The existence of solutions to variational problems of slow growth

We consider the existence of solutions, in the space W1,1(Ω)W1,1(Ω), to the problemminimize ∫ΩL(∇v(x))dxonϕ+W01,1(Ω) where L is of slow (linear or at most quadratic) growth. We present a necessary and sufficient condition in order that, for any smooth boundary datum ϕ and for any bounded Ω with smooth boundary, the minimum problem be solvable.