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

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  • 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.
    Mathematics Subject Classification: Primary: 49Q10, 53B20, 53C21; Secondary: 53A10, 35N25, 58C40.

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  • [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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