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) 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 $n⩾2$. The volume of such domains is close to the volume of the manifold. If the first eigenfunction ${\phi }_0$ 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 ${\phi }_0$. If ${\phi }_0$ 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.

[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