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.

Brandolini B., Freitas P., Nitsch C., & Trombetti C. (2011). Sharp estimates and saturation phenomena for a nonlocal eigenvalue problem. ADVANCES IN MATHEMATICS, 228(4), 2352-2365 [10.1016/j.aim.2011.07.007].

Sharp estimates and saturation phenomena for a nonlocal eigenvalue problem

Brandolini B.
;
2011

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.
Settore MAT/05 - Analisi Matematica
Brandolini B., Freitas P., Nitsch C., & Trombetti C. (2011). Sharp estimates and saturation phenomena for a nonlocal eigenvalue problem. ADVANCES IN MATHEMATICS, 228(4), 2352-2365 [10.1016/j.aim.2011.07.007].
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/10447/494003
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