Integral representations of lower semicontinuous envelopes and Lavrentiev phenomenon for non continuous Lagrangians

We consider the functional F∞(u)=∫Ωf(x,u(x),∇u(x))dxu∈φ+W01,∞(Ω,R)where Ω is an open bounded Lipschitz subset of RN and φ∈W1,∞(Ω). We do not assume neither convexity or continuity of the Lagrangian w.r.t. the last variable. We prove that, under suitable assumptions, the lower semicontinuous envelope of F∞ both in φ+W1,∞(Ω) and in the larger space φ+W1,p(Ω) can be represented by means of the bipolar f∗∗ of f. In particular we can also exclude Lavrentiev Phenomenon between W1,∞(Ω) and W1,1(Ω) for autonomous Lagrangians.