A free boundary problem for p-Laplacian in the plane

We consider the following free boundary problem in an unbounded domain Ω in two dimensions: Δ p u = 0 in Ω, u = 0 , ∂ u ∂ n = g 0 on J 0 , u = 1 , ∂ u ∂ n = g 1 on J 1 , where ∂ Ω = J 0 ∪ J 1 . We prove that if 0 < u < 1 in Ω, J i is the graph of a function in C loc 1 , α ( R ) and g i is a constant for each i = 0 , 1 , then the free boundary ∂Ω must be two parallel straight lines and the solution u must be a linear function. The proof is based on maximum principle.