Integrals which are convex functionals. II

Formulas are derived in this paper for the conjugates of convex integral functionals on Banach spaces of measurable or continuous vector-valued functions. These formulas imply the weak compactness of certain convex sets of summable functions, and they thus have applications in the existence theory and duality theory for various optimization problems. They also yield formulas for the subdifferentials of integral functionals, as well as characterizations of supporting hyper-planes and normal cones. Let T be an arbitrary set, let j? ~ be a σ-field of subsets of T (the "measurable " sets), and let dt denote a nonnegative, σ-finite measure on ^ \\ We shall be interested in functionals of the form (1.1) If(u)- \\ f(t, u(t))dt, ueL, where L is a linear space of measurable functions from T to Rn, and / is a function from Γ x Rn to R1 U {+ °°} such that the function ft = f(t, •) is convex on Rn for every te T. A functional of the form / / is obviously convex (with values in R1 U {+°°}), provided that it is well-defined in the sense that, for every u e L, f(t, u(t)) is a measur-able function of t which majorizes at least one summable function of t. In our preceding paper with the same title [16], rather general spaces L were considered, but here the cases L = L~(T) and L = LX%{T) dominate. (We denote by Lζ(T) = Lζ(T, ^ dt), 1 ^ p ^ +oo, the Banach space consisting of all (equivalence classes of) measurable functions u: T'-+ Rn such that the realvalued function t — • | u(t) |, where | | denotes the Euclidean norm in Rn, belongs to Lp(T,^~,dt) the norm of the function t-+\(t) \\ in LP(T, J^dt) being the Lζ(T)-norm of u.) We assume as in [16] that / is a normal convex integrand on T x Rn, in other words, (a) ft is for each t a lower semicontinuous convex function from Rn to Rι U {+°°} which is not identically +°o, and (b) there exists a countable collection U of measurable functions from T to Rn, such that /(ί, u(t)) is measurable in t for every u e U, and U(t) Π D(t) is dense in the (nonempty, convex) set (1.2) D(t) = {xeRn \(t, x) < + oo}