Conforming Finite Elements - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 45

Clearly, γv is the trace of v on the boundary, i.e., the restriction of v to the
boundary. We know that the evaluation of an H 1 function at a single point does not
always make sense. Theorem 3.1 asserts that the restriction of v to the boundary
is at least an L 2 function.
We delay the proof of the trace theorem until the end this section.

Boundary-Value Problems with Natural Boundary Conditions

3.2 Theorem. Suppose the domain satisfies the hypotheses of the trace theorem.
Then the variational problem

1
2 a(v, v)

J(v) : = − (f, v) 0 , − (g, v) 0 , −→ min !

has exactly one solution u ∈ H 1 (). The solution of the variational problem lies in
C 2 () ∩ C 1 ( ¯ ) if and only if there exists a classical solution of the boundary-value
problem

Lu = f in ,

( 3 . 4 )

ν i a ik ∂ k u = g on ,

i,k

in which case the two solutions are identical. Here ν : = ν(x) is the outward-

pointing normal defined almost everywhere on .
Proof. Since a is an H 1 -elliptic bilinear form, the existence of a unique mini-
mum u ∈ H 1 () follows from the Lax-Milgram Theorem. In particular, u is
characterized by

for all v ∈ H 1 (). a(u, v) = (f, v) 0 , + (g, v) 0 , ( 3 . 5 )

Now suppose (3.5) is satisfied for u ∈ C 2 () ∩ C 1 ( ¯ ) .For v ∈ H 0 () ,
γv =

0, and we deduce from (3.5) that

for all v ∈ H 0 ().

a(u, v) = (f, v) 0

By (2.21), u is also a solution of the Dirichlet problem, where we define the
boundary condition using u . Thus, in the interior we have

Lu = f in . ( 3 . 6 )

For v ∈ H 1 () , Green's formula (2.9) yields

v∂ i (a ik ∂ k u) dx =− ∂ i va ik ∂ k udx + va ik ∂ k uν i ds.