Extremal domains for the first eigenvalue in a general compact Riemannian manifold

Erwann  Delay; Pieralberto  Sicbaldi

doi:10.3934/dcds.2015.35.5799

Article Contents

2015, Volume 35, Issue 12: 5799-5825. Doi: 10.3934/dcds.2015.35.5799

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Extremal domains for the first eigenvalue in a general compact Riemannian manifold

Erwann Delay^1, and
Pieralberto Sicbaldi^2,

1.
Laboratoire de Mathématiques d'Avignon, Faculté des Sciences, 33 rue Louis Pasteur, F-84000 Avignon
2.
Institut de Mathématiques de Marseille, 39 rue Joliot-Curie, 13453 Marseille Cedex 13

Received: December 31, 2013

Revised: August 31, 2014

Published: November 2015

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Abstract

We prove the existence of extremal domains with small prescribed volume for the first eigenvalue of the Laplace-Beltrami operator in any compact Riemannian manifold. This result generalizes a results of F. Pacard and the second author where the existence of a nondegenerate critical point of the scalar curvature of the Riemannian manifold was required.

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Mathematics Subject Classification: Primary: 49Q10, 53B20, 53C21; Secondary: 53A10, 35N25, 58C40.

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