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

TL;DR

This paper investigates a boundary version of Talenti's inequality for the fractional Laplacian in the unit ball, revealing conditions for its validity and demonstrating its failure in the fractional radial setting.

Contribution

It provides new sufficient conditions for the validity of a boundary Talenti inequality and shows its universal failure in fractional radial cases, extending previous work.

Findings

Boundary Talenti inequality does not always hold for fractional Laplacian.

Universal failure of classical Talenti inequality in fractional radial context.

Provides conditions under which the inequality is valid or invalid.

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.