December  2015, 35(12): 5799-5825. doi: 10.3934/dcds.2015.35.5799

Extremal domains for the first eigenvalue in a general compact Riemannian manifold

1. 

Laboratoire de Mathématiques d'Avignon, Faculté des Sciences, 33 rue Louis Pasteur, F-84000 Avignon, France

2. 

Institut de Mathématiques de Marseille, 39 rue Joliot-Curie, 13453 Marseille Cedex 13, France

Received  January 2014 Revised  September 2014 Published  May 2015

We prove the existence of extremal domains with small prescribed volume for the first eigenvalue of the Laplace-Beltrami operator in any compact Riemannian manifold. This result generalizes a results of F. Pacard and the second author where the existence of a nondegenerate critical point of the scalar curvature of the Riemannian manifold was required.
Citation: Erwann Delay, Pieralberto Sicbaldi. Extremal domains for the first eigenvalue in a general compact Riemannian manifold. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5799-5825. doi: 10.3934/dcds.2015.35.5799
References:
[1]

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

[2]

O. Druet, Asymptotic expansion of the Faber-Krahn profile of a compact Riemannian manifold, C. R. Math. Acad. Sci. Paris, 346 (2008), 1163-1167. doi: 10.1016/j.crma.2008.09.022.

[3]

A. El Soufi and S. Ilias, Domain deformations and eigenvalues of the Dirichlet Laplacian in Riemannian manifold, Illinois J. Math., 51 (2007), 645-666.

[4]

G. Faber, Beweis, dass unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt, Sitzungsber. - Bayer. Akad. Wiss. München, Math.-Phys. Kl., (1923), 169-172.

[5]

P. R. Garadedian and M. Schiffer, Variational problems in the theory of elliptic partial differetial equations, J. Rat. Mech. An., 2 (1953), 137-171.

[6]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der mathematischen Wissenschaften, a Series of Comprehensive Studies in Mathematics, Vol. 224, $2^{nd}$ Edition, Springer, 1983. doi: 10.1007/978-3-642-61798-0.

[7]

K. Gro$\beta$e-Brauckmann, New surfaces of constant mean curvature, Math. Z., 214 (1993), 527-565. doi: 10.1007/BF02572424.

[8]

D. Henry, Perturbation of the Boundary in Boundary-Value Problems of Partial Differential Equations, London Mathematical Society, Lecture Note Series, 318, Cambridge University Press, 2005. doi: 10.1017/CBO9780511546730.

[9]

L. Karp and M. Pinsky, The first eigenvalue of a small geodesic ball in a Riemannian manifold, Sci. Math. (2), 111 (1987), 229-239.

[10]

E. Krahn, Über eine von Raleigh formulierte Minimaleigenschaft der Kreise, Math. Ann., 94 (1925), 97-100. doi: 10.1007/BF01208645.

[11]

E. Krahn, Uber Minimaleigenschaften der Kugel in drei und mehr dimensionen, Acta Comm. Univ. Tartu (Dorpat), A9 (1926), 1-44.

[12]

T. Lang and R. Wong, "Best possible'' upper bounds for the first two positive zeros of the Bessel function $J_\nu(x)$: The infinite case, J. Comput. Appl. Math., 71 (1996), 311-329. doi: 10.1016/0377-0427(95)00220-0.

[13]

J. M. Lee and T. H. Parker, The Yamabe Problem, Bulletin of the American Mathematical Society, 17 (1987), 37-91. doi: 10.1090/S0273-0979-1987-15514-5.

[14]

S. Nardulli, The isoperimetric profile of a smooth Riemannian manifold for small volumes, Ann. Global Anal. Geom., 36 (2009), 111-131. doi: 10.1007/s10455-008-9152-6.

[15]

F. Pacard and P. Sicbaldi, Extremal domains for the first eigenvalue of the Laplace-Beltrami operator, Ann. Inst. Fourier, 59 (2009), 515-542. doi: 10.5802/aif.2438.

[16]

F. Pacard and X. Xu, Constant mean curvature sphere in Riemannian manifolds, Manuscripta Math., 128 (2009), 275-295. doi: 10.1007/s00229-008-0230-7.

[17]

M. Ritoré, Superficies Con Curvatura Media Constante, Tesis doctoral, Universidad de Granada, 1994.

[18]

M. Ritoré, Examples of constant mean curvature surfaces obtained from harmonic maps to the two sphere, Math. Z., 226 (1997), 127-146. doi: 10.1007/PL00004326.

[19]

A. Ros and P. Sicbaldi, Geometry and Topology for some overdetermined elliptic problems, J. Diff. Eq., 255 (2013), 951-977. doi: 10.1016/j.jde.2013.04.027.

[20]

F. Schlenk and P. Sicbaldi, Bifurcating extremal domains for the first eigenvalue of the Laplacian, Adv. Math., 229 (2012), 602-632. doi: 10.1016/j.aim.2011.10.001.

[21]

R. Schoen and S. T. Yau, Lectures on Differential Geometry, International Press, 1994.

[22]

J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.

