Nondegeneracy implies the existence of parametrized families of free boundaries

Article Mathematics, Applied

Existence and stability of solutions for a fourth order overdetermined problem

Alexandra Gilsbach et al.

Summary: This article examines a Serrin-type overdetermined boundary value problem for the biharmonic operator, proving the existence of a solution on the unit ball and in an open and bounded domain for small perturbations of the overdetermining condition, with stability estimates established. The approach is inspired by recent results and applies relevant research findings.

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS (2022)

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Article Mathematics, Applied

The simultaneous asymmetric perturbation method for overdetermined free boundary problems

Lorenzo Cavallina

Summary: This paper introduces a new method for finding nontrivial solutions to overdetermined problems with fixed and free boundaries by applying the implicit function theorem. The method's novelty lies in considering perturbations of different regularity levels on each boundary, allowing for the construction of solutions that would have been out of reach via simpler techniques. Additionally, the method improves the regularity gap between the free boundary and the given domain boundary, and can be used to construct solutions to specific problems near radially symmetric configurations.

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS (2022)

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Article Mathematics, Applied

Nontrivial solutions to Serrin's problem in annular domains

Nikola Kamburov et al.

Summary: Researchers have constructed a class of nontrivial bounded, real analytic domains bifurcating from annuli that satisfy a certain boundary value problem. They found that the condition is both sufficient and necessary, and proved that the constructed domains are self-Cheeger.

ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE (2021)

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Article Mathematics, Applied

Linear stability estimates for Serrin's problem via a modified implicit function theorem

Alexandra Gilsbach et al.

Summary: In this study, we investigate Serrin's classical overdetermined problem under a perturbation of the Neumann boundary condition. The existence of a solution for a constant Neumann boundary condition is established when the underlying domain is a ball. By developing a new implicit function theorem and conducting a detailed analysis of the linearized operator, we prove the existence and local uniqueness of a domain admitting a solution to the perturbed overdetermined problem, while also establishing an optimal linear stability estimate for the shape of the domain.

CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS (2021)

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Article Mathematics, Applied

Hyperbolic Solutions to Bernoulli's Free Boundary Problem

Antoine Henrot et al.

Summary: The study of Bernoulli's free boundary problem involves elliptic and hyperbolic solutions, with the former being stable and tractable, and the latter being unstable and less understood. By introducing a new implicit function theorem and solving non-local geometric flows, the existence of hyperbolic and elliptic solutions is addressed by clarifying the spectral structure of the linearized operator through harmonic analysis.

ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS (2021)

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Article Automation & Control Systems