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.
|Number of pages||14|
|Journal||Advances in Mathematics|
|Publication status||Published - 2011|
All Science Journal Classification (ASJC) codes
- General Mathematics