We prove the Faber–Krahn inequality for the first eigenvalue of the fractional Dirichlet
-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
-Laplacian
with respect to the convergence, in the Hausdorff distance, of convex domains.