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, Volume 31 (2014) no. 6, p. 1231-1265
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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.

@article{AIHPC_2014__31_6_1231_0,
     author = {Sicbaldi, Pieralberto},
     title = {Extremal domains of big volume for the first eigenvalue of the Laplace--Beltrami operator in a compact manifold},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {31},
     number = {6},
     year = {2014},
     pages = {1231-1265},
     doi = {10.1016/j.anihpc.2013.09.001},
     zbl = {1304.58011},
     mrnumber = {3280066},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2014__31_6_1231_0}
}
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, Volume 31 (2014) no. 6, pp. 1231-1265. doi : 10.1016/j.anihpc.2013.09.001. http://www.numdam.org/item/AIHPC_2014__31_6_1231_0/

[1] A.D. Alexandrov, Uniqueness theorems for surfaces in the large, I, Vestn. Leningr. Univ., Math. 11 (1956), 5 -17 | MR 86338

[2] T. Aubin, Nonlinear Analysis on Manifolds. Monge–Ampère Equations, Grundlehren Math. Wiss. vol. 252 , Springer-Verlag, New York (1982) | Zbl 0512.53044

[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 1217590 | Zbl 0811.49028

[4] E. Delay, P. Sicbaldi, Extremal domains for the first eigenvalue in a general compact Riemannian manifold, preprint. | MR 3393256

[5] O. Druet, Sharp local isoperimetric inequalities involving the scalar curvature, Proc. Am. Math. Soc. 130 no. 8 (2002), 2351 -2361 | MR 1897460 | Zbl 1067.53026

[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 2464258 | Zbl 1158.53031

[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 2342681 | Zbl 1124.49035

[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 49.0342.03

[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 54819 | Zbl 0050.10002

[10] E. Krahn, Über eine von Raleigh formulierte Minimaleigenschaft der Kreise, Math. Ann. 94 (1924), 97 -100 | JFM 51.0356.05 | MR 1512244

[11] E. Krahn, Über Minimaleigenschaften der Kugel in drei und mehr Dimensionen, Acta Comment. Univ. Tartu (Dorpat) A 9 (1926), 1 -44 | JFM 52.0510.03

[12] R. Mazzeo, F. Pacard, Constant scalar curvature metrics with isolated singularities, Duke Math. J. 99 no. 3 (1999), 353 -418 | MR 1712628 | Zbl 0945.53024

[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 2074431 | Zbl 1059.53049

[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 1763040 | Zbl 0948.35003

[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 | Numdam | MR 2521426 | Zbl 1166.53029

[17] F. Pacard, X. Xu, Constant mean curvature sphere in Riemannian manifolds, Manuscr. Math. 128 no. 3 (2009), 275 -295 | MR 2481045 | Zbl 1165.53038

[18] A. Ros, P. Sicbaldi, Geometry and topology of some overdetermined elliptic problems, J. Differ. Equ. 255 no. 5 (2013), 951 -977 | MR 3062759 | Zbl 1284.35297

[19] R. Schoen, S.T. Yau, Lectures on Differential Geometry, International Press (1994) | MR 1333601 | Zbl 0830.53001

[20] J. Serrin, A symmetry theorem in potential theory, Arch. Ration. Mech. Anal. 43 (1971), 304 -318 | MR 333220 | Zbl 0222.31007

[21] F. Schlenk, P. Sicbaldi, Bifurcating extremal domains for the first eigenvalue of the Laplacian, Adv. Math. 229 (2012), 602 -632 | MR 2854185 | Zbl 1233.35147

[22] P. Sicbaldi, New extremal domains for the Laplacian in flat tori, Calc. Var. Partial Differ. Equ. 37 (2010), 329 -344 | MR 2592974 | Zbl 1188.35122

[23] T.J. Willmore, Riemannian Geometry, Oxford Sci. Publ. (1996) | MR 1261641 | Zbl 0797.53002

[24] R. Ye, Foliation by constant mean curvature spheres, Pac. J. Math. 147 no. 2 (1991), 381 -396 | MR 1084717 | Zbl 0722.53022

[25] D.Z. Zanger, Eigenvalue variation for the Neumann problem, Appl. Math. Lett. 14 (2001), 39 -43 | MR 1793700 | Zbl 0977.58028