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⊂RnΩ⊂Rn is a C1,αC1,α domain, f∈C1,α(Ω¯) for some 0<α<10<α<1 and uεuε verifiesdivA(x,uε,∇uε)=βε(uε)in Ω,0⩽uε⩽1 in Ω,uε=f on ∂Ω, where ε>0ε>0, βε(t)=1εβ(tε) and0⩽β(t)⩽Bχ{00, with some positive constants B and M  , then there exists a constant C>0C>0 independent of ε   such that ‖∇uε‖L∞(Ω¯)⩽C.