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1 Vector-Valued Functions Partial differential equations 2 last updated May 19, 2017 1 Vector-valued functions [2, Section VII.1,2,3], [3, Chapter II, IV] ∗ ∗ In the whole section X denotes a Banach space with norm kukX , X is dual of X, hx ; xiX∗;X is the duality between x∗ 2 X∗ and x 2 X, T > 0, I = [0;T ]. Some theorems hold also for unbounded or open interval I. Check it. 1.1 Vector-valued integrable functions { Bochner integral Definition 1. Function u : I ! X is called 1. simple, if there is N 2 N, Aj ⊂ I, j 2 f1;:::;Ng Lebesgue measurable and xj 2 X, j 2 f1;:::;Ng PN such that u(t) = j=1 χAj (t)xj. 2. simple integrable on I~ ⊂ I if for all j 2 f1;:::;Ng: jAj \ I~j ≤ +1 or xj = o. 3. measurable (strongly measurable), if there are un(t) simple such that un(t) ! u(t) (strongly in X) for a.e. t 2 I Definition 2. Function u : I ! X is called (Bochner) integrable, provided it is strongly measurable R and there exist un simple integrable such that I ku(t) − un(t)kX dt ! 0 for n ! 1. The (Bochner) integral of u : I ! X is defined as follows: R PN 1. I u(t) dt = j=1 xjλ(Aj), if u(t) is simple R R 2. I u(t) dt = limn!1 I un(t) dt, if u(t) is (Bochner) integrable Remark 1. One has to check these definitions are correct (i.e. independent of xj, Aj in the first part, and of un(t) in the second part). R R One also proves that k I u(t) dtkX ≤ I ku(t)kX dt for any u(t) integrable. R Theorem 1 (1-Bochner). Function u : I ! X is Bochner integrable iff u is measurable and I ku(t)kX dt < 1. Theorem 2 (2-Lebesgue). Let un : I ! X be measurable, un(t) ! u(t) for a.e. t 2 I, and let there exist g : I ! R integrable such that kun(t)k ≤ g(t) for a.e. t and all n. Then u is Bochner integrable R R R R and I un(t) dt ! I u(t) dt; in fact one even has k I un(t) − u(t) dtkX ≤ I kun(t) − u(t)kX dt ! 0, n ! 1. Definition 3. For p 2 [1; 1), u : I ! X we set Z p p L (I; X) = u(t): I ! X; u(t) is measurable and ku(t)kX dt < 1 ; I 1 Z p p kukLp(I;X) = ku(t)kX d t I 1 For p = 1 we set 1 L (I;X) = u(t): I ! X; u(t) is measurable and t 7! ku(t)kX is essentially bounded ; kuk = ess-supt2I ku(t)kX : Essential boundedness means: there is c > 0 such that ku(t)kX ≤ c pro a.e. t 2 I, ess-supt2I ku(t)kX = inffM 2 R; λ(ft 2 I; ku(t)kX > Mg) = 0g. p Theorem 3 (3). [2, Section VII.3, Theorem 14] Let p 2 [1; +1]. (L (I;X); k · kLp(I;X)) is a Banach space (we identify the function equal a. e.). 2 Let X be a Hilbert space with scalar product h; i then L (I;X) with the scalar product hu; viL2(I;X) = R I hu(t); v(t)i d t is a Hilbert space. Let I bounded, 1 ≤ p ≤ q ≤ +1, then Lq(I;X) ,! Lp(I;X). If p 2 [1; +1), the simple integrable functions are dense subspace of Lp(I;X). [2, Section VII.3, Theorem 15] 1 Remark 2. If X = R then the simple functions are dense in L (I; R). This is not in general true if X is infinite dimensional. Definition 4. Let u 2 L1( ;X), ' 2 D( ). We define for t 2 u ? '(t) = R u(t − s)'(s) d s. R R R R Definition 5. Let J be an interval. Let u : J ! X. We say that u is continuous (u 2 C(J; X)) if for any U ⊂ X open u−1(U) is open in J. ◦ We say that u is differentiable in t 2 J if the limit limh!0(u(t + h) − u(t))=h exists in X. We denote it u0(t). If u is differentiable at every t 2 J ◦ then u0 : J ◦ ! X. We say that u 2 C1(J; X) if u; u0 can be extended to functions in C(J; X). (1) 0 (k) (k−1) 0 We define first derivative of u as u = u and for k 2 N, k > 1 we define u = (u ) . We say that u 2 Ck(J; X) if u; u0 2 Ck−1(J; X). 