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

Filippo Morabito (1) , Pieralberto Sicbaldi (2)

1 Department of Mathematical Sciences, KAIST
2 I2M - Institut de Mathématiques de Marseille

We prove the existence of a countable family of Delaunay type domains \Omega_j in M^n x R, where M^n is the Riemannian manifold S^n or H^n and n is at least 2, bifurcating from the cylinder B^n x R (where B^n is a geodesic ball of radius 1 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. The domains \Omega_j are rotationally symmetric and periodic with respect to the R-axis of the cylinder and as j converges to 0 the domain \Omega_j converges to the cylinder B^n x R.

Domaines: Géométrie différentielle [math.DG]; Equations aux dérivées partielles [math.AP]

Filippo Morabito, Pieralberto Sicbaldi. DELAUNAY TYPE DOMAINS FOR AN OVERDETERMINED ELLIPTIC PROBLEM IN S^n x R AND H^n x R. ESAIM: Control, Optimisation and Calculus of Variations, 2016, 22, pp.1-28. ⟨10.1051/cocv/2014064⟩. ⟨hal-01266515⟩