Serrin's problem and Alexandrov's Soap Bubble Theorem: enhanced stability via integral identities

Magnanini, Rolando; Poggesi, Giorgio

doi:10.1512/iumj.2020.69.7925

Cited by 34 publications

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“…This fact suggests that δ(ψ) measures the deviation of ψ from Q, or that of Ω from a ball. This observation was further extended by Magnanini and Poggesi [17][18][19], and they showed that the integral identity (1.6)…”

Section: Introductionmentioning

confidence: 61%

Stability analysis of an overdetermined fourth order boundary value problem via an integral identity

Okamoto,

Onodera

2021

Preprint

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We consider an overdetermined fourth order boundary value problem in which the boundary value of the Laplacian of the solution is prescribed, in addition to the homogeneous Dirichlet boundary condition. It is known that, in the case where the prescribed boundary value is a constant, this overdetermined problem has a solution if and only if the domain under consideration is a ball. In this paper, we study the shape of a domain admitting a solution to the overdetermined problem when the prescribed boundary value is slightly perturbed from a constant. We derive an integral identity for the fourth order Dirichlet problem and a nonlinear weighted trace inequality, and the combination of them results in a quantitative stability estimate which measures the deviation of a domain from a ball in terms of the perturbation of the boundary value.

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Section: Introductionmentioning

confidence: 61%

Stability analysis of an overdetermined fourth order boundary value problem via an integral identity

Okamoto,

Onodera

2021

Preprint

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“…The key is, that the soap bubble theorem can also be proved by integral methods, see [Ros87] and also compare [Rei82]. With the help of integral formulae, which are valid for solutions f of the torsion Dirichlet problem ∆f = 1 in Ω MP20a,MP20b] were able to prove that what they call Cauchy-Schwarz deficit…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…In the Euclidean space, under a C 0 -condition on | ∇f | this was answered affirmatively in [ABR99, BNST08a,CMV16]. When it comes to pinching in terms of an integral quantity Magnanini/Poggesi [MP20b] have shown the following identity for solutions of (1.5):…”

Section: Defining Functions and Level Setsmentioning

confidence: 91%

Stability from rigidity via umbilicity

Scheuer¹

2021

Preprint

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We consider a range of geometric stability problems for hypersurfaces of spaceforms. One of the key results is an estimate relating the Hausdorff distance to a geodesic sphere of an embedded hypersurface to the L 2 -norm of the traceless Hessian operator of a defining function for the open set bounded by the hypersurface. As application, this result allows for a unified treatment of many old and new stability problems arising in geometry and analysis. Those problems ask for spherical closeness of a hypersurface, given a geometric constraint. Examples include stability in Alexandroff's soap bubble theorem in space forms, Serrin's overdetermined problem, a Steklov problem involving the bi-Laplace operator and non-convex Alexandroff-Fenchel inequalities.

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“…Another optimal stability estimate for proximity to a single sphere was obtained in [90] by using a different approach. Other quantitative studies regarding the proximity to a single ball can be found in [62,97,98,99] where a different deficit is considered.…”

Section: Theorem 15 ([1])mentioning

confidence: 99%

The method of moving planes: a quantitative approach

Ciraolo,

Roncoroni

2018

Preprint

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We review classical results where the method of the moving planes has been used to prove symmetry properties for overdetermined PDE's boundary value problems (such as Serrin's overdetermined problem) and for rigidity problems in geometric analysis (like Alexandrov soap bubble Theorem), and we give an overview of some recent results related to quantitative studies of the method of moving planes, where quantitative approximate symmetry results are obtained.

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Serrin's problem and Alexandrov's Soap Bubble Theorem: enhanced stability via integral identities

Cited by 34 publications

References 0 publications

Stability analysis of an overdetermined fourth order boundary value problem via an integral identity

Stability analysis of an overdetermined fourth order boundary value problem via an integral identity

Stability from rigidity via umbilicity

The method of moving planes: a quantitative approach

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