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 & Continuous Dynamical Systems - A, 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.   Google Scholar

[2]

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

[3]

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

[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, (1923), 169.   Google Scholar

[5]

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

[6]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Grundlehren der mathematischen Wissenschaften, (1983).  doi: 10.1007/978-3-642-61798-0.  Google Scholar

[7]

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

[8]

D. Henry, Perturbation of the Boundary in Boundary-Value Problems of Partial Differential Equations,, London Mathematical Society, (2005).  doi: 10.1017/CBO9780511546730.  Google Scholar

[9]

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

[10]

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

[11]

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

[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.  doi: 10.1016/0377-0427(95)00220-0.  Google Scholar

[13]

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

[14]

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

[15]

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

[16]

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

[17]

M. Ritoré, Superficies Con Curvatura Media Constante,, Tesis doctoral, (1994).   Google Scholar

[18]

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

[19]

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

[20]

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

[21]

R. Schoen and S. T. Yau, Lectures on Differential Geometry,, International Press, (1994).   Google Scholar

[22]

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

[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.  doi: 10.1016/j.anihpc.2013.09.001.  Google Scholar

[24]

T. J. Willmore, Riemannian Geometry,, Oxford Science Publications, (1993).   Google Scholar

[25]

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

show all references

References:
[1]

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

[2]

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

[3]

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

[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, (1923), 169.   Google Scholar

[5]

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

[6]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Grundlehren der mathematischen Wissenschaften, (1983).  doi: 10.1007/978-3-642-61798-0.  Google Scholar

[7]

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

[8]

D. Henry, Perturbation of the Boundary in Boundary-Value Problems of Partial Differential Equations,, London Mathematical Society, (2005).  doi: 10.1017/CBO9780511546730.  Google Scholar

[9]

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

[10]

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

[11]

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

[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.  doi: 10.1016/0377-0427(95)00220-0.  Google Scholar

[13]

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

[14]

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

[15]

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

[16]

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

[17]

M. Ritoré, Superficies Con Curvatura Media Constante,, Tesis doctoral, (1994).   Google Scholar

[18]

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

[19]

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

[20]

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

[21]

R. Schoen and S. T. Yau, Lectures on Differential Geometry,, International Press, (1994).   Google Scholar

[22]

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

[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.  doi: 10.1016/j.anihpc.2013.09.001.  Google Scholar

[24]

T. J. Willmore, Riemannian Geometry,, Oxford Science Publications, (1993).   Google Scholar

[25]

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

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