On measure differential inclusions, weak and strong formulations

On measure differential inclusions 73

3. On measure differential inclusions, weak and strong formulations

Let Q : I → n , with I ⊂ closed interval, be a given multifunction with nonempty closed convex values and let µ be a Borel measure on I , of bounded variation. In [23] Stewart considered the two formulations of measure differential inclusions. Strong formulation. S dµ a dλ t ∈ Qt λ –a.e. in I dµ s d|µ s | t ∈ Q ∞ t µ s –a.e. in I where µ = µ a + µ s be the Lebesgue decomposition of measure µ. Weak formulation. W R I φ dµ R I φ dλ ∈ cl co [ t ∈I ∩ Supp φ Qt for every φ ∈ , where denotes the set of all continuous functions φ : → + , with compact support, such that R I φ dλ 6= 0. Stewart proved that the two formulations are equivalent, under suitable assumptions on Q see Theorem 2, by means of a transfinite induction process. We provide here a direct proof of the equivalence, under weaker assumption. Moreover, for our convenience, we introduce also the following local version of weak for- mulation. Local-weak formulation. Let t ∈ I be fixed. There exists h = ht 0 such that for every 0 h h, LW R B h φ dµ R B h φ dλ ∈ cl co [ t ∈B h Qt for every φ ∈ such that Supp φ ⊂ B h . Of course, if µ satisfies W, then LW holds for every t ∈ I . Rather surprising also the convers hold, as we shall show in the following Theorem 3. In other words, also this last formulation proves to be equivalent to the previous ones. T HEOREM 1. Every solution of S is also a solution of W. Proof. Let φ ∈ be given. Note that R I φ dµ = R I φ dµ a + R I φ dµ s moreover Z I φ dµ a = Z I dµ a dλ φ dλ = Z I ∩ Supp φ dµ a dλ dλ φ 5 Z I φ dµ s = Z I dµ s d |µ s | φ d |µ s | = Z I ∩ Supp φ dµ s d |µ s | dµ s,φ 6 where λ φ and µ s,φ are the Borel measures defined respectively by λ φ E = Z E φ dλ µ s,φ E = Z E φ d |µ s | E ⊂ I . 74 P. Brandi – A. Salvadori From 5, in force of the assumption and taking Theorem 1.3 in [1] into account, we get φ a : = R I φ dµ a R I φ dλ = R I ∩ Supp φ dµ a dλ dλ φ λ φ I ∩ Supp φ ∈ cl co [ t ∈I ∩ Supp φ Qt . 7 In the case R I φ d |µ s | = 0, then R I φ dµ s = 0 and the assertion is an immediate consequence of 7. Let us put 7 ′ Q φ : = cl co [ t ∈I ∩ Supp φ Qt . Let us assume now that R I φ d |µ s | 6= 0. Then from 6, in force of the assumption we get, as before R I φ dµ s R I φ d |µ s | = R I ∩ Supp φ dµ s d|µ s | dµ s,φ µ s,φ I ∩ Supp φ ∈ cl co [ t ∈I ∩ Supp φ Q ∞ t ⊂  cl co [ t ∈I ∩ Supp φ Qt   ∞ = [Q φ ] ∞ and since the right-hand side is a cone, we deduce φ s : = R I φ dµ s R I φ dλ = R I φ dµ s R I φ d |µ s | · R I φ d |µ s | R I φ dλ ∈ [Q φ ] ∞ . 8 From 7 and 8 we have that R I φ dµ R I φ dλ = φ a + φ s with φ a ∈ Q φ φ s ∈ [Q φ ] ∞ and, by virtue of P 3 , we conclude that R I φ dµ R I φ dλ ∈ Q φ = cl co [ t ∈ Supp φ Qt which proves the assertion. T HEOREM 2. Let µ be a solution of LW in t ∈ I . a If Q has properties Q at t and the derivative dµ a dλ t exists, then dµ a dλ t ∈ Qt . b If Q ∞ has properties Q at t and the derivative dµ s d|µ s | t exists, then dµ s d |µ s | t ∈ Q ∞ t . On measure differential inclusions 75 Proof. Let S µ denote the set where measure µ s is concentrated, i.e. S µ = {t ∈ I : µ s {t} 6= 0}. Since µ s is of bounded variation, then S µ is denumerable; let us put S µ = {s n , n ∈ } . Let us fix a point t ∈ I . The case where t is an end-point for I is analogous. The proof will proceed into steps. Step 1. Let us prove first that for every B h = Bt , h ⊂ I with 0 h ht and such that ∂ B h ∩ S µ = φ, we have µ B h − S µ 2h = µ a B h 2h ∈ cl co [ t ∈B h Qt . 9 Let n ∈ be fixed. For every 1 ≤ i ≤ n, we consider a constant 0 r i = r i n ≤ 1 n2 i such that Bs i , r i ∩ Bs j , r j = φ, i 6= j, 1 ≤ i, j ≤ n. Moreover, we put I n = n [ i=1 B s i , r i . Fixed a constant 0 η min {h, r i , 1 ≤ i ≤ n}, we denote by I n,η = n [ i=1 B s i , r i − η and consider the function φ n,η t =    t ∈ I − B h ∪ I n,η 1 t ∈ B h−η − I n linear otherwise Of course φ n,η ∈ thus, by virtue of the assumption, we have R n,η : = R I φ n,η dµ R I φ n,η dλ ∈ cl co [ t ∈B h Qt . 10 Note that, put C n,η = B h − I n,η ∪ B h−η − I n , we have R n,η = R B h −I n,η φ n,η dµ R B h −I n,η φ n,η dλ = µ B h−η − I n + R C n,η φ n,η dµ λ B h−η − I n + R C n,η φ n,η dλ . 