Geodesic
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
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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 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.

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     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},
     year = {2014},
     publisher = {Elsevier},
     volume = {31},
     number = {6},
     doi = {10.1016/j.anihpc.2013.09.001},
     mrnumber = {3280066},
     zbl = {1304.58011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/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

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