Higher-Order Parabolic Equations with VMO Assumptions and General Boundary Conditions with Variable Leading Coefficients

Abstract

We prove weighted mixed Lp(Lq)-estimates, with p,q∈(1,∞)⁠, and the corresponding solvability results for higher-order elliptic and parabolic equations on the half space Rd+1+ and on general C2m−1,1 domains with general boundary conditions, which satisfy the Lopatinskii–Shapiro condition. We assume that the elliptic operators A have leading coefficients that are in the class of vanishing mean oscillations both in the time and the space variables and that the boundary operators have variable leading coefficients. The proofs are based on and generalize the estimates recently obtained by the authors in [6].