Symmetrization for fractional elliptic problems: a direct approach

We provide new direct methods to establish symmetrization results in the form of a mass concentration (that is, integral) comparison for fractional elliptic equations of the type (-Δ)su=f(0<1) in a bounded domain Ω , equipped with homogeneous Dirichlet boundary conditions. The classical pointwise Talenti rearrangement inequality in [47] is recovered in the limit s→ 1. Finally, explicit counterexamples constructed for all s∈(0 , 1) highlight that the same pointwise estimate cannot hold in a nonlocal setting, thus showing the optimality of our results.