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arXiv · 2007.13195

Symmetrization for fractional elliptic problems: a direct approach

Vincenzo Ferone·Bruno Volzone
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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.

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Vincenzo Ferone, Bruno Volzone. 2020-07-26. Symmetrization for fractional elliptic problems: a direct approach. https://doi.org/10.1007/s00205-020-01601-8

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