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

We prove mixed $L_p(L_q)$-estimates, with $p,q\in (1,\mathrm{\infty })$, for higher-order elliptic and parabolic equations on the half space ${\mathbb{R}}_{+}^{d+1}$ 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.