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Local Regularity for Fractional Heat Equations

  • Umberto Biccari
  • Mahamadi Warma
  • Enrique ZuazuaEmail author
Chapter
Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI, volume 17)

Abstract

We prove the maximal local regularity of weak solutions to the parabolic problem associated with the fractional Laplacian with homogeneous Dirichlet boundary conditions on an arbitrary bounded open set \(\Omega \subset {\mathbb {R}}^N\). Proofs combine classical abstract regularity results for parabolic equations with some new local regularity results for the associated elliptic problems.

Keywords

Fractional Laplacian Heat equation Dirichlet boundary condition Weak solutions Local regularity 

2010 Mathematics Subject Classification

35B65 35R11 35S05 

Notes

Acknowledgments

The work of Umberto Biccari was partially supported by the Advanced Grant DYCON (Dynamic Control) of the European Research Council Executive Agency, by the MTM2014-52347 and MTM2017-92996 Grants of the MINECO (Spain) and by the Air Force Office of Scientific Research under the Award No: FA9550-15-1-0027. The work of Mahamadi Warma was partially supported by the Air Force Office of Scientific Research under the Award No: FA9550-15-1-0027. The work of Enrique Zuazua was partially supported by the Advanced Grant DYCON (Dynamic Control) of the European Research Council Executive Agency, FA9550-15-1-0027 of AFOSR, FA9550-14-1-0214 of the EOARD-AFOSR, the MTM2014-52347 and MTM2017-92996 Grants of the MINECO (Spain) and ICON of the French ANR

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Umberto Biccari
    • 1
    • 2
  • Mahamadi Warma
    • 3
  • Enrique Zuazua
    • 1
    • 2
    • 4
    • 5
    Email author
  1. 1.DeustoTechUniversity of DeustoBilbao, Basque CountrySpain
  2. 2.Facultad de IngenieríaUniversidad de DeustoBilbao, Basque CountrySpain
  3. 3.University of Puerto Rico (Rio Piedras Campus)College of Natural Sciences, Department of MathematicsSan JuanUSA
  4. 4.Departamento de MatemáticasUniversidad Autónoma de MadridMadridSpain
  5. 5.Sorbonne UniversitesUPMC Univ Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis LionsParisFrance

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