Finite Element Methods for Parabolic Problems - Computational Methods for Quantitative Finance - page 29

We equip W m,p (G) with the norm

m

p
W m,p (G) =

p
L p (G) .

0 D n u

u

n =

The normed space W m,p (G) is complete and hence a Banach space for 1 ≤ p ≤∞ .

W 1 ,p (G) are “essentially” continuous. Functions u ∈

Theorem 3.1.4 Let G b e bounded and u ∈ W 1 ,p (G) . Then , there exists a c ontin-
uous function C 0 (G) such that u

u ∈ = u a . e . on G and for all x 1 ,x 2 ∈ G there

holds

x 2

u (ξ ) d ξ. u(x 2 ) − u(x 1 ) = (3.2)

x 1

Proof Fix y 0 ∈ G and set for any g ∈ L p (G) ,

x

v(x) := g(t) d t, x ∈ G.

y 0

Then, v ∈ C 0 (G) and

x

g(t) d t ϕ (x) d x vϕ d x =

G G y 0

y 0 y 0 b x

g(t)ϕ (x) d t d x g(t)ϕ (x) d t d x. =− +

a x y 0 y 0

C 0 (G) ,

Fubini's theorem implies, ∀ ϕ ∈

y 0

g(t) t
a

b

g(t) b
t

vϕ d x ϕ (x) d x d t ϕ (x) d x d t =− +

G a y 0

=− g(t)ϕ(t) d t. (3.3)

G

:= x

y 0 u (ξ ) d ξ . With ( 3.3 ) we obtain

We set u(x)

uϕ d x u ϕ d x, C 0 (G), =− ∀ ϕ ∈

G G

and hence with the definition of the weak derivative,

(u − u)ϕ d x = ∀ ϕ ∈ C 0 (G). 0 ,

G

T herefore, it follows that for a.e. x ∈ G ,wehave u(x) − u(x) = C . Putting u :=

u + C , we obtain the result.