Up-to boundary regularity for a singular perturbation problem of p-Laplacian type

In this paper we are interested in establishing up-to boundary uniform estimates for the one phase singular perturbation problem involving a nonlinear singular/degenerate elliptic operator. Our main result states: if Omega subset of R-n is a C-1,C-a domain, f is an element of C-1,C-a (Omega) for some 0 < a < 1 and u(epsilon) verifies div A(x, u(epsilon), del u(epsilon)) = beta(epsilon)(u(epsilon)) in Omega, 0 <= u(epsilon) <= 1 in Omega, u(epsilon) = f on partial derivative Omega, where epsilon > 0, beta(epsilon) (t) = epsilon/1 beta (epsilon/t) and 0 <= beta(t) <= B chi({0 < t < 1}), (R)integral beta(epsilon)(t) dt = M > 0, with some positive constants B and M, then there exists a constant C > 0 independent of E such that vertical bar vertical bar del u epsilon vertical bar vertical bar(L infinity(Omega)) <= C. (c) 2005 Elsevier Inc. All rights reserved.