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(∇u) = 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 ∈ 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 → ∞, 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→∞ un(t) dt = limn→∞ (Ak )xk . 0 0

And thus the space L p(0, T; X) of all functions u with

1   p ZT    kukL p (0,T;X) :=  ku(t)kX dt < ∞.   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 |u (t)|, 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 < ∞ and m ∈ 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 ∈ X and n ∈ 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 ≤ ∞,

is also continuous. (d) The closure

∞ 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 ∈ N, 1 < p < ∞ and p0 its Hölder conjugate.

Z

kvk−m,p = sup u(x)v(x) dx . ∈ 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 ∈ L 1(0, T; X). If w satisfies

Z T Z T (n) n ∞ ϕ (t)u(t) dt = (−1) ϕ(t)w(t), for all ϕ ∈ 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 < ∞, 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 ∈ L p(0, T; X) with the generalised derivative

p0 ? ∂t u ∈ 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 ) =   ∈ p( ; 1,p( )) ∈ q( ; −1,p( )) u L 0, T W0 B ∂t u L 0, 1 W B .

Wolfgang Stefani Bochner Spaces 12/12