Optimal regularity of mixed Dirichlet-conormal boundary value problems for parabolic operators

TL;DR

This paper establishes the regularity and unique solvability of mixed Dirichlet-conormal boundary value problems for parabolic operators in complex domains, extending known results to more general boundary conditions and geometries.

Contribution

It provides the first regularity results in Sobolev spaces for these problems in domains with time-dependent separations and Reifenberg-flat boundaries.

Findings

Proves unique solvability for p in a specific range depending on boundary separation.

Establishes optimal p-range for the case when m=0, matching known Laplace equation results.

Extends regularity theory to domains with time-dependent and Reifenberg-flat boundaries.

Abstract

We obtain the regularity of solutions in Sobolev spaces for the mixed Dirichlet-conormal problem for parabolic operators in cylindrical domains with time-dependent separations, which is the first of its kind. Assuming the boundary of the domain to be Reifenberg-flat and the separation to be locally sufficiently close to a Lipschitz function of $m$ variables, where $m=0,\ldots,d-2$, with respect to the Hausdorff distance, we prove the unique solvability for $p\in (2(m+2/(m+3),2(m+2)/(m+1)))$. In the case when $m=0$, the range $p\in(4/3,4)$ is optimal in view of the known results for Laplace equations.