Symmetrization for fractional elliptic problems: a direct approach

We provide new direct methods to establish symmetrization results in the form of mass concentration (i.e., integral) comparison for fractional elliptic equations of the type $(-\mathrm{\Delta }{)}^{s}u=f$ $(0<s<1)$ in a bounded domain $\mathrm{\Omega }$, equipped with homogeneous boundary conditions. The classical pointwise Talenti rearrangement inequality is recovered in the limit $s\to 1$. Finally, explicit counterexamples constructed for all $s\in (0,1)$ highlight that the same pointwise estimate cannot hold in a nonlocal setting, thus showing the optimality of our results.