Bochner Spaces Wolfgang Stefani LMU München Hüttenseminar 26.-29.06.2014 Wolfgang Stefani Bochner Spaces 1/12 Motivation We consider a parabolic system of partial differential equations @t u − div DF(ru) = 0 on (0; T) × B = QT u(t; ·) = 0 on [0; T] × @B u(0; ·) = u0(·) a. e., and postulate N u : QT ! R ; F : RdN ! R; 1;2 u0 2 W (B): What is the appropriate solutions space? Wolfgang Stefani Bochner Spaces 2/12 The Bochner Integral I Let (X; k · kX ) be a Banach space. A stair function is a v :[0; T] ! X, XN v(t) = χAk (t)xk ; k=1 S for Ak : Ai \ Aj = ; and k Ak = [0; T]: If there exists a sequence (un)n of stair functions with un(t) ! u(t); in X; RT and kun(t) − u(t)kX dt ! 0; for n ! 1; 0 it is Bochner measurable. Wolfgang Stefani Bochner Spaces 3/12 The Bochner Integral II Now we can define RT RT L1 n n u dt = limn!1 un(t) dt = limn!1 (Ak )xk : 0 0 And thus the space L p(0; T; X) of all functions u with 1 0 1 p BZT C B C kukL p (0;T;X) := B ku(t)kX dtC < 1: @B AC 0 Wolfgang Stefani Bochner Spaces 4/12 The space Cm([0; T]; X) Let u be m-times differentiable, m X (i) kukCm(X) = max ju (t)j; 0≤t≤T i=0 and denote their space Cm([0; T]; X). This space is Banach. Wolfgang Stefani Bochner Spaces 5/12 Properties Let X; Y be Banach spaces, 1 ≤ p < 1 and m 2 N. Then a) Cm([0; T]; X) is dense in L p(0; T; X) and the embedding Cm([0; T]; X) ,! L p(0; T; X); is continuous. b) the set of all polynomials v :[0; T] ! X, i.e. n X i v(t) = ait ; i=1 p with ai 2 X and n 2 N is dense in both C([0; T]; X) and L (0; T; X). Wolfgang Stefani Bochner Spaces 6/12 Properties cont. (c) if the embedding X ! Y is continuous, the embedding L r (0; T; X) ,! L q(0; T; Y); for 1 ≤ q ≤ r ≤ 1; is also continuous. (d) The closure 1 k·kLp (X) C0 ([0; T]; X) can be identified by L p(0; T; X): Wolfgang Stefani Bochner Spaces 7/12 Negative Sobolev Spaces W −k;q Let m 2 N; 1 < p < 1 and p0 its Hölder conjugate. Z kvk−m;p = sup u(x)v(x) dx : 2 m;p Q u W0 (Ω); kukm;p ≤1 0 Now we denote W −m;p(Ω) as the closure of L p (Ω), i.e. k·k− 0 p0 m;p ≡ m;p ? ≡ −m;p L (Ω) W0 (Ω) W (Ω): Wolfgang Stefani Bochner Spaces 8/12 Generalised derivative Let X be Banach spaces, u; w 2 L 1(0; T; X). If w satisfies Z T Z T (n) n 1 ' (t)u(t) dt = (−1) '(t)w(t); for all ' 2 C0 ([0; T]; R); 0 0 then it is the n-th generalised derivative of u on (0; T). We denote u(n) = w. Wolfgang Stefani Bochner Spaces 9/12 Dual Space of L p(0; T; X) 1 1 Let X be a reflexive Banach space and p + p0 = 1. If 1 < p < 1, then L p(0; T; X) is reflexive and we have ? 0 L p(0; T; X) L p (0; T; X ?); i.e. its dual space is isometrically isomorphic to the Hölder conjugate Bochner space of the dual X ?. Wolfgang Stefani Bochner Spaces 10/12 Construction of the Solution Space If we have u 2 L p(0; T; X) with the generalised derivative p0 ? @t u 2 L (0; T; X ) form a real Banach space with the norm kukW p;q = kukL p (0;T;X) + k@t ukL q(0;T;Z): We denote this space W p;q(0; T; X; Z). Wolfgang Stefani Bochner Spaces 11/12 Application 1;p −1;p We set X = W0 (B) and Z = W (B). Then we get p;q 1;p −1;p W (0; 1; W0 ; W ) = 2 p( ; 1;p( )) 2 q( ; −1;p( )) u L 0; T W0 B @t u L 0; 1 W B : Wolfgang Stefani Bochner Spaces 12/12.