| Issue |
ESAIM: COCV
Volume 22, Number 1, January-March 2016
|
|
|---|---|---|
| Page(s) | 1 - 28 | |
| DOI | https://doi.org/10.1051/cocv/2014064 | |
| Published online | 09 July 2015 | |
Delaunay type domains for an overdetermined elliptic problem in 
n × ℝ and ℍn × ℝ∗,∗∗
1
KAIST, Korea Advanced Institute of Science and Technology,
Department of Mathematical Sciences, 291 Daehak-ro, Yuseong-gu, 305701, Daejeon, South Korea
morabito@kaist.ac.kr
2
Korea Institute for Advanced Study, School of
Mathematics, 87 Hoegi-ro,
Dongdaemun-gu, 130-722, Seoul,
South Korea
3
Aix-Marseille Université, CNRS – Ecole Centrale Marseille, I2M,
UMR 7373, 13453
Marseille,
France
pieralberto.sicbaldi@univ-amu.fr
Received:
10
April
2014
Revised:
2
October
2014
We prove the existence of a countable family of Delaunay type domains 
t ∈ℕ, where 
n is the Riemannian manifold n or ℍn and n ≥ 2, bifurcating from the cylinder Bn × ℝ (where Bn is a geodesic ball in 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
has a bounded positive solution for some positive constant λ, where g is the standard metric in
n × 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 × ℝ.
Mathematics Subject Classification: 58J32 / 58J05 / 58J55 / 53C30 / 53A10 / 35B32 / 35R01 / 49Q05 / 49Q10 / 33CXX
Key words: Overdetermined elliptic problems / homogeneous manifolds / bifurcation / Laplace–Beltrami operator / Delaunay surfaces
The work which led to this article started while the first named author was affiliated to Korea University, Mathematics Department, Seoul, South Korea
The first author was supported by Basic Science Research Program through the National Research Foundation of South Korea (NRF) funded by the Ministry of Education, Grant NRF-2013R1A1A1013521. The second author was supported by the GDRE Geometric Analysis (France-Spain), the University of Marseille and the Grant ANR-11-IS01-0002.
© EDP Sciences, SMAI, 2015
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