Symmetrization for fractional elliptic and parabolic equations and an isoperimetric application

2015/06/23 by Yannick Sire, Sire, Yannick, Juan Luis Vazquez +3 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.1506.07199

arXiv admin note: substantial text overlap with arXiv:1303.2970

arxiv created 2015/06/23 · arxiv updated 2015/06/25

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 \mathbb 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.

Citations

• Symmetrization for Linear and Nonlinear Fractional Parabolic Equations of Porous Medium Type

Cited by

• First Eigenvalue and Torsional Rigidity: Isoperimetric Inequalities for the Fractional Laplacian

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