Conforming Finite Elements - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 40

2.7 Example. Let H = 2 be the space of infinite sequences (x 1 ,x 2 ,...) , equipped
with the norm x

: = m x m . The form

2

∞

2 − m x m y m

a(x,y) : =

m = 1

= m = 1 2 − m x m defines a is positive and continuous but not coercive, and , x :

1

continuous linear functional. However, J(x) = does not attain a

minimum in 2 . Indeed, a necessary condition for a minimal solution in this case
is that x m =

2 a(x,x) − , x

1 , 2 ,... , and this contradicts m x m < ∞ 1 for m = .

With the above preparations, we can now make the concept of a solution of
the boundary-value problem more precise.

2.8 Definition. A function u ∈ H 0 () is called a weak solution of the second
order elliptic boundary-value problem

Lu = f in ,

( 2 . 17 )

u = 0 n ∂,

with homogeneous Dirichlet boundary conditions, provided that

for all v ∈ H 0 (), a(u, v) = (f, v) 0

( 2 . 18 )

where a is the associated bilinear form defined in (2.11).

In other cases we shall also refer to a function as a weak solution of an ellip-
tic boundary-value problem provided it is a solution of an associated variational
problem. - Throughout the above, we have implicitly assumed that the coefficient
functions are sufficiently smooth. For the following theorem, a ij ∈ L ∞ () and
f ∈ L 2 () suffice.

2.9 Existence Theorem. Let L be a second order uniformly elliptic partial differ-
ential operator. Then the Dirichlet problem (2.17) always has a weak solution in
H 0 (). It is a minimum of the variational problem

1
2 a(v, v)

− (f, v) 0 −→ min !

over H 0 ().