A rigidity problem on the round sphere

Abstract

Abstract. We consider a class of overdetermined problems in rotationally symmetric spaces, which reduce to the classical Serrin's overdetermined problem in the case of the Euclidean space. We prove some general integral identities for rotationally symmetric spaces which imply a rigidity result in the case of the round sphere.

Wulff shape characterizations in overdetermined anisotropic elliptic problems

“…Indeed, by using some integral identity and just one basic pointwise inequality (Cauchy-Schwarz) on the Hessian of the solution D 2 u, the authors prove that D 2 u is a multiple of the identity matrix, which easily leads to the conclusion. This strategy has been also used in [12] to extend Serrin's result to the Finsler Laplacian, in [9] for the exterior Serrin's problem in anisotropic spaces (see below for a more detalied description) and in [14] for an overdetermined problem on the round sphere.…”

A Serrin-type symmetry result on model manifolds: an extension of the Weinberger argument

“…• Condition (b) appears in other articles on the subject, see for instance [4] by Ciraolo and Vezzoni. • The "compatibility" condition (3) describes a property of the solution in relation to the geometry of the model.…”

Overdetermined problems for fully nonlinear equations in space forms

“…In space forms, Molzon [11] used a Pfunction to prove symmetry results for equation ∆u = V (see the definition of V in Section 2) in S n + and used the moving plane method to obtain the symmetry results for equation ∆u = −1 in space forms; Kumaresan and Prajapat [9] used the moving plane method to prove the results for equation ∆u+g(u) = 0, where g is a C 1 function, under the condition u > 0. For more papers about space forms, the interested readers may refer to [13,4,5,6] and references therein.…”