Delaunay type domains for an overdetermined elliptic problem in 𝕊 n × and n ×
ESAIM: Control, Optimisation and Calculus of Variations, Volume 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
@article{COCV_2016__22_1_1_0,
     author = {Morabito, Filippo and Sicbaldi, Pieralberto},
     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},
     volume = {22},
     number = {1},
     year = {2016},
     doi = {10.1051/cocv/2014064},
     zbl = {1336.58015},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2014064/}
}
TY  - JOUR
AU  - Morabito, Filippo
AU  - Sicbaldi, Pieralberto
TI  - Delaunay type domains for an overdetermined elliptic problem in $\mathrm{\mathbb{S}}^n \times{} \mathrm{\mathbb{R}}$ and $\mathbb{H}^{n} \times{} \mathbb{R}$
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2016
SP  - 1
EP  - 28
VL  - 22
IS  - 1
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv/2014064/
DO  - 10.1051/cocv/2014064
LA  - en
ID  - COCV_2016__22_1_1_0
ER  - 
%0 Journal Article
%A Morabito, Filippo
%A Sicbaldi, Pieralberto
%T Delaunay type domains for an overdetermined elliptic problem in $\mathrm{\mathbb{S}}^n \times{} \mathrm{\mathbb{R}}$ and $\mathbb{H}^{n} \times{} \mathbb{R}$
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2016
%P 1-28
%V 22
%N 1
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv/2014064/
%R 10.1051/cocv/2014064
%G en
%F COCV_2016__22_1_1_0
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, Volume 22 (2016) no. 1, pp. 1-28. doi : 10.1051/cocv/2014064. http://archive.numdam.org/articles/10.1051/cocv/2014064/

U. Abresch and H. Rosenberg, A Hopf differential for constant mean curvature surfaces in S 2 ×R and H 2 ×R. Acta Math. 193 (2004) 141–174. | DOI | Zbl

A.D. Alexandrov, Uniqueness theorems for surfaces in the large. (Russian) Vestnik Leningrad Univ. Math. 11 (1956) 5–17.

H.W. Alt and L.A. Caffarelli, Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math. 325 (1981) 105–144. | Zbl

L. Bessières, G. Besson, M. Boileau, S. Maillot and J. Porti, Geometrisation of 3-manifolds. Vol. 13 of EMS Tracts Math. European Mathematical Society, Zurich (2010). | Zbl

H. Berestycki, L.A. Caffarelli and L. Nirenberg, Monotonicity for elliptic equations in unbounded Lipschitz domains. Commun. Pure Appl. Math. 50 (1997) 1089–1111. | DOI | Zbl

I. Chavel, Eigenvalues in Riemannian geometry. Academic Press, Orlando, Florida (1984). | Zbl

C. Delaunay, Sur la surface de révolution dont la courbure moyenne est constante. With a note appended by M. Sturm. J. Math. Pures Appl. Sér. 1 6 (1841) 309–320.

Digital Library of Mathematical Functions. Available on http://dlmf.nist.gov/

A. Erdély, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher Transcendental Functions, vol. I. McGraw-Hill Book Company (1953). | Zbl

D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order. In Vol. 224 of A Series of Comprehensive Studies in Mathematics, Grundlehren der mathematischen Wissenschaften, 3rd edition. Springer-Verlag, Berlin-Heidelberg-New York (1977, 1983, 1998). | Zbl

F. Hélein, L. Hauswirth and F. Pacard, A note on some overdetermined problems. Pacific J. Math. 250 (2011) 319–334. | DOI | Zbl

M.A. Karlovitz, Some solutions to overdetermined boundary value problems on subsets of spheres. University of Maryland at College Park (1990).

T. Kato, Perturbation Theory for Linear Operators. Springer-Verlag, Berlin-Heidelberg-New York (1987). | Zbl

H. Kielhofer, Bifurcation Theory, An Introduction with Applications to PDEs. Appl. Math. Sci. 156 (2004). | Zbl

S. Kumaresan and J. Prajapat, Serrin’s result for hyperbolic space and sphere. Duke Math. J. 91 (1998) 17–28. | DOI | Zbl

N.N. Lebedev, Special functions and their applications. Dover Publications (1972). | Zbl

W.H. Meeks and H. Rosenberg, The theory of minimal surfaces in M×R. Comment. Math. Helv. 80 (2005) 811–858. | DOI | Zbl

W.H. Meeks and H. Rosenberg, Stable minimal surfaces in M×R. J. Differ. Geom. 68 (2004) 515–534. | Zbl

R. Molzon, Symmetry and overdetermined boundary value problems. Forum Math. 3 (1991) 143–156. | DOI | Zbl

F. Olver, Asymptotics and special functions. AK Peters (1997). | Zbl

R. Pedrosa and M. Ritoré, Isoperimetric domains in the Riemannian product of a circle with a simply connected space form and applications to free boundary problems. Indiana Univ. Math. J. 48 (1999) 1357–1394. | DOI | Zbl

G. Perelman, The entropy formula for the Ricci flow and its geometric applications. Preprint (2002). | arXiv | Zbl

G. Perelman, Ricci flow with surgery on three-manifolds. Preprint (2003). | arXiv | Zbl

G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. Preprint (2003) | arXiv | Zbl

P. Pucci and J. Serrin, The maximum principle. Progress in Nonlinear Differential Equations and Their Applications. Birkhauser, Basel (2007). | Zbl

A. Ros and P. Sicbaldi, Geometry and Topology for some overdetermined elliptic problems. J. Differ. Equ. 255 (2013) 951–977. | DOI | Zbl

F. Schlenk and P. Sicbaldi, Bifurcating extremal domains for the first eigenvalue of the Laplacian. Adv. Math. 229 (2012) 602–632. | DOI | Zbl

J. Serrin, A Symmetry Theorem in Potential Theory. Arch. Rational Mech. Anal. 43 (1971) 304–318. | DOI | Zbl

P. Sicbaldi, New extremal domains for the first eigenvalue of the Laplacian in flat tori. Calc. Var. Partial Differ. Equ. 37 (2010) 329–344. | DOI | Zbl

J. Smoller, Shock Waves and Reaction-Diffusion Equations. In Vol. 258 of A Series of Comprehensive Studies in Mathematics, Grundlehren der mathematischen Wissenschaften, 2nd edition. Springer-Verlag, Berlin-Heidelberg-New York (1994). | Zbl

I.S. Sokolnikoff, Mathematical theory of elasticity. McGraw-Hill Book Company, Inc., New York-Toronto-London (1956). | Zbl

M. Traizet, Classification of the solutions to an overdetermined elliptic problem in the plane. Geom. Funct. Anal. 24 (2014) 690–720. | DOI | Zbl

Cited by Sources: