Symmetrization for fractional elliptic problems: a direct approach

TL;DR

This paper introduces new direct methods for symmetrization in fractional elliptic equations, establishing integral comparison results and demonstrating the limits of pointwise inequalities in nonlocal problems.

Contribution

The authors develop novel direct approaches for symmetrization in fractional elliptic problems, extending classical results and clarifying the limitations of pointwise estimates in nonlocal contexts.

Findings

Classical pointwise Talenti inequality recovered as s→1

New integral comparison results for fractional elliptic equations

Counterexamples show pointwise estimates do not hold for all s

Abstract

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 $(-\Delta)^{s}u=f$ $(0<s<1)$ in a bounded domain $\Omega$, equipped with homogeneous boundary conditions. The classical pointwise Talenti rearrangement inequality is recovered in the limit $s\rightarrow1$. 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.