11 If we let η → 0, we get B h−η − I n ր B h − I n I n,η ր I n and hence C n,η ց ∂ B h = {t − h, t + h} . As a consequence, we have see e.g. [14] lim η→ µ B h−η − I n = µB h − I n lim η→ λ B h−η − I n = λB h − I n lim η→ |µ|C n,η = lim η→ λ C n,η = 0 12 76 P. Brandi – A. Salvadori and hence 12 ′ lim η→ Z C n,η φ n,η dµ = lim η→ Z C n,η φ n,η dλ = 0 . From 11, 12 and 12 ′ , we obtain lim η→ R n,η = µ B h − I n λ B h − I n = µ a B h − I n + µ s B h − I n λ B h − I n . 13 Note that since λ I n = n X i=1 2r i ≤ 2 n n X i=1 1 2 i 2 n we have lim n→+∞ λ I n = lim n→+∞ µ a I n = 0 . 14 Moreover |µ s B h − I n | ≤ |µ s |B h − I n ≤ |µ s |S µ − I n = X nn |µ s |{s n } and, recalling that µ has bounded variation 14 ′ lim n→+∞ |µ s B h − I n | ≤ lim n→+∞ X nn |µ s |{s n } = 0 . Finally, from 13, 14 and 14 ′ we conclude that lim n→+∞ lim η→ R n,η = µ a B h 2h that, by virtue of 10, proves 9. Step 2. Let us prove now part a. We recall that dµ a dλ t = lim h→0 µ a B h 2h 15 By virtue of step 1, for every fixed h 0 such that B h ⊂ I , we have µ a B h 2h ∈ cl co [ t ∈B h Qt ⊂ cl co [ t ∈B h Qt λ –a.e. 0 h h and hence, by letting h → 0, and taking 15 into account, we get dµ a dλ t ∈ cl co [ t ∈B h Qt . By virtue of the arbitrariness of h 0 and in force of assumption Q, we conclude that dµ a dλ t ∈ \ h0 cl co [ t ∈B h Qt = Qt . On measure differential inclusions 77 Step 3. For the proof of part b let us note that dµ s d |µ s | t = µ s {t } |µ s |{t } 16 since µ s {t } = R {t } dµ s = R {t } dµ s d|µ s | d |µ s | = dµ s d|µ s | t |µ s |{t }. Let h 0 be fixed in such a way that B h = Bt , h ⊂ I . For every 0 η h we consider the continuous function defined by φ η t =    1 t ∈ B η 2 t ∈ I − B η linear otherwise. Note that see e.g. [14] µ s {t } = lim η→ µ B η 2 = lim η→ µ B η . 17 Moreover we have µ B η 2 = Z B η φ η dµ = Z I φ η dµ − Z B η −B η 2 φ η dµ = R I φ η dµ R I φ η dλ · Z I φ η dλ − Z B η −B η 2 φ η dµ . 18 By assumption we know that R I φ η dµ R I φ η dλ ∈ cl co [ t ∈B η Qt ⊂ cl co [ t ∈B h Qt let us put Q h : = cl co [ t ∈B h Qt . Since lim η→ Z I φ η dµ = 0, by virtue of P 4 we get lim η→ R I φ η dµ R I φ η dλ · Z I φ η dλ ∈ [Q h ] ∞ . 19 Furthermore, by virtue of 17 we have Z B η −B η 2 φ η dµ ≤ |µ|B η − |µ|B η 2 η→ longr i ght arr ow 0 20 thus, from 18 and taking 17, 19 and 20 into account, we obtain µ {t } ∈ [Q h ] ∞ for every h 0such that B h = Bt , h ⊂ I . Finally, recalling P 5 we deduce that µ {t } ∈ \ h0 [Q h ] ∞ = \ h0 Q h ∞ = Q ∞ t and taking 16 into account, since Q ∞ t is a cone, the assertion follows. 78 P. Brandi – A. Salvadori D EFINITION 4. Let µ be a given measure. We will say that a property P holds λ, µ s – a.e. if property P is satisfied for every point t with the exception perhaps of a set N with λ N + µ s N = 0. From Theorem 2 the following result can be deduced. T HEOREM 3. Assume that i Q has properties Q λ–a.e. ii Q ∞ has properties Q µ s –a.e. Then every measure µ which is a solution of LW λ, µ s –a.e. is also a solution of S. As we will observe in Section 6, the present equivalence result [among the three formula- tions S, W, LW] improves the equivalence between strong and weak formulation proved by Stewart, by means of a transfinite process in [23]. It is easy to see that Theorem 3 admits the following generalization. T HEOREM 4. Let Q h : I → m , h ≥ 0 be a net of multifunctions and let µ be a Borel measure. Assume that i Q t = \ h0 Q h t λ –a.e.; ii [Q ] ∞ t = \ h0 [Q h ] ∞ t µ s –a.e.; iii for λ, µ s –a.e. t there exists h = ht 0 such that for every 0 h h R B h φ dµ R B h φ dλ ∈ Q h t for every φ ∈ such that Supp φ ⊂ B h . Then µ is a solution of S. Proof. Let t ∈ I be fixed in such a way that all the assumptions hold. Following the proof of step 1 in Theorem 3, from assumption iii we deduce that µ a B h 2h ∈ Q h t and hence from assumption i as in step 2 we get dµ a dλ t ∈ \ h0 Q h t = Q t . Finally, analogously to the proof of step 3, from asumptions iii and ii we obtain µ {t } ∈ \ h0 [Q h ] ∞ t = Q ∞ t On measure differential inclusions 79 and since Q ∞ t is a cone, we get dµ s d |µ s | t = µ {t } |µ|{t } ∈ Q ∞ t .

4. The main closure theorem