Higher-order parabolic equations with VMO assumptions and general boundary conditions with variable leading coefficients

TL;DR

This paper establishes weighted mixed $L_{p}(L_{q})$-estimates for higher-order elliptic and parabolic equations with VMO coefficients and variable boundary conditions, extending previous results to more general settings.

Contribution

It generalizes existing estimates to equations with VMO coefficients and variable boundary conditions on domains with general boundaries.

Findings

Proved weighted mixed $L_{p}(L_{q})$-estimates for higher-order equations.

Extended previous estimates to VMO coefficient cases.

Handled general boundary conditions satisfying the Lopatinskii--Shapiro condition.

Abstract

We prove weighted mixed $L_{p}(L_{q})$-estimates, with $p,q\in(1,\infty)$, for higher-order elliptic and parabolic equations on the half space $\mathbb{R}^{d+1}_{+}$ and on domains 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 both in the time and the space variables, and that the boundary conditions have variable leading coefficients. The proofs are based on and generalize the estimates recently obtained by the authors in [DG17]. [DG17] H.Dong and C.Gallarati. Higher order elliptic and parabolic equations with VMO assumptions and general boundary conditions. Accepted for publication in Journal of Functional Analysis. See: arXiv:1702.03254.