Integrals which are convex functionals. II

Theorem 4 of [16], under which the "singular" component is trivial, so that // is identically + °o outside of L ι n(T) and hence reduces entirely to If*. This property implies the weak compactness in L ι n{T) of all the convex level sets of the functional //* (see Corollaries 2A and 2B). The sufficient condition for weak compactness which is obtained in this way may be regarded as a generalization of Nagumo's Theo- rem [12] in the calculus of variations, and it is also related to recent work of

THEOREM 1. Assume that f(t, u(t)) is majorized by a summable

Theorem 2] that If and If* are well-defined on L~(T) and L ι n(T), respectively, and that formulas (1.6) and (1.7) hold for p — oo and q — 1. Thus for every w* G L ι n(T) and every measurable subset S of T one has ( /*(«, u*(t))dt = supjί [<u(t), iι*(t)> (2.6)

Theorem 1 is satisfied if and only if WΦ 0, in which event If on LZ{T) is the support function of W (regarded as a subset of Z£(T)*), so that If is the indicator of the closure of W in the weak topology induced on L~(T)* by Z£(T). Thus, if WΦ 0, the latter closure of W is by Corollary 1A the direct sum of W and a certain "singular cone." A different version of this result will be given in Corollaries 5B and 5C. Another important case of Corollary 1A occurs when C is all of Ln(T). We shall cover

THEOREM 2. Assume that ΰeL~(T) and r > 0 have the property that f(t, ΰ(t) + x) is a summable function of t whenever \x\ < r, x e R n. Then f*(t,u*(t)) is majorized by a summable function of t for at least one u* e L ι n(T), so that the hypothesis of Theorem 1 is satisfied. Moreover, in this case If is continuous (in the I/Z{T) norm) at u whenever \\u — ΰ\\ < r, and in formula (2.4) one has (2.13) δ*(vs)^v3(u) + r\\v,\\.

Theorem 2 is thereby proved. The following corollary generalizes Theorem 4 of [16], where we imposed the more restrictive condition that f(t, x) be (finite and) essentially bounded as a function of t for every x e R n.

COROLLARY 2A. Assume that f(t, x) is a summable function of teT for every xeR n. Then If on L~(T) and If* on L ι n(T) are well- defined convex functionals conjugate to each other with respect to the pairing (1.5), and If is finite and continuous throughout L™(T). Furthermore, the conjugate function 1/ on L~(T)* reduces to If* on Ln(T), in the sense that i;(v) = iAuη if v is an "absolutely continuous 97 functional corresponding to a function u* e L

COROLLARY 2B (Weak Compactness). Let g be a normal convex integrand on T x R n whose conjugate g* has the property that g*(t, y) is summable in t for every y e R n. Then Ig is a well-defined convex functional from Un(T) to R ι U {+ 00}, not identically +°°. Moreover, for every aeL~(T) and aeR 1 the convex level set

COROLLARY 2C. The subgradient formula given in Corollary IB is also valid if there exists a function ΰeL~(T) satisfying the hy- pothesis of Theorem 2. Moreover, for every such ΰ the subgradient set dlf(ΰ) may actually be identified with a nonempty, weakly compact subset of L ι n(T): a functional veLζ(T)* belongs to dlf(u) if and only if the "singular" component of v vanishes, and function u* e L\{T) corresponding to v satisfies (1.12).

THEOREM 3. Let F and G be arbitrary, locally convex Hausdorff topological vector spaces with duals F* and G*. Let g be a lower

COROLLARY 3A. Under the hypothesis of Theorem 3, if there is a point of the range of A at which g is finite and continuous,, then for every yeF the subgradient set dk{y)(zF* is the image of the subgradient set dg(Ay) in G* under the transformation A*.

THEOREM 4. Assume that ΰ e Cn(T) and r > 0 have the property that f(t, ΰ(t) + x) is a summable function of t whenever \x\ < r, x 6 R n. Then Jf is a well-defined, lower semicontinuous, convex functional from Cn(T) to R 1 U {+ °°}, and Jf is finite and continuous at every ueCn(T) such that \\u — ΰ\\ < r. Furthermore, Jf on Mn(T) satisfies

COROLLARY 4A. Assume that f(t, x) is a summable function of t for every xeR n. Then the functional Jf on Cn{T) is well-defined, finite, continuous and convex. The conjugate functional J* on Mn(T) is given by

COROLLARY 4B. Under the hypothesis of Theorem 4, a measure μeMn(T) belongs to the subgradient set djf(u), where ueCn(T), if and only if there exists an absolutely continuous measure μ'eCn{T) satisfying (3.11) —(ί) e dft(u(t)) for almost every t , dt such that the linear functional on Cn{T) corresponding to μ — μ r attains its maximum over the set E at u. In fact, if u has the property that the function f(t, u(t) + x) is summable in t for every x in some neighborhood of 0 in R

LEMMA 1. The recession function h of f* is a normal convex integrand on T x R n.

