The main interest in all this is in the application to bounded open sets. Recall the
following definition.
Definition 41.2.1A bounded open subset, Ω, of ℝ^{n}has a C^{m,1}boundary if it satisfiesthe following conditions. For each p ∈ Γ ≡Ω∖ Ω, there exists an open set, W, containingp, an open interval
(0,b)
, a bounded open box U^{′}⊆ ℝ^{n−1}, and an affine orthogonaltransformation, R_{W}consisting of a distance preserving linear transformation followed bya translation such that
′
RW W = U × (0,b), (41.2.30)
(41.2.30)
RW (W ∩ Ω) = {u ∈ ℝn : u′ ∈ U ′,0 < un < ϕW (u′)} (41.2.31)
(41.2.31)
where ϕ_{W}∈ C^{m,1}
(--)
U ′
meaning ϕ_{W}is the restriction to U^{′}of a function, still denoted byϕ_{W}which is in C^{m,1}
( )
ℝn−1
and
inf{ϕ (u′) : u′ ∈ U′} > 0
W
The following picture depicts the situation.
PICT
For the situation described in the above definition, let h_{W} : U^{′}→ Γ ∩ W be defined
by
hW (u′) ≡ R−W1 (u ′,ϕW (u′)),gW (x) ≡ (RW x)′,HW (u) ≡ R −W1(u′,ϕW (u′)− un).
where x^{′}≡
(x1,⋅⋅⋅,xn− 1)
for x =
(x1,⋅⋅⋅,xn)
. Thus g_{W}∘ h_{W} = id on U^{′} and
h_{W}∘ g_{W} = id on Γ ∩ W. Also note that H_{W} is defined on all of ℝ^{n} is C^{m,1}, and has an
inverse with the same properties. To see this, let G_{W}
(u)
=
′ ′
(u ,ϕW (u )− un)
. Then
H_{W} = R_{W}^{−1}∘ G_{W} and G_{W}^{−1} =
′ ′
(u ,ϕW (u) − un )
and so H_{W}^{−1} = G_{W}^{−1}∘ R_{W}.
Note also that as indicated in the picture,
R (W ∩ Ω) = {u ∈ ℝn : u′ ∈ U′ and 0 < u < ϕ (u ′)}.
W n W
Since Γ = ∂Ω is compact, there exist finitely many of these open sets, W, denoted by
{Wi}
_{i=1}^{q} such that Γ ⊆∪_{i=1}^{q}W_{i}. Let the corresponding sets, U^{′} be denoted by U_{i}^{′} and
let the functions, ϕ be denoted by ϕ_{i}. Also let h_{i} = h_{Wi},G_{Wi} = G_{i} etc. Now
let
Φi : GiRi(Ω ∩W ) ≡ Vi → Ω∩ Wi
be defined by
Φi(y) ≡ R−i1∘ G −i1(y).
Thus Φ_{i},Φ_{i}^{−1}∈ C^{m,1}
n
(ℝ )
. The following picture might be helpful.
PICT
Therefore, by Lemma 38.3.3 on Page 4622, it follows that for t ∈ [m,m + 1),
Φ ∗∈ ℒ(Ht (W ∩Ω ),Ht(V )).
i i i
Assume
aij (x) vivj ≥ δ |v|2. (41.2.32)
(41.2.32)
Lemma 41.2.2Let W be one of the sets described in the above definition andlet m ≥ 1. Let W_{1}⊆W_{1}⊆ W where W_{1}is an open set. Suppose also that
1
u ∈ H (Ω(),)
αrs ∈ C0,1 Ω ,
f ∈ L2 (Ω ),
1
hk ∈ H (Ω),
Now the function on the right in 41.2.36 is in C^{0,1}
(U-)
. This is because of the
assumption that m ≥ 1 in the statement of the lemma. This function is therefore a finite
product of bounded functions in C^{0,1}
Therefore, by Lemma 41.1.1, there exists a constant, C, independent of f,g, and w
such that
( || ||2 ∑ || ||2 )
||w||2H2(Φ−1(W1∩Ω )) ≤ C ||||^f||||2 + ||w||2H1 (U) + ||||^hl|||| 1 .
L (U) l H (U)
Therefore,
( )
2 2 2 ∑ 2
||u||H2(W1∩Ω) ≤ C ||f||L2(W∩Ω ) + ||w||H1(W∩Ω ) + ||hk||H1(W ∩Ω)
( k )
2 2 ∑ 2
≤ C ||f||L2(Ω) + ||w||H1 (Ω) + ||hk||H1 (Ω) .
k
which proves the lemma.
With this lemma here is the main result.
Theorem 41.2.3Let Ω be a bounded open set with C^{1,1}boundary as in Definition41.2.1, let f ∈ L^{2}
(Ω)
,h_{k}∈ H^{1}
(Ω )
, and suppose that for all x ∈Ω,
aij (x) vivj ≥ δ |v|2.
Suppose also that u ∈ H^{1}
(Ω )
and
∫ ∫ ∫
aij(x)u (x)v (x)dx + h (x) v (x)dx = f (x )v(x)dx
Ω ,i ,j Ω k ,k Ω
for all v ∈ H^{1}
(Ω )
. Then u ∈ H^{2}
(Ω)
and for some C independent of f,g, andu,
( )
2 2 2 ∑ 2
||u||H2(Ω) ≤ C ||f||L2(Ω) + ||u||H1 (Ω) + ||hk||H1 (Ω) .
k
Proof: Let the W_{i} for i = 1,
⋅⋅⋅
,l be as described in Definition 41.2.1. Thus
∂Ω ⊆∪_{j=1}^{l}W_{j}. Then let C_{1}≡ ∂Ω ∖∪_{i=2}^{l}W_{i}, a closed subset of W_{1}. Let D_{1} be an open
set satisfying
---
C1 ⊆ D1 ⊆ D1 ⊆ W1.
Then D_{1},W_{2},
⋅⋅⋅
,W_{l} cover ∂Ω. Let C_{2} = ∂Ω ∖
(D ∪ (∪l W ))
1 i=3 i
. Then C_{2} is a closed
subset of W_{2}. Choose an open set, D_{2} such that
---
C2 ⊆ D2 ⊆ D2 ⊆ W2.
Thus D_{1},D_{2},W_{3}
⋅⋅⋅
,W_{l} covers ∂Ω. Continue in this way to get D_{i}⊆ W_{i}, and
∂Ω ⊆∪_{i=1}^{l}D_{i}, and D_{i} is an open set. Now let
D ≡ Ω ∖∪l D-.
0 i=1 i
Also, let D_{i}⊆ V_{i}⊆V_{i}⊆ W_{i}. Therefore, D_{0},V_{1},
⋅⋅⋅
,V_{l} covers Ω. Then the same
estimation process used above yields
( ∑ )
||u ||H2(D0) ≤ C ||f||2L2(Ω) + ||u||2H1(Ω) + ||hk||2H1(Ω) .
k