Parabolic equations in simple convex polytopes with time irregular coefficients

TL;DR

This paper establishes $W^{1,2}_p$-estimates and solvability results for second-order parabolic equations in convex polytopes with irregular time-dependent coefficients, covering Dirichlet and Neumann boundary conditions.

Contribution

It provides new solvability and regularity results for parabolic equations with irregular coefficients in convex polytopes, extending previous work to time irregular settings.

Findings

Proves $W^{1,2}_p$-estimates for Dirichlet problems in convex polytopes.

Establishes solvability for Neumann problems in half spaces with irregular coefficients.

Extends results to equations with coefficients measurable in tangential directions with small mean oscillations.

Abstract

We prove the $W^{1,2}_p$-estimate and solvability for the Dirichlet problem of second-order parabolic equations in simple convex polytopes with time irregular coefficients, when $p\in (1,2]$. We also consider the corresponding Neumann problem in a half space when $p\in [2,\infty)$. Similar results are obtained for equations in a half space with coefficients which are measurable in a tangential direction and have small mean oscillations in the other directions.