Sharp estimates and saturation phenomena for a nonlocal eigenvalue problem |
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| Authors: | B Brandolini P Freitas C Nitsch C Trombetti |
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| Institution: | aDipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, Complesso Monte S. Angelo, via Cintia, 80126 Napoli, Italy;bDepartment of Mathematics, Human Kinetics Faculty, Technical University of Lisbon, and Group of Mathematical Physics of the University of Lisbon, Complexo Interdisciplinar, Av. Prof. Gama Pinto 2, P-1649-003 Lisboa, Portugal |
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| Abstract: | We determine the shape which minimizes, among domains with given measure, the first eigenvalue of a nonlocal operator consisting of a perturbation of the standard Dirichlet Laplacian by an integral of the unknown function. We show that this problem displays a saturation behaviour in that the corresponding value of the minimal eigenvalue increases with the weight affecting the average up to a (finite) critical value of this weight, and then remains constant. This critical point corresponds to a transition between optimal shapes, from one ball as in the Faber–Krahn inequality to two equal balls. |
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| Keywords: | MSC: primary 35P15 secondary 49R50 |
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