Chinese Annals of Mathematics, Series B

, Volume 38, Issue 2, pp 661–686 | Cite as

Symmetrization for fractional elliptic and parabolic equations and an isoperimetric application

Article

Abstract

This paper develops further the theory of symmetrization of fractional Laplacian operators contained in recent works of two of the authors. This theory leads to optimal estimates in the form of concentration comparison inequalities for both elliptic and parabolic equations. The authors 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, an original proof of the corresponding fractional Faber-Krahn inequality is derived. A more classical variational proof of the inequality is also provided.

Keywords

Symmetrization Fractional Laplacian Nonlocal elliptic and parabolic equations Faber-Krahn inequality 

2000 MR Subject Classification

35B45 35R11 35J61 35K55 

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Copyright information

© Fudan University and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Yannick Sire
    • 1
  • Juan Luis Vázquez
    • 2
  • Bruno Volzone
    • 3
  1. 1.Johns Hopkins UniversityBaltimoreUSA
  2. 2.Departamento de MatemáticasUniversidad Autónoma de MadridMadridSpain
  3. 3.Dipartimento di IngegneriaUniversit degli Studi di Napoli “Parthenope”NapoliItalia

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