Philipp Svinger

In this article we establish the optimal $C^s$ boundary regularity for solutions to nonlocal parabolic equations in divergence form in $C^{1,\alpha}$ domains and prove a higher order boundary Harnack principle in this setting. Our approach applies to a broad class of nonlocal operators with merely Hölder continuous coefficients, but our results are new even in the translation invariant case. As an application, we obtain sharp two-sided estimates for the associated Dirichlet heat kernel. Notably, our estimates cover nonlocal operators with time-dependent coefficients, which had remained open in the literature.

We prove optimal Hölder boundary regularity for a non-local operator with a singular, symmetric kernel that depends on the distance to the boundary of the underlying domain. Additionally, we prove higher boundary regularity of solutions.