A uniqueness result for a class of non-strictly convex variational problems

Let Ω be a smooth domain in R2R2, we prove that if g:[0,+∞)→[0,+∞]g:[0,+∞)→[0,+∞] is convex with g(0)0t>0 then there exists an unique minimizer u∈C0,1(Ω)u∈C0,1(Ω) of the functional u↦∫Ωg(|∇u|)dxdy among all Lipschitz-continuous functions that assume the same value of u on ∂Ω.