Symmetry and Overdetermined Boundary Value Problems

Abstract

A polarized version of Bochner's identity is used to obtain symmetry results for an overdetermined boundary value problem on constant curvature manifolds. The identity is also used to prove a Rellich identity for Dirichlet eigenvalues of the Laplacian. Finally, a reflection argument which was developed by Alexandrov is used to obtain symmetry results for overdetermined boundary value problems on constant curvature manifolds.1980 Mathematics Subject Classification (1985 Revision): 53C20. l IntroductionIn this paper we consider some overdetermined boundary value problems for the Laplacian on spaces of constant curvature. Suppose M is a manifold of constant curvature and Ω c M is a domain with C 2 boundary. Let u e C 2 ( ) be a function such that Au=ffor some given radially Symmetrie function / and suppose in addition u satisfies the boundary conditions w = 0 on δ Ω and (d/dn)(u) = k (constant) on 3Ω. We want to then show that Ω is a metric ball in M. The given radially Symmetrie function,/, and the manifold M dictate the result one obtains s well s the method used. An example due to C. Berenstein and M. Karlowitz, [3], will help illustrate this point.On the Standard sphere, S" 1 , they construct a domain Ω c S" with smooth boundary and a smooth function u such that Au = -i on Ω, w = 0 on 5 Ω and (djdn)(u) = k on du. Furthermore the domain Ω is not radially Symmetrie. On the other hand if Ω is contained in the hemisphere and we consider the equation Au=f where/ = cos r and r is the geodesic distance from a fixed point or/ = -l, then in fact Ω is radially Symmetrie. Moreover, the methods used to prove these two results are quite different. The first technique we use involves an integral formula obtained from a polarized version of Bochner's identity. The second technique is a reflection argument due to Alexandrov [1].

Symmetry for an overdetermined boundary problem in a punctured domain

“…Not surprisingly, it is also for these spaces that satisfactory symmetry theorems have been proved by several authors, usually by means of the moving plane method (see e.g. [18,23,33] and the references therein). The existence of an isoperimetric inequality is a completely open problem today for other Riemannian manifolds, starting with complex hyperbolic space [5].…”

Schiffer’s Problem and an Isoperimetric Inequality for the First Buckling Eigenvalue of Domains on $$\mathbb{S}^{2}$$

“…This follows from the seminal paper of Serrin [15]. Serrin's result has been extended to the hyperbolic space H n and the hemisphere S n + (see [8,7]. In the case of convex domains, a nice alternative proof, not using the maximum principle, was given recently by Chatelain and Henrot [3].…”

A symmetry result for the p-Laplacian in a punctured manifold

“…Essentially the only technique available to tackle such problems is Serrin's method of moving planes, which has been successfully implemented in the hyperbolic space H n and the hemisphere S n + [21,25]. The problem is still wide open for arbitrary Riemannian manifolds, and even the aforementioned extensions to constant curvature spaces are not at all straightforward.…”