41.2. THE CASE OF BOUNDED OPEN SETS 1361

Then there exists a constant, C, independent of f ,u, and g such that

||u||2H2(Ω∩W1)≤C

(|| f ||2L2(Ω)+ ||u||

2H1(Ω)+∑

k||hk||2H1(Ω)

). (41.2.34)

Proof: LetE ≡

{v ∈ H1 (Ω∩W ) : spt(v)⊆W

}u restricted to W ∩Ω is in H1 (Ω∩W ) and∫

Ω∩Wai j (x)u,iv, jdx+

∫Ω

hk (x)v,k (x)dx =∫

Ω

f (x)v(x)dx for all v ∈ E. (41.2.35)

Now let i (y) = x. For this particular W, denote Φi more simply by Φ, Ui ≡Φi (Ω∩Wi)by U, and Vi by V. Denoting the coordinates of V by y, and letting u(x)≡ w(y) and v(x)≡z(y) , it follows that in terms of the new coordinates, 41.2.35 takes the form∫

Uai j (Φ(y))

∂w∂yr

∂yr

∂xi∂ z∂ys

∂ys

∂x j |detDΦ(y)|dy

+∫

Uhk (Φ(y))

∂ z∂yl

∂yl

∂xk |detDΦ(y)|dx

=∫

Uf (Φ(y))z(y) |detDΦ(y)|dy

Let

αrs (y)≡ ai j (Φ(y))

∂yr

∂xi∂ys

∂x j |detDΦ(y)| , (41.2.36)

h̃l (y)≡ hk (Φ(y))∂yl

∂xk |detDΦ(y)| , (41.2.37)

andf̃ (y)≡Φ

∗ f |detDΦ|(y)≡ f (Φ(y)) |detDΦ(y)| . (41.2.38)

Now the function on the right in 41.2.36 is in C0,1(U). This is because of the assump-

tion that m≥ 1 in the statement of the lemma. This function is therefore a finite product ofbounded functions in C0,1

(U).

The function h̃l defined in 41.2.37 is in H1 (U) and∣∣∣∣∣∣h̃l

∣∣∣∣∣∣H1(U)

≤C∑k||hk||H1(Ω∩W )

again because m≥ 1.Finally, the right side of 41.2.38 is a function in L2 (U) by Lemma 38.3.3 on Page 1312

and the observation that |detDΦ(·)| ∈C0,1(U)

which follows from the assumption of thelemma that m≥ 1 so Φ ∈C1,1 (Rn). Also∣∣∣∣∣∣ f̃ ∣∣∣∣∣∣

L2(U)≤C || f ||L2(Ω∩W ) .