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Boundary regularity for the fractional heat equation

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.

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Correspondence to Xavier Ros-Oton.

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This work is part of the bachelor’s degree thesis of X. Fernández-Real.

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Fernández-Real, X., Ros-Oton, X. Boundary regularity for the fractional heat equation. RACSAM 110, 49–64 (2016). https://doi.org/10.1007/s13398-015-0218-6

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  • DOI: https://doi.org/10.1007/s13398-015-0218-6

Keywords

  • Fractional Laplacian
  • Fractional heat equation
  • Boundary regularity

Mathematics Subject Classification

  • 35B65
  • 47G20