Serrin-Type Overdetermined Problems: an Alternative Proof

Brandolini, Barbara; Nitsch, Carlo; Salani, Paolo; Trombetti, Cristina

doi:10.1007/s00205-008-0119-3

Cited by 94 publications

(90 citation statements)

References 9 publications

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“…Weinberger's approach was used in [2,3,4] to extend the symmetry result to more general elliptic equations. Still by using integral identities, Serrin's theorem was extended to Hessian equations in [1]. The proof is in the same fashion as the one of Weinberger, but without using any P -function.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 98%

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A rigidity problem on the round sphere

Ciraolo

Vezzoni

2016

Commun. Contemp. Math.

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Section: Introduction and Statement Of The Resultsmentioning

confidence: 98%

“…A crucial tool for proving Serrin's result in [1] and [10] is the Pohožaev identity, which in the Euclidean space is given by n − 2 2…”

Section: A Pohožaev-type Identity In Rotationally Symmetric Spacesmentioning

confidence: 99%

A rigidity problem on the round sphere

Ciraolo

Vezzoni

2016

Commun. Contemp. Math.

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“…Next, we will make use of the fact that on every level set L t = {u = t} the following identity holds (see Brandolini and co. [2]):…”

Section: A Maximum Principlementioning

confidence: 99%

“…and Philippin and Safoui [8,9] (when p = N and gh = const.) for maximum principles or Payne and Schaefer [7] and Brandolini and co. [2] (when gh = const.) for symmetry results.…”

Section: Introductionmentioning

confidence: 99%

Maximum principles and symmetry results for a class of fully nonlinear elliptic PDEs

Enache

2010

Nonlinear Differ. Equ. Appl.

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Abstract. This paper is concerned with a class of boundary value problems for fully nonlinear elliptic PDEs involving the p-Hessian operator. We first derive a maximum principle for a suitable function involving the solution u (x) and its gradient. This maximum principle is then applied to obtain some sharp estimates for the solution and the magnitude of its gradient. We also investigate some symmetry properties of Ω or u (x) under specific boundary condition or geometry of Ω. Mathematics Subject Classification (2000). Primary 35B50, 35R35; Secondary 35B45.

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“…H. F. Weinberger gave in [15] an alternative proof by means of elementary arguments; mainly by describing the Laplacian in polar coordinates and applying the minimum principle and Green's identity. In the last decade, there have been generalizations of the problem; for instance to the case when the Laplacian is replaced by a quasilinear or nonlinear elliptic operator; to the case when the elliptic problem is stated on an exterior domain, or to the case when the overdetermined boundary condition is placed only in a part of the boundary, see [1,5,11,10,13] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Discrete Serrin's problem

Araúz

Carmona

Encinas

2015

Linear Algebra and its Applications

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We consider here the discrete analogue of Serrin's problem: if the equilibrium measure of a network with boundary satisfies that its normal derivative is constant, what can be said about the structure of the network and the symmetry of the equilibrium measure? In the original Serrin's problem, the conclusion is that the domain is a ball and the solution is radial. To study the discrete Serrin's problem, we first introduce the notion of radial function and then prove a generalization of the minimum principle, which is one of the main tools in the continuous case. Moreover, we obtain similar results to those of the continuous case for some families of networks with a balllike structure, which include spider networks with radial conductances, distance-regular graphs or, more generally, regular layered networks.

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Serrin-Type Overdetermined Problems: an Alternative Proof

Cited by 94 publications

References 9 publications

A rigidity problem on the round sphere

A rigidity problem on the round sphere

Maximum principles and symmetry results for a class of fully nonlinear elliptic PDEs

Discrete Serrin's problem

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