DELAUNAY TYPE DOMAINS FOR AN OVERDETERMINED ELLIPTIC PROBLEM IN S-n x R AND H-n x R

Abstract

We prove the existence of a countable family of Delaunay type domains Omega(t) subset of M-n x R, t is an element of N, where M-n is the Riemannian manifold S-n or H-n and n >= 2, bifurcating from the cylinder B-n x R (where B-n is a geodesic ball in M-n) for which the first eigenfunction of the Laplace-Beltrami operator with zero Dirichlet boundary condition also has constant Neumann data at the boundary. In other words, the overdetermined problem {Delta(g) u + lambda u = 0 in ohm(t) u = 0 on partial derivative ohm(t) g(del u,v) = const. on partial derivative ohm(t) has a bounded positive solution for some positive constant lambda, where g is the standard metric in M-n x R. The domains Omega(t) are rotationally symmetric and periodic with respect to the R-axis of the cylinder and the sequence {Omega(t)}(t) converges to the cylinder B-n x R.