#### Vol. 288, No. 2, 2017

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The Faber–Krahn inequality for the first eigenvalue of the fractional Dirichlet $p$-Laplacian for triangles and quadrilaterals

Vol. 288 (2017), No. 2, 425–434
##### Abstract

We prove the Faber–Krahn inequality for the first eigenvalue of the fractional Dirichlet $p$-Laplacian for triangles and quadrilaterals of a given area. The proof is based on a nonlocal Pólya–Szegő inequality under Steiner symmetrization and the continuity of the first eigenvalue of the fractional Dirichlet $p$-Laplacian with respect to the convergence, in the Hausdorff distance, of convex domains.

##### Keywords
fractional $p$-Laplacian, Dirichlet condition, Faber–Krahn inequality, polygonal domains, Steiner symmetrization, Riesz's inequality
##### Mathematical Subject Classification 2010
Primary: 49Q20, 52B60
Secondary: 47A75, 47J10