41.2. THE CASE OF BOUNDED OPEN SETS 1365

Now the function on the right in 41.2.43 is in Ck,1(U). This is because of the assump-

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

(U).

The function h̃l defined in 41.2.44 is in Hk (U) and∣∣∣∣∣∣h̃l

∣∣∣∣∣∣Hk(U)

≤C∑s||hs||Hk(Ω∩W )

again because m≥ k.Finally, the right side of 41.2.45 is a function in Hk−1 (U) by Lemma 38.3.3 on Page

1312 and the observation that |detDΦ(·)| ∈Ck−1,1(U)

which follows from the assumptionof the lemma that m≥ k so Φ ∈Ck−1,1 (Rn). Also∣∣∣∣∣∣ f̃ ∣∣∣∣∣∣

Hk−1(U)≤C || f ||Hk−1(Ω∩W ) .

Therefore, 41.2.42 is of the form∫U

αrs (y)w,rz,sdy+

∫U

h̃lz,ldy =∫

Uf̃ zdy, (41.2.46)

for all z in H1 (U) having support in V.Claim: There exists r > 0 independent of y ∈U such that for all y ∈U ,

αrs (y)vrvs ≥ r |v|2 .

Proof of the claim: If this is not so, there exist vectors, vn, |vn| = 1, and yn ∈U suchthat αrs (yn)vn

r vns ≤ 1

n . Taking a subsequence, there exists y ∈ U and |v| = 1 such thatαrs (y)vrvs = 0 contradicting 41.2.32.

Therefore, by Corollary 41.1.2, there exists a constant, C, independent of f ,g, and wsuch that

||w||2Hk+1(Φ−1(W1∩Ω)) ≤C

(∣∣∣∣∣∣ f̃ ∣∣∣∣∣∣2Hk−1(U)

+ ||w||2Hk(U)+∑l

∣∣∣∣∣∣h̃l

∣∣∣∣∣∣2Hk(U)

).

Therefore,

||u||2Hk+1(W1∩Ω) ≤ C(|| f ||2Hk−1(W∩Ω)+ ||w||

2Hk(W∩Ω)+∑

s||hs||2Hk(W∩Ω)

)≤ C

(|| f ||2Hk−1(Ω)+ ||w||

2Hk(Ω)+∑

s||hs||2Hk(Ω)

).

which proves the lemma.Now here is a theorem which generalizes the one above in the case where more regu-

larity is known.

Theorem 41.2.6 Let Ω be a bounded open set with Ck,1 boundary as in Definition 41.2.1,let f ∈ Hk−1 (Ω) ,hs ∈ Hk (Ω), and suppose that for all x ∈Ω,

ai j (x)viv j ≥ δ |v|2 .