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Boundary regularity for the fractional heat equation

  • Xavier Fernández-Real
  • Xavier Ros-OtonEmail author
Original Paper

Abstract

We study the regularity up to the boundary of solutions to fractional heat equation in bounded \(C^{1,1}\) domains. More precisely, we consider solutions to \(\partial _t u + (-\Delta )^su=0 \hbox { in }\Omega ,\ t > 0\), with zero Dirichlet conditions in \(\mathbb {R}^n{\setminus } \Omega \) and with initial data \(u_0\in L^2(\Omega )\). Using the results of the second author and Serra for the elliptic problem, we show that for all \(t>0\) we have \(u(\cdot , t)\in C^s(\mathbb {R}^n)\) and \(u(\cdot , t)/\delta ^s \in C^{s-\epsilon }(\overline{\Omega })\) for any \(\epsilon > 0\) and \(\delta (x) = \hbox {dist}(x,\partial \Omega )\). Our regularity results apply not only to the fractional Laplacian but also to more general integro-differential operators, namely those corresponding to stable Lévy processes. As a consequence of our results, we show that solutions to the fractional heat equation satisfy a Pohozaev-type identity for positive times.

Keywords

Fractional Laplacian Fractional heat equation Boundary regularity 

Mathematics Subject Classification

35B65 47G20 

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Copyright information

© Springer-Verlag Italia 2015

Authors and Affiliations

  1. 1.Facultat de Matemàtiques i EstadísticaUniversitat Politècnica de CatalunyaBarcelonaSpain
  2. 2.Department of MathematicsUniversity of Texas at AustinAustinUSA

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