On a fractional boundary version of Talenti's inequality in the unit ball

Abstract

Inspired by recent work of Ferone and Volzone arXiv:2007.13195, we derive sufficient conditions for the validity and non-validity of a boundary version of Talenti's comparison principle in the context of Dirichlet-Poisson problems for the fractional Laplacian $(-\Delta)^s$ in the unit ball $\Omega= B_1(0) \subset \mathbb{R}^N$. In particular, our results imply a universial failure of the classical pointwise Talenti inequality in the fractional radial context which sheds new light on the one-dimensional counterexamples given in arXiv:2007.13195.