Compactness methods in the theory of homogenization II: Equations in non‐divergence form

We prove C^{0, α}, C^{1, α} and C^{1, 1} a priori estimates for solutions of boundary value problems for elliptic operators with periodic coefficients of the form Σ in,^j=1a^{i j}(x/ϵ)δ²/δx_iδx_j. The constants in these estimates are independent of the small parameter ϵ, and hence our results imply strengthened versions of the classical averaging theorems. These results generalize to a wide class of linear operators in non‐divergence form, including equations with lower‐order terms and nonuniformly oscillating coefficients, as well as to certain nonlinear problems, which we discuss in the last section.