Parabolic equations with partially VMO coefficients and boundary value problems in Sobolev spaces with mixed norms

TL;DR

This paper investigates second order parabolic equations with mixed norm Sobolev space solutions, focusing on coefficients with variable regularity, and establishes solvability results for boundary value problems in half-spaces.

Contribution

It introduces solvability results for parabolic equations with partially VMO coefficients in Sobolev spaces with mixed norms, including boundary value problems in half-spaces.

Findings

Unique solvability in the whole space established.

Boundary value problems in half-spaces are solvable under the given coefficient conditions.

Extension of classical results to equations with partially VMO coefficients.

Abstract

Second order parabolic equations in Sobolev spaces with mixed norms are studied. The leading coefficients (except $a^{11}$) are measurable in both time and one spatial variable, and VMO in the other spatial variables. The coefficient $a^{11}$ is measurable in time and VMO in the spatial variables. The unique solvability of equations in the whole space is applied to solving Dirichlet and oblique derivative problems for parabolic equations defined in a half-space.