Extremal domains of big volume for the first eigenvalue of the Laplace–Beltrami operator in a compact manifold

Sicbaldi, Pieralberto

Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014), p. 1231-1265 / Harvested from Numdam

Résumé

We prove the existence of new extremal domains for the first eigenvalue of the Laplace–Beltrami operator in some compact Riemannian manifolds of dimension $n ⩾ 2$ . 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 $n ⩾ 4$ , these domains are close to the complement of geodesic balls centered at a nondegenerate critical point of the scalar curvature.

Publié le : 2014-01-01

MR 3280066 | Zbl 1304.58011

DOI : https://doi.org/10.1016/j.anihpc.2013.09.001

@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},
     volume = {31},
     year = {2014},
     pages = {1231-1265},
     doi = {10.1016/j.anihpc.2013.09.001},
     mrnumber = {3280066},
     zbl = {1304.58011},
     language = {en},
     url = {http://dml.mathdoc.fr/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, Tome 31 (2014) pp. 1231-1265. doi : 10.1016/j.anihpc.2013.09.001. http://gdmltest.u-ga.fr/item/AIHPC_2014__31_6_1231_0/

Bibliographie

[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