1 k We say that u 2 C (J; X) if for all k 2 N u 2 C (J; X). By subscript c we always mean that the functions has compact support (in J). Theorem 4 (4). Let p 2 [1; 1). 1 PN 1. The set f' : I ! Xj9N 2 N; xj 2 X; 'j 2 Cc (I) for j 2 f1;:::;Ng such that ' = j=1 'jxjg is dense in Lp(I;X). 1 2. If Y is dense subset of X, the set f' : I ! Xj9N 2 N; yj 2 Y; 'j 2 Cc (I) for j 2 f1;:::;Ng PN p 1 p such that ' = j=1 'jxjg is dense in L (I;X). In particular Cc (I;Y ) is dense in L (I;X). 3. If X is separable, then Lp(I;X) is separable. 1 R p 4. Let 2 C (R), spt( ) ⊂ (−1; 1), = 1, ≥ 0, n(t) = n (nt) for n 2 N. Let u 2 L (I;X) R p be extended by 0 outside I. Then u ? n ! u in L (I;X) as n ! +1. Remark 3. If X is separable, then also Lp(I; X) is separable for p < 1. But none of these holds for p = 1. p R p Corollary 5 (5). If p 2 [1; +1), > 0 and u 2 L (U(I; );X). Then I ku(x + t) − u(x)kX ! 0 as t ! 0. p Lemma 6 (6). Let p 2 [1; +1], un ! u in L (I;X), then there is a subsequence funk g of fung such that unk (t) ! u(t) in X for a.e. t 2 I. 2 Theorem 7 (7-H¨older'sinequality.). Let u 2 Lp(I; X), v 2 Lp0 (I; X∗), where p, p0 are H¨olderconjugate. Then t 7! hv(t); u(t)i is measurable and 1 1 Z Z p Z 0 p p0 p j hv(t); u(t)i j ≤ ku(t)kX dt kv(t)kX∗ dt I I I p0 ∗ R p ∗ Remark 4. If v 2 L (I;X ), p 2 [1; +1] then Φv(u) = I hv; ui satisfies Φv 2 (L (I;X)) . Theorem 8 (8-Dual space to Lp(I; X).). Let X be reflexive, separable and p 2 [1; 1). Denote X = 0 Lp(I; X). Then for any F 2 X ∗ there is v 2 Lp (I;X∗) such that Z hF; uiX ∗;X = hv(t); u(t)iX∗;X dt 8u 2 X : I 0 Moreover, v is uniquely defined, and its norm in Lp (I; X∗) equals to the norm of F in X ∗. Remark 5. If X is reflexive, separable, and p 2 (1; 1), then Lp(I; X) is also reflexive, separable. Any sequence bounded in Lp(I; X) has a weakly convergent subsequence. 1.2 Weakly differentiable function Definition 6. We say that u : I ! X is weakly differentiable if u 2 L1(I;X) and there is g 2 L1(I;X) such that Z Z 8' 2 D(I): u'0 = − g': I I d We call g a weak derivative of u and we write d t u = g or ut = g. Theorem 9 (9). Let u; g 2 L1(I; X). Then the following are equivalent: 1. ut = g d ∗ ∗ ∗ ∗ 2. dt hx ; u(t)i = hx ; g(t)i in the sense of distributions on (0;T ), for every x 2 X fixed. R t 3. There exist x0 2 X such that u(t) = x0 + 0 g(s) ds for a.e. t 2 I. end of the first lecture 24.2.2017 Theorem 10 (10). Let u : I ! X be weakly differentiable. 1. The weak derivative is linear. 1 d d 2. If η : I ! R is C , then uη : I ! X is weakly differentiable, and dt u(t)η(t) = dt u(t)η(t) + u(t)η0(t) for a.e. t 2 I. 1 Theorem 11 (11). If u : I ! X be in Lloc(I;X), 2 D(I), J = ft 2 I; t − spt ⊂ Ig. Then 1 ◦ ◦ 0 u ? 2 C (J ) and 8t 2 J :(u ? )t(t) = u ? . If moreover u is weakly differentiable then ◦ (u ? )t(t) = (ut ? )(t) for t 2 J . Notation. Symbol X,! Y means embedding: X ⊂ Y and there is c > 0 such that kukY ≤ ckukX for all u 2 X. Symbol X,!,! Y means compact embedding: X,! Y and any sequence bounded in X has a subsequence converging strongly in Y . 3 p q Definition 7. The claim u 2 L (I;X), ut 2 L (I;Y ) means that there is a Banach space Z such that p q p X,! Z, Y,! Z, i.e.
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