Symmetrization methods for local and nonlocal operators

Symmetrization methods have been largely used in order to get the best constants in functional inequalities and to get sharp estimates for solutions to boundary value problems for elliptic and parabolic equations. They are based on the properties of rearrangement of measurable functions. After a brief discussion about such properties, some boundary value problems for equations involving Laplacian-type operators will be considered in detail, both in the case of the classical local Laplacian and in the case of the fractional s-Laplacian.