Regularity for the Dirichlet Problem

Abstract

This chapter is devoted to the study of uniform boundary regularity estimates for the Dirichlet problem $$ \left\{\begin{array}{rc}{\mathcal{L}_\varepsilon(\mathit{u}_\varepsilon)\,=\,\mathit{F}} & \mathrm{in}\,\,\Omega, \mathit{u}_\varepsilon\,=\,\mathit{g} & \mathrm{on}\,\,\partial\Omega, \end{array}\right.$$ (5.0.1) where \( \mathcal{L}_\varepsilon\,\,=\,\,\mathrm{-div}(\mathit{A}(\mathit{x}/\varepsilon)\nabla)\). Assuming that the coefficient matrix \( \mathit{A}\,\,=\,\,\mathit{A}(\mathit{y}) \) is elliptic, periodic, and belongs to VMO(\( \mathbb{R}^\mathit{d}\)), we establish uniform boundary Holder and W1, p estimates in C1 domains Ω. We also prove uniform boundary Lipschitz estimates in C1, α domains under the assumption that A is elliptic, periodic, and Holder continuous. As in the previous chapter for interior estimates, boundary Holder and Lipschitz estimates are proved by a compactness method. The boundaryW1, p estimates are obtained by combining the boundary Holder estimates with the interior W1, p estimates, via the real-variable method introduced in Section 4.2.