Mixed boundary value problems for parabolic equations in Sobolev spaces with mixed-norms

TL;DR

This paper develops $L_{q,p}$-estimates and proves solvability for mixed boundary value problems of parabolic equations in complex, irregular domains with time-dependent boundaries, advancing the mathematical understanding of such problems.

Contribution

It introduces new $L_{q,p}$-estimates and solvability results for parabolic equations with mixed boundary conditions in Reifenberg-flat domains with rough, time-dependent separation.

Findings

Established $L_{q,p}$-estimates for the problems.

Proved solvability in Reifenberg-flat domains.

Addressed rough, time-dependent boundary conditions.

Abstract

We establish $L_{q,p}$-estimates and solvability for mixed Dirichlet-conormal problems for parabolic equations in a cylindrical Reifenberg-flat domain with a rough time-dependent separation.