Regularity theory for general stable operators: parabolic equations

Abstract

We establish sharp interior and boundary regularity estimates for solutions to $\partial_t u - L u = f(t, x)$ in $I\times Ω$, with $I \subset \mathbb{R}$ and $Ω\subset\mathbb{R}^n$. The operators $L$ we consider are infinitessimal generators of stable Lévy processes. These are linear nonlocal operators with kernels that may be very singular. On the one hand, we establish interior estimates, obtaining that $u$ is $C^{2s+α}$ in $x$ and $C^{1+\fracα{2s}}$ in $t$, whenever $f$ is $C^α$ in $x$ and $C^{\fracα{2s}}$ in $t$. In the case $f\in L^\infty$, we prove that $u$ is $C^{2s-ε}$ in $x$ and $C^{1-ε}$ in $t$, for any $ε> 0$. On the other hand, we study the boundary regularity of solutions in $C^{1,1}$ domains. We prove that for solutions $u$ to the Dirichlet problem the quotient $u/d^s$ is Hölder continuous in space and time up to the boundary $\partialΩ$, where $d$ is the distance to $\partialΩ$. This is new even when $L$ is the fractional Laplacian.