Symmetrization for fractional elliptic and parabolic equations and an isoperimetric application

Abstract

We develop further the theory of symmetrization of fractional Laplacian operators contained in recent works of two of the authors. The theory leads to optimal estimates in the form of concentration comparison inequalities for both elliptic and parabolic equations. In this paper we extend the theory for the so-called restricted fractional Laplacian defined on a bounded domain Ω of R N with zero Dirichlet conditions outside of Ω. As an application, we derive an original proof of the corresponding fractional Faber-Krahn inequality. We also provide a more classical variational proof of the inequality.

Eigenvalue Bounds for the Fractional Laplacian: A Review

“…(We also mention that a version of (2.1) for functions on a interval appears in [59].) An alternative proof of Theorem 2.1, based on a comparison result for the corresponding heat equations, can be found in [98]. For results related to and generalizing Theorem 2.1, see [15].…”

The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion

“…It turns out that Steiner symetrization works and it does much better for fractional FDE than for the fractional PME range. This work was followed by recent collaboration with Y. Sire and B. Volzone to apply the techniques to the fractional Faber-Krahn inequality, [220].…”

Uniqueness of minimal energy solutions for a semilinear problem involving the fractional Laplacian

“…This operator is classical and have been studied by several authors. See for instance [1,4,5,7,9,8,20,25], etc.…”