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 ${\partial }_{t}u+(-\Delta {)}^{s}u=0\text{ in }\Omega ,t>0$, 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 $\delta (x)=\text{dist}(x,\partial \Omega )$. 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.

This work is part of the bachelor's degree thesis of the first author