Oblique derivative problem for non-divergence parabolic equations with discontinuous in time coefficients

TL;DR

This paper establishes weighted coercive estimates for solutions to oblique derivative problems for non-divergence parabolic equations with discontinuous in time coefficients, with applications to bounded domains with smooth boundaries.

Contribution

It provides new weighted coercive estimates in anisotropic Sobolev spaces for such equations, including cases with boundary conditions of class ${\cal C}^{1,\delta}$.

Findings

Weighted coercive estimates for solutions in anisotropic Sobolev spaces.

Application of estimates to bounded domains with smooth boundaries.

Extension to equations with discontinuous in time coefficients.

Abstract

We consider an oblique derivative problem for non-divergence parabolic equations with discontinuous in $t$ coefficients in a half-space. We obtain weighted coercive estimates of solutions in anisotropic Sobolev spaces. We also give an application of this result to linear parabolic equations in a bounded domain. In particular, if the boundary is of class ${\cal C}^{1,\delta}$, $\delta\in (0,1]$, then we present a coercive estimate of solutions in weighted anisotropic Sobolev spaces, where the weight is a power of the distance to the boundary.