Conforming Finite Elements - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 49

Note that the cone condition excludes cusps in the domain. The domain

2

0 <y<x 5 < 1 : ={ (x, y) ∈ R ; }

has a cusp at the origin, and H 1 () contains the function

u(x, y) = x − 1 ,

whose trace is not square-integrable over .
We would like to point out that Green's formula (2.9) also holds for functions
u, w ∈ H 1 () , provided that satisfies the hypotheses of the trace theorem.
The space H 1 () is isomorphic to a direct sum

H 1 () ∼ H 0 () ⊕ γ(H 1 ()).

Specifically, every u ∈ H 1 () can be decomposed as

u = v + w,

according to the following rule. Let w be the solution of the variational problem
| w |

2
1

→ min ! More exactly, suppose

H 0 (), ( ∇ w, ∇ v) 0 , = 0 for all v ∈

w − u ∈ H 0 ().

= u − w ∈ H 0 () . Here γ is an injective mapping on the set of functions
w which appear in the decomposition.

Let v :

We now consider the connection with continuous functions. As usual, the
norm u ∞ = u ∞ , is based on the essential supremum of | u | over . [It is

not a Sobolev norm.]

2 be a convex polygonal domain, or a domain with
Lipschitz continuous boundary. Then H 2 () is compactly imbedded in C( ¯ ) , and

3.4 Remarks. (1) Let ⊂ R

for all v ∈ H 2 (),

v ∞ ≤ c v 2 ( 3 . 11 )

for some number c = c() .
(2) For every open connected domain ⊂ R

2 , H 2 () is compactly imbedded

in C() .
The above results are not the sharpest possible in this framework. Because of
their importance, and because they follow simply from the trace theorem, we now
give their proofs.