We prove mixed Lp(Lq)-estimates, with p,q∈(1,∞), 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 both in the time variable and the space variable. In the proof, we apply and extend the techniques developed by Krylov [24] as well as Dong and Kim in [13] to produce mean oscillation estimates for equations on the half space with general boundary conditions.@en

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].@en