Theorem 2]. To state the main result of this section, we recall that the multifunction D: t —> D(t) is said to be lower semicontinuous from T to R n if, whenever U is an open subset of R n and tQ is an element of T such that D(tQ) Π U Φ 0 , there exists a neighborhood V of tQ such that D(t) n U Φ 0 for every teV. We shall say that D is fully lower semicontinuous if D is lower semicontinuous and, in ad- dition, one has x0 e cl D(tQ) whenever there are neighborhoods U and

THEOREM 5. Assume that T is a compact space with no nonempty open sets of measure zero, and that the multifunction D: t —> D(t) is fully lower semicontinuous, with int D(t) Φ 0 for every t. Assume further that \ I /(ί, x) \dt< + oo JV whenever V is an open subset of T and x is a point of R n having a neighborhood U such that U c D(t) for all te V.

LEMMA 2. Let D: £ —> D(t) aR n be a multifunction such that D(t) is for each t a convex set with int D(t) Φ 0 , and let (4.5) G = {(t, x)\xe int D(t)} c Γ x R n . (a) D is lower semicontinuous if and only if G = int G; (b) D is fully lower semicontinuous if and only if G — int cl G.

Theorem 4 is satisfied if the ΰ in that hypothesis is taken to be any function satisfying (4.3). In this way we obtain the fact that Jf is a well-defined, lower semicontinuous convex functional from Cn(T) to R ι U {+ °°} such that Jf is finite and continuous at each element u of Cn(T) satisfying (4.3). Our next step is to show that the functions satisfying (4.3) con- stitute the interior of the set E in (3.6). They are certainly contained in int E by the above. On the other hand, let ΰ be a func

Theorem 6 furnishes an integral representation of <?£- which can be substituted in (4.13): (4.14) δUμ - μ') = ( δξw(d(μ ~ μ')ldθ f)dθ f , where Q(t) = cl D(t), and θ f is any nonnegative measure in M^T) with respect to which μ — μ' is absolutely continuous. Since μ f is absolutely continuous with respect to dt, we can take dθ' be of the form dt + dθ, where θ is an arbitrary, singular, nonnegative measure in M^T) with respect to which the singular part v of μ is absolutely continuous. By virtue o

THEOREM 6. Assume T is compact. Let Q: T—*R n be a lower semicontinuous multifunction such that Q(t) is for every t a non- empty, closed convex set. Let ft = f(t, •) be the indicator of Q(t), so that /* is the support function of Q{t), and let (4.19) K= {ue Cn(T) \ u{t) e Q(t), VteT} .

Theorem 3.2]. To show that / and /* are normal convex integrands, it suffices by [17, Theorem 3] to show that the multifunction Q is measurable. The measurability of Q follows in fact from lower semicontinuity, as has been observed by Castaing: for any closed set W(zR n, one has {teT\Q(t)Π WΦ 0 } = U Π{teT\Q(t)f] W?Φ0}

Theorem 5 about subgradients. Let N: T—> R n be a multifunction such that N(t) is for each t a convex cone containing the origin. A measure v e Mn{T) will be called N-valued, if one has (dv/dθ)(t) e N(t) except for a set of θ- measure zero, where M s a nonnegative measure in Mγ(T) with respect to which v is absolutely continuous. (Note that this defi- nition does not depend on the particular θ, inasmuch as N(t) is closed under multiplication by nonnegative scalars.) We shall be interested in the

COROLLARY 5A. Under the hypothesis of Theorem 5, a measure μeMn(T) belongs to the subgradίent set dJf(u), where ueCn(T), if and only if dμ/dt satisfies (3.11'), u(t) belongs to cl D(t) for every t, and the singular component v of μ is N-valued in the above sense, where N(t) is the normal cone (5.1) to cl D(t) at u{t).

COROLLARY 5B. Let Q:T-+R n be a measurable multifunction such that Q(t) is for each t a nonempty, closed, convex set, and let ft = f(t, •) be the support function of Q(t). Assume that the hypothesis of Theorem 5 is satisfied by f (which is a normal convex integrand). Let W be the convex subset of Mn{T) consisting of all the absolutely continuous measures μ such that (5.5) ^~{t) e Q(t) for almost every t. dt Then Jf is the support function of W on Cn{T), and the following conditions on a measure

COROLLARY 5C. The conclusions of Corollary 5B remain valid if, instead of assuming that Q is measurable, and that the hypothesis of Theorem 5 is satisfied by f, one assumes the following: T is com- pact (with no nonempty open sets of measure zero),the set D(t) has a nonempty interior which is independent of t, and for each x in this interior f(t, x) is summable (measurable) function of t.

COROLLARY 5D. Let the assumptions of either Corollary 5B or

Corollary 5C be satisfied, and let W denote the set of all measures μ G Mn(T) for which condition (b) of Corollary 5B holds. Then the set (5.8) Wa,a = iμ e W I ί adμ ^ o\ is weak* compact in Mn(T) for any aeCn(T) satisfying (5.9) -a(t) e int D(t) for every teT , and any real number a.

COROLLARY 6A. Under the hypothesis of Theorem 6, the normal cone to the convex set K at a point ue K is the set of all N-valued measures μeMn(T), where N{t) is for each t the normal cone to Q(t) at u(t). Added in proof. See [23] for additional references and results along the lines of this paper. REFERENCES 1. N. Bourbaki, Espaces Vectorίels Topologiques, Hermann et Cie., Paris, 1953. 2. C. Castaing, Sur les multi-applications measurables, These, Faculte des Sciences, Caen, 1967. This has partly