Interior estimates

In this chapter we establish interior Hölder (C^0,α) estimates, W^1,p estimates, and Lipschitz (C^0,1) estimates, that are uniform in ε > 0, for solutions of Lε (u_ε) = F, where Lε = -div(A(x/ε)∇). As a result, we obtain uniform size estimates of Γe(x, y)∇_xΓ _ε (x, y)∇_yΓ _ε (x, y), and ∇_x∇_yΓ _ε (x, y), where Γ _ε (x, y) denotes the matrix of fundamental solutions for L_ε in ℝ^d. This in turn allows us to derive asymptotic expansions, [Formula presented.]. Thus no uniform regularity beyond Lipschitz estimates should be expected (unless div(A) = 0, which would imply X^β _j = 0).