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.

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.

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     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},
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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, 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/

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