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 Ω Rn is a C1,α domain, for some 0<α<1 and uε verifies where ε>0, and with some positive constants B and M, then there exists a constant C>0 independent of ε such that .