Symmetrization for fractional elliptic problems: a direct approach

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 $s\to 1$. 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.

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Cited by in corpus (3)

• Symmetrization results for general nonlocal linear ellipitic and parabolic problems
• Variational method for fractional Hamiltonian system in bounded domain
• On a fractional boundary version of Talenti's inequality in the unit ball