Delaunay type domains for an overdetermined elliptic problem in 𝕊 n × and n ×
ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 1, pp. 1-28.

We prove the existence of a countable family of Delaunay type domains

Ω t 𝕄 n × ,
t , where 𝕄 n is the Riemannian manifold 𝕊 n or n and n2, bifurcating from the cylinder B n × (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
Δ g u + λ u = 0 in Ω t u = 0 on Ω t g ( u , ν ) = const. on Ω t
has a bounded positive solution for some positive constant λ, where g is the standard metric in 𝕄 n × . The domains Ω t are rotationally symmetric and periodic with respect to the -axis of the cylinder and the sequence {Ω t } t converges to the cylinder B n × .

DOI : 10.1051/cocv/2014064
Classification : 58J32, 58J05, 58J55, 53C30, 53A10, 35B32, 35R01, 49Q05, 49Q10, 33CXX
Mots clés : Overdetermined elliptic problems, homogeneous manifolds, bifurcation, Laplace–Beltrami operator, Delaunay surfaces
Morabito, Filippo 1, 2 ; Sicbaldi, Pieralberto 3

1 KAIST, Korea Advanced Institute of Science and Technology, Department of Mathematical Sciences, 291 Daehak-ro, Yuseong-gu, 305701, Daejeon, South Korea
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
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     title = {Delaunay type domains for an overdetermined elliptic problem in $\mathrm{\mathbb{S}}^n \times{} \mathrm{\mathbb{R}}$ and $\mathbb{H}^{n} \times{} \mathbb{R}$},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {1--28},
     publisher = {EDP-Sciences},
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Morabito, Filippo; Sicbaldi, Pieralberto. Delaunay type domains for an overdetermined elliptic problem in $\mathrm{\mathbb{S}}^n \times{} \mathrm{\mathbb{R}}$ and $\mathbb{H}^{n} \times{} \mathbb{R}$. ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 1, pp. 1-28. doi : 10.1051/cocv/2014064. http://www.numdam.org/articles/10.1051/cocv/2014064/

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