A Relaxation Result for Autonomous Integral Functionals with Discontinuous Non-coercive Integrand

a L∗∗(y(t), y′(t)) dt : y ∈ AC ([a, b],RN) , y(a) = A, y(b) = B } · (P ∗∗) It is well known that inf F = inf F ∗∗ if L is super-linear and continuous. Recently Cellina in [5] proved that the same conclusion holds true assuming, instead of superlinearity, a weaker growth condition that we will call (GA). Roughly, a convex function L(x, ξ) satisfies (GA) if the intersection of the supporting hyperplane to its epigraph at (ξ, L(x, ξ)) with the ordinate axis tends to −∞ as |ξ| tends to +∞, uniformly with respect to x in compact sets. This condition implies, but is not equivalent to, a sort of conical growth: we say that L