Extremal domains of big volume for the first eigenvalue of the Laplace–Beltrami operator in a compact manifold
Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 6, pp. 1231-1265.

We prove the existence of new extremal domains for the first eigenvalue of the Laplace–Beltrami operator in some compact Riemannian manifolds of dimension n2. The volume of such domains is close to the volume of the manifold. If the first eigenfunction φ 0 of the Laplace–Beltrami operator over the manifold is a nonconstant function, these domains are close to the complement of geodesic balls centered at a nondegenerate critical point of φ 0 . If φ 0 is a constant function and n4, these domains are close to the complement of geodesic balls centered at a nondegenerate critical point of the scalar curvature.

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     title = {Extremal domains of big volume for the first eigenvalue of the {Laplace{\textendash}Beltrami} operator in a compact manifold},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1231--1265},
     publisher = {Elsevier},
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     year = {2014},
     doi = {10.1016/j.anihpc.2013.09.001},
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Sicbaldi, Pieralberto. Extremal domains of big volume for the first eigenvalue of the Laplace–Beltrami operator in a compact manifold. Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 6, pp. 1231-1265. doi : 10.1016/j.anihpc.2013.09.001. http://www.numdam.org/articles/10.1016/j.anihpc.2013.09.001/

[1] A.D. Alexandrov, Uniqueness theorems for surfaces in the large, I, Vestn. Leningr. Univ., Math. 11 (1956), 5 -17 | MR

[2] T. Aubin, Nonlinear Analysis on Manifolds. Monge–Ampère Equations, Grundlehren Math. Wiss. vol. 252 , Springer-Verlag, New York (1982) | Zbl

[3] G. Buttazzo, G. Dal Maso, An existence result for a class of Shape Optimization Problems, Arch. Ration. Mech. Anal. 122 (1993), 183 -195 | MR | Zbl

[4] E. Delay, P. Sicbaldi, Extremal domains for the first eigenvalue in a general compact Riemannian manifold, preprint. | MR

[5] O. Druet, Sharp local isoperimetric inequalities involving the scalar curvature, Proc. Am. Math. Soc. 130 no. 8 (2002), 2351 -2361 | MR | Zbl

[6] O. Druet, Asymptotic expansion of the Faber–Krahn profile of a compact Riemannian manifold, C. R. Acad. Sci. Paris, Ser. I 346 (2008), 1163 -1167 | MR | Zbl

[7] A. El Soufi, S. Ilias, Domain deformations and eigenvalues of the Dirichlet Laplacian in Riemannian manifold, Ill. J. Math. 51 (2007), 645 -666 | MR | Zbl

[8] 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 | JFM

[9] P.R. Garabedian, M. Schiffer, Variational problems in the theory of elliptic partial differential equations, J. Ration. Mech. Anal. 2 (1953), 137 -171 | MR | Zbl

[10] E. Krahn, Über eine von Raleigh formulierte Minimaleigenschaft der Kreise, Math. Ann. 94 (1924), 97 -100 | EuDML | JFM | MR

[11] E. Krahn, Über Minimaleigenschaften der Kugel in drei und mehr Dimensionen, Acta Comment. Univ. Tartu (Dorpat) A 9 (1926), 1 -44 | JFM

[12] R. Mazzeo, F. Pacard, Constant scalar curvature metrics with isolated singularities, Duke Math. J. 99 no. 3 (1999), 353 -418 | MR | Zbl

[13] F. Pacard, Lectures on Connected sum constructions in geometry and nonlinear analysis, http://www.math.polytechnique.fr/~pacard/Publications/Lecture-Part-I.pdf

[14] F. Pacard, F. Pimentel, Attaching handles to constant mean curvature one surfaces in hyperbolic 3-space, J. Inst. Math. Jussieu 3 no. 3 (2004), 421 -459 | MR | Zbl

[15] F. Pacard, T. Rivière, Linear and Nonlinear Aspects of Vortices: The Ginzburg Landau Model, Prog. Nonlinear Differ. Equ. Appl. vol. 39 , Birkäuser (2000) | MR | Zbl

[16] F. Pacard, P. Sicbaldi, Extremal domains for the first eigenvalue of the Laplace–Beltrami operator, Ann. Inst. Fourier 59 no. 2 (2009), 515 -542 | EuDML | Numdam | MR | Zbl

[17] F. Pacard, X. Xu, Constant mean curvature sphere in Riemannian manifolds, Manuscr. Math. 128 no. 3 (2009), 275 -295 | MR | Zbl

[18] A. Ros, P. Sicbaldi, Geometry and topology of some overdetermined elliptic problems, J. Differ. Equ. 255 no. 5 (2013), 951 -977 | MR | Zbl

[19] R. Schoen, S.T. Yau, Lectures on Differential Geometry, International Press (1994) | MR | Zbl

[20] J. Serrin, A symmetry theorem in potential theory, Arch. Ration. Mech. Anal. 43 (1971), 304 -318 | MR | Zbl

[21] F. Schlenk, P. Sicbaldi, Bifurcating extremal domains for the first eigenvalue of the Laplacian, Adv. Math. 229 (2012), 602 -632 | MR | Zbl

[22] P. Sicbaldi, New extremal domains for the Laplacian in flat tori, Calc. Var. Partial Differ. Equ. 37 (2010), 329 -344 | MR | Zbl

[23] T.J. Willmore, Riemannian Geometry, Oxford Sci. Publ. (1996) | MR | Zbl

[24] R. Ye, Foliation by constant mean curvature spheres, Pac. J. Math. 147 no. 2 (1991), 381 -396 | MR | Zbl

[25] D.Z. Zanger, Eigenvalue variation for the Neumann problem, Appl. Math. Lett. 14 (2001), 39 -43 | MR | Zbl

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