(Fractional) principal frequencies and inradius

Abstract

We consider the first eigenvalue of the Dirichlet-Laplacian in a general open set. We seek for a lower bound for this object, in terms of the inradius (i.e., the radius of a largest ball contained in the set). At first, we review some classical results, by focusing in particular on planar sets. We then present the natural counterparts of these results for the case of the fractional Dirichlet-Laplacian. We obtain results for open planar sets with a given topology, both in the trivial and in the non-trivial case. The results obtained are sharp, in many respects. Some of the results presented have been obtained in collaboration with Francesca Bianchi (Parma).

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