Sharp estimates and saturation phenomena for a nonlocal eigenvalue problem

Barbara Brandolini, Barbara Brandolini, Carlo Nitsch, Cristina Trombetti, Freitas

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


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.
Original languageEnglish
Pages (from-to)2352-2365
Number of pages14
JournalAdvances in Mathematics
Publication statusPublished - 2011

All Science Journal Classification (ASJC) codes

  • General Mathematics


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