We prove the existence of new extremal domains for the first eigenvalue of the Laplace–Beltrami operator in some compact Riemannian manifolds of dimension . The volume of such domains is close to the volume of the manifold. If the first eigenfunction 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 . If is a constant function and , 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{\textendash}Beltrami} operator in a compact manifold}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {1231--1265}, publisher = {Elsevier}, volume = {31}, number = {6}, year = {2014}, doi = {10.1016/j.anihpc.2013.09.001}, mrnumber = {3280066}, zbl = {1304.58011}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2013.09.001/} }
TY - JOUR AU - Sicbaldi, Pieralberto TI - Extremal domains of big volume for the first eigenvalue of the Laplace–Beltrami operator in a compact manifold JO - Annales de l'I.H.P. Analyse non linéaire PY - 2014 SP - 1231 EP - 1265 VL - 31 IS - 6 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2013.09.001/ DO - 10.1016/j.anihpc.2013.09.001 LA - en ID - AIHPC_2014__31_6_1231_0 ER -
%0 Journal Article %A Sicbaldi, Pieralberto %T Extremal domains of big volume for the first eigenvalue of the Laplace–Beltrami operator in a compact manifold %J Annales de l'I.H.P. Analyse non linéaire %D 2014 %P 1231-1265 %V 31 %N 6 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2013.09.001/ %R 10.1016/j.anihpc.2013.09.001 %G en %F 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/articles/10.1016/j.anihpc.2013.09.001/
[1] Uniqueness theorems for surfaces in the large, I, Vestn. Leningr. Univ., Math. 11 (1956), 5 -17 | MR
,[2] Nonlinear Analysis on Manifolds. Monge–Ampère Equations, Grundlehren Math. Wiss. vol. 252 , Springer-Verlag, New York (1982) | Zbl
,[3] 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] Sharp local isoperimetric inequalities involving the scalar curvature, Proc. Am. Math. Soc. 130 no. 8 (2002), 2351 -2361 | MR | Zbl
,[6] 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] Domain deformations and eigenvalues of the Dirichlet Laplacian in Riemannian manifold, Ill. J. Math. 51 (2007), 645 -666 | MR | Zbl
, ,[8] 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] Variational problems in the theory of elliptic partial differential equations, J. Ration. Mech. Anal. 2 (1953), 137 -171 | MR | Zbl
, ,[10] Über eine von Raleigh formulierte Minimaleigenschaft der Kreise, Math. Ann. 94 (1924), 97 -100 | EuDML | JFM | MR
,[11] Über Minimaleigenschaften der Kugel in drei und mehr Dimensionen, Acta Comment. Univ. Tartu (Dorpat) A 9 (1926), 1 -44 | JFM
,[12] Constant scalar curvature metrics with isolated singularities, Duke Math. J. 99 no. 3 (1999), 353 -418 | MR | Zbl
, ,[13] Lectures on Connected sum constructions in geometry and nonlinear analysis, http://www.math.polytechnique.fr/~pacard/Publications/Lecture-Part-I.pdf
,[14] 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] Linear and Nonlinear Aspects of Vortices: The Ginzburg Landau Model, Prog. Nonlinear Differ. Equ. Appl. vol. 39 , Birkäuser (2000) | MR | Zbl
, ,[16] 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] Constant mean curvature sphere in Riemannian manifolds, Manuscr. Math. 128 no. 3 (2009), 275 -295 | MR | Zbl
, ,[18] Geometry and topology of some overdetermined elliptic problems, J. Differ. Equ. 255 no. 5 (2013), 951 -977 | MR | Zbl
, ,[19] Lectures on Differential Geometry, International Press (1994) | MR | Zbl
, ,[20] A symmetry theorem in potential theory, Arch. Ration. Mech. Anal. 43 (1971), 304 -318 | MR | Zbl
,[21] Bifurcating extremal domains for the first eigenvalue of the Laplacian, Adv. Math. 229 (2012), 602 -632 | MR | Zbl
, ,[22] New extremal domains for the Laplacian in flat tori, Calc. Var. Partial Differ. Equ. 37 (2010), 329 -344 | MR | Zbl
,[23] Riemannian Geometry, Oxford Sci. Publ. (1996) | MR | Zbl
,[24] Foliation by constant mean curvature spheres, Pac. J. Math. 147 no. 2 (1991), 381 -396 | MR | Zbl
,[25] Eigenvalue variation for the Neumann problem, Appl. Math. Lett. 14 (2001), 39 -43 | MR | Zbl
,Cited by Sources: