Regularity theory for general stable operators: parabolic equations

TL;DR

This paper develops sharp interior and boundary regularity estimates for solutions to parabolic equations involving stable Lévy operators, extending known results to more singular kernels and boundary behaviors.

Contribution

It provides new interior and boundary regularity results for solutions to nonlocal parabolic equations with general stable operators, including boundary quotient regularity.

Findings

Solutions are $C^{2s+eta}$ in space and $C^{1+rac{eta}{2s}}$ in time under smooth data.

For bounded data, solutions are $C^{2s- ext{epsilon}}$ in space and $C^{1- ext{epsilon}}$ in time.

The boundary quotient $u/d^s$ is Hölder continuous up to the boundary in $C^{1,1}$ domains.

Abstract

We establish sharp interior and boundary regularity estimates for solutions to $\partial_t u - L u = f(t, x)$ in $I\times \Omega$, with $I \subset \mathbb{R}$ and $\Omega \subset\mathbb{R}^n$. The operators $L$ we consider are infinitessimal generators of stable L\'evy 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 $C^{2s+\alpha}$ in $x$ and $C^{1+\frac{\alpha}{2s}}$ in $t$, whenever $f$ is $C^{\alpha}$ in $x$ and $C^{\frac{\alpha}{2s}}$ in $t$. In the case $f\in L^\infty$, we prove that $u$ is $C^{2s-\epsilon}$ in $x$ and $C^{1-\epsilon}$ in $t$, for any $\epsilon > 0$. 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 H\"older continuous in space and time up to the boundary $\partial\Omega$, where $d$ is the distance to $\partial\Omega$. This is new even when $L$ is the fractional Laplacian.