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 \((-\Delta )^{s}u=f \, (0<s<1)\) in a bounded domain \(\Omega \), equipped with homogeneous Dirichlet boundary conditions. The classical pointwise Talenti rearrangement inequality in [47] is recovered in the limit \(s\rightarrow 1\). 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.

中文翻译：

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分数椭圆问题的对称化：一种直接方法

我们提供了一种新的直接方法来建立对称结果的形式，形式为\（（-\ Delta）^ {s} u = f \，（0 <s <1）\）在有界Dirichlet边界条件的有界\（\ Omega \）中。[47]中的经典点式塔伦蒂重排不等式在极限\（s \ rightarrow 1 \）中恢复。最后，为所有\（s \ in（0,1）\）构造的显式反例突出表明，相同的逐点估计不能在非局部设置中成立，因此显示了我们结果的最优性。