For a locally Lipschitz integral functional I (f) on L-1(T, R-n) associated with a measurable integrand f, the limiting subdifferential and the approximate subdifferential never coincide at a point x0 where f(t, .) is not subdifferentially regular at x0(t) for a.e. t is an element of T. The coincidence of both subdifferentials occurs on a dense set of L-1(T, R-n) if and only if f(t, .) is convex for a.e. t is an element of T. Our results allow us to characterize Aubin's Lipschitz-like property as well as the convexity of multivalued mappings between L-1-spaces. New necessary optimality conditions for some Bolza problems are also obtained.