Boundary behaviour for a singular perturbation problem

In this paper we study the boundary behaviour of the family of solutions {uε}{uε} to singular perturbation problem Δuε=βε(uε),∣uε∣≤1Δuε=βε(uε),∣uε∣≤1 in B1+={xn>0}∩{∣x∣<1}, where a smooth boundary data ff is prescribed on the flat portion of ∂B1+. Here βε(⋅)=1εβ(⋅ε),β∈C0∞(0,1),β≥0,∫01β(t)=M>0 is an approximation of identity. If ∇f(z)=0∇f(z)=0 whenever f(z)=0f(z)=0 then the level sets ∂{uε>0}∂{uε>0} approach the fixed boundary in tangential fashion with uniform speed. The methods we employ here use delicate analysis of local solutions, along with elaborated version of the so-called monotonicity formulas and classification of global profiles.