Parabolic and Elliptic Systems in Divergence Form with Variably Partially BMO Coefficients

Abstract

We establish the solvability of second order divergence-type parabolic systems in Sobolev spaces. The leading coefficients are assumed to be merely measurable in one spatial direction on each small parabolic cylinder with the spatial direction allowed to depend on the cylinder. In the other orthogonal directions and the time variable, the coefficients have locally small mean oscillations. We also obtain the corresponding $W^1_p$-solvability of second order elliptic systems in divergence form. This type of system arises from the problems of linearly elastic laminates and composite materials. Our results are new even for scalar equations, and the proofs differ from and simplify the methods used previously in [H. Dong and D. Kim, Arch. Ration. Mech. Anal., 196 (2010), pp. 25–70]. As an application, we improve a result by Chipot, Kinderlehrer, and Vergara-Caffarelli [Arch. Ration. Mech. Anal., 96 (1986), pp. 81–96] on gradient estimates for elasticity system $D_\alpha (A^{\alpha\beta}(x_1) D_\beta {\textit{\textbf{u}}\,})={\textit{\textbf{f}}\,}$, which typically arises in homogenization of layered materials. We relax the condition on ${\textit{\textbf{f}}\,}$ from $H^k, k\geq d/2$, to $L_p$ with $p>d$.