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

[1]

David L. Finn. Noncompact manifolds with constant negative scalar curvature and singular solutions to semihnear elliptic equations. Conference Publications, 1998, 1998 (Special) : 262-275. doi: 10.3934/proc.1998.1998.262

[2]

Randa Ben Mahmoud, Hichem Chtioui. Prescribing the scalar curvature problem on higher-dimensional manifolds. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1857-1879. doi: 10.3934/dcds.2012.32.1857

[3]

Marco Ghimenti, Anna Maria Micheletti, Angela Pistoia. The role of the scalar curvature in some singularly perturbed coupled elliptic systems on Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2535-2560. doi: 10.3934/dcds.2014.34.2535

[4]

A. M. Micheletti, Angela Pistoia. Multiple eigenvalues of the Laplace-Beltrami operator and deformation of the Riemannian metric. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 709-720. doi: 10.3934/dcds.1998.4.709

[5]

Micol Amar, Roberto Gianni. Laplace-Beltrami operator for the heat conduction in polymer coating of electronic devices. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1739-1756. doi: 10.3934/dcdsb.2018078

[6]

Mark Pollicott. Closed geodesic distribution for manifolds of non-positive curvature. Discrete & Continuous Dynamical Systems - A, 1996, 2 (2) : 153-161. doi: 10.3934/dcds.1996.2.153

[7]

Yaiza Canzani, Dmitry Jakobson, Igor Wigman. Scalar curvature and $Q$-curvature of random metrics. Electronic Research Announcements, 2010, 17: 43-56. doi: 10.3934/era.2010.17.43

[8]

Andrea Bonito, Roland Glowinski. On the nodal set of the eigenfunctions of the Laplace-Beltrami operator for bounded surfaces in $R^3$: A computational approach. Communications on Pure & Applied Analysis, 2014, 13 (5) : 2115-2126. doi: 10.3934/cpaa.2014.13.2115

[9]

Simone Calogero, Stephen Pankavich. On the spatially homogeneous and isotropic Einstein-Vlasov-Fokker-Planck system with cosmological scalar field. Kinetic & Related Models, 2018, 11 (5) : 1063-1083. doi: 10.3934/krm.2018041

[10]

Colin Guillarmou, Antônio Sá Barreto. Inverse problems for Einstein manifolds. Inverse Problems & Imaging, 2009, 3 (1) : 1-15. doi: 10.3934/ipi.2009.3.1

[11]

Feng Luo. A combinatorial curvature flow for compact 3-manifolds with boundary. Electronic Research Announcements, 2005, 11: 12-20.

[12]

Alberto Farina, Enrico Valdinoci. A pointwise gradient bound for elliptic equations on compact manifolds with nonnegative Ricci curvature. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1139-1144. doi: 10.3934/dcds.2011.30.1139

[13]

Vittorio Martino. On the characteristic curvature operator. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1911-1922. doi: 10.3934/cpaa.2012.11.1911

[14]

M. Ben Ayed, Mohameden Ould Ahmedou. On the prescribed scalar curvature on $3$-half spheres: Multiplicity results and Morse inequalities at infinity. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 655-683. doi: 10.3934/dcds.2009.23.655

[15]

Igor Rivin and Jean-Marc Schlenker. The Schlafli formula in Einstein manifolds with boundary. Electronic Research Announcements, 1999, 5: 18-23.

[16]

Osama Khalil. Geodesic planes in geometrically finite manifolds. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 881-903. doi: 10.3934/dcds.2019037

[17]

Giuseppina Barletta, Roberto Livrea, Nikolaos S. Papageorgiou. A nonlinear eigenvalue problem for the periodic scalar $p$-Laplacian. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1075-1086. doi: 10.3934/cpaa.2014.13.1075

[18]

Paul W. Y. Lee, Chengbo Li, Igor Zelenko. Ricci curvature type lower bounds for sub-Riemannian structures on Sasakian manifolds. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 303-321. doi: 10.3934/dcds.2016.36.303

[19]

Misha Bialy. On Totally integrable magnetic billiards on constant curvature surface. Electronic Research Announcements, 2012, 19: 112-119. doi: 10.3934/era.2012.19.112

[20]

Mohammad Shafiee. A note on $2$-plectic homogeneous manifolds. Journal of Geometric Mechanics, 2015, 7 (3) : 389-394. doi: 10.3934/jgm.2015.7.389

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]