Symmetrization in a periodic setting: local and nonlocal Pólya-Szegő inequalities

The celebrated Pólya-Szegő inequality states that the rearrangement decreases the L^p norm of the gradient under appropriate assumptions. We discuss some periodic versions of this inequality and compare local versus nonlocal results. Finally, I shall mention an application to a periodic nonlocal geometric problem dealing with so-called nonlocal Delaunay cylinders. This problem is the fractional analogue of what could be called the periodic isoperimetric problem.