We prove the existence of a countable family of Delaunay type domains Ωt ⊂ double-struck Mn x ℝ, t ∈ ℕ, where double-struck Mn is the Riemannian manifold Sn or ℍn and n ≥ 2, bifurcating from the cylinder Bn x ℝ (where Bn is a geodesic ball in double-struck Mn) 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 {▵g u + λ u = 0 in Ωt u = 0 on ∂Ωt g(▿u, ν) = const. on ∂Ωt has a bounded positive solution for some positive constant λ, where g is the standard metric double-struck Mn x ℝ. The domains Ωt are rotationally symmetric and periodic with respect to the ℝ-axis of the cylinder and the sequence {Ωt}t converges to the cylinder Bn x ℝ.
Delaunay type domains for an overdetermined elliptic problem in Sn × R and Hn × R
Morabito, Filippo;
2016-01-01
Abstract
We prove the existence of a countable family of Delaunay type domains Ωt ⊂ double-struck Mn x ℝ, t ∈ ℕ, where double-struck Mn is the Riemannian manifold Sn or ℍn and n ≥ 2, bifurcating from the cylinder Bn x ℝ (where Bn is a geodesic ball in double-struck Mn) 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 {▵g u + λ u = 0 in Ωt u = 0 on ∂Ωt g(▿u, ν) = const. on ∂Ωt has a bounded positive solution for some positive constant λ, where g is the standard metric double-struck Mn x ℝ. The domains Ωt are rotationally symmetric and periodic with respect to the ℝ-axis of the cylinder and the sequence {Ωt}t converges to the cylinder Bn x ℝ.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