[23]

P. Sicbaldi, Extremal domains of big volume for the first eigenvalue of the Laplace-Beltrami operator in a compact manifold, Ann. Inst. Poincaré (C) An. non linéaire, 31 (2014), 1231-1265. doi: 10.1016/j.anihpc.2013.09.001.

[24]

T. J. Willmore, Riemannian Geometry, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1993.

[25]

R. Ye, Foliation by constant mean curvature spheres, Pacific J. Math., 147 (1991), 381-396. doi: 10.2140/pjm.1991.147.381.

show all references

References:
[1]

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

[2]

O. Druet, Asymptotic expansion of the Faber-Krahn profile of a compact Riemannian manifold, C. R. Math. Acad. Sci. Paris, 346 (2008), 1163-1167. doi: 10.1016/j.crma.2008.09.022.

[3]

A. El Soufi and S. Ilias, Domain deformations and eigenvalues of the Dirichlet Laplacian in Riemannian manifold, Illinois J. Math., 51 (2007), 645-666.

[4]

G. Faber, Beweis, dass unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt, Sitzungsber. - Bayer. Akad. Wiss. München, Math.-Phys. Kl., (1923), 169-172.

[5]

P. R. Garadedian and M. Schiffer, Variational problems in the theory of elliptic partial differetial equations, J. Rat. Mech. An., 2 (1953), 137-171.

[6]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der mathematischen Wissenschaften, a Series of Comprehensive Studies in Mathematics, Vol. 224, $2^{nd}$ Edition, Springer, 1983. doi: 10.1007/978-3-642-61798-0.

[7]

K. Gro$\beta$e-Brauckmann, New surfaces of constant mean curvature, Math. Z., 214 (1993), 527-565. doi: 10.1007/BF02572424.

[8]

D. Henry, Perturbation of the Boundary in Boundary-Value Problems of Partial Differential Equations, London Mathematical Society, Lecture Note Series, 318, Cambridge University Press, 2005. doi: 10.1017/CBO9780511546730.

[9]

L. Karp and M. Pinsky, The first eigenvalue of a small geodesic ball in a Riemannian manifold, Sci. Math. (2), 111 (1987), 229-239.

[10]

E. Krahn, Über eine von Raleigh formulierte Minimaleigenschaft der Kreise, Math. Ann., 94 (1925), 97-100. doi: 10.1007/BF01208645.

[11]

E. Krahn, Uber Minimaleigenschaften der Kugel in drei und mehr dimensionen, Acta Comm. Univ. Tartu (Dorpat), A9 (1926), 1-44.

[12]

T. Lang and R. Wong, "Best possible'' upper bounds for the first two positive zeros of the Bessel function $J_\nu(x)$: The infinite case, J. Comput. Appl. Math., 71 (1996), 311-329. doi: 10.1016/0377-0427(95)00220-0.

[13]

J. M. Lee and T. H. Parker, The Yamabe Problem, Bulletin of the American Mathematical Society, 17 (1987), 37-91. doi: 10.1090/S0273-0979-1987-15514-5.

[14]

S. Nardulli, The isoperimetric profile of a smooth Riemannian manifold for small volumes, Ann. Global Anal. Geom., 36 (2009), 111-131. doi: 10.1007/s10455-008-9152-6.

[15]

F. Pacard and P. Sicbaldi, Extremal domains for the first eigenvalue of the Laplace-Beltrami operator, Ann. Inst. Fourier, 59 (2009), 515-542. doi: 10.5802/aif.2438.

[16]

F. Pacard and X. Xu, Constant mean curvature sphere in Riemannian manifolds, Manuscripta Math., 128 (2009), 275-295. doi: 10.1007/s00229-008-0230-7.

[17]

M. Ritoré, Superficies Con Curvatura Media Constante, Tesis doctoral, Universidad de Granada, 1994.

[18]

M. Ritoré, Examples of constant mean curvature surfaces obtained from harmonic maps to the two sphere, Math. Z., 226 (1997), 127-146. doi: 10.1007/PL00004326.

[19]

A. Ros and P. Sicbaldi, Geometry and Topology for some overdetermined elliptic problems, J. Diff. Eq., 255 (2013), 951-977. doi: 10.1016/j.jde.2013.04.027.

[20]

F. Schlenk and P. Sicbaldi, Bifurcating extremal domains for the first eigenvalue of the Laplacian, Adv. Math., 229 (2012), 602-632. doi: 10.1016/j.aim.2011.10.001.

[21]

R. Schoen and S. T. Yau, Lectures on Differential Geometry, International Press, 1994.

[22]

J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.

[23]

P. Sicbaldi, Extremal domains of big volume for the first eigenvalue of the Laplace-Beltrami operator in a compact manifold, Ann. Inst. Poincaré (C) An. non linéaire, 31 (2014), 1231-1265. doi: 10.1016/j.anihpc.2013.09.001.

[24]

T. J. Willmore, Riemannian Geometry, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1993.

[25]

R. Ye, Foliation by constant mean curvature spheres, Pacific J. Math., 147 (1991), 381-396. doi: 10.2140/pjm.1991.147.381.

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