Higher-order elliptic and parabolic equations with VMO assumptions and general boundary conditions

TL;DR

This paper establishes mixed $L_{p}(L_{q})$-estimates for higher-order elliptic and parabolic equations with VMO coefficients and general boundary conditions satisfying the Lopatinskii--Shapiro condition, on the half space.

Contribution

It introduces new mean oscillation estimates for such equations with VMO coefficients and general boundary conditions, extending previous regularity results.

Findings

Proved mixed $L_{p}(L_{q})$-estimates for elliptic and parabolic equations.

Established mean oscillation estimates for equations with general boundary conditions.

Validated the estimates under VMO assumptions on coefficients.

Abstract

We prove mixed $L_{p}(L_{q})$-estimates, with $p,q\in(1,\infty)$, for higher-order elliptic and parabolic equations on the half space $\R^{d+1}_{+}$ with general boundary conditions which satisfy the Lopatinskii--Shapiro condition. We assume that the elliptic operators $A$ have leading coefficients which are in the class of vanishing mean oscillations in both the time and the space variable. In the proof, we produce mean oscillation estimates for equations on the half space with general boundary conditions.