Regularity for the dirichlet problem

This chapter is devoted to the study of uniform boundary regularity estimates for the Dirichlet problem [Formula presented.] where Lε =div(A(x/ε)∇). Assuming that the coefficient matrix A = A(y) is elliptic, periodic, and belongs to VMO(ℝ^d), we establish uniform boundary Hölder and W^{1, p} estimates in C¹ domains Ω. We also prove uniform boundary Lipschitz estimates in C^{1, α} domains under the assumption that A is elliptic, periodic, and Hölder continuous. As in the previous chapter for interior estimates, boundary Hölder and Lipschitz estimates are proved by a compactness method. The boundaryW^{1, p} estimates are obtained by combining the boundary Hölder estimates with the interior W^{1, p} estimates, via the real-variable method introduced in Section 4.2.