1362 CHAPTER 41. ELLIPTIC REGULARITY

Therefore, 41.2.35 is of the form∫U

αrs (y)w,rz,sdy+

∫U

h̃lz,ldy =∫

Uf̃ zdy, (41.2.39)

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 Lemma 41.1.1, there exists a constant, C, independent of f ,g, and w suchthat

||w||2H2(Φ−1(W1∩Ω)) ≤C

(∣∣∣∣∣∣ f̃ ∣∣∣∣∣∣2L2(U)

+ ||w||2H1(U)+∑l

∣∣∣∣∣∣h̃l

∣∣∣∣∣∣2H1(U)

).

Therefore,

||u||2H2(W1∩Ω) ≤ C

(|| f ||2L2(W∩Ω)+ ||w||

2H1(W∩Ω)+∑

k||hk||2H1(W∩Ω)

)

≤ C

(|| f ||2L2(Ω)+ ||w||

2H1(Ω)+∑

k||hk||2H1(Ω)

).

which proves the lemma.With this lemma here is the main result.

Theorem 41.2.3 Let Ω be a bounded open set with C1,1 boundary as in Definition 41.2.1,let f ∈ L2 (Ω) ,hk ∈ H1 (Ω), and suppose that for all x ∈Ω,

ai j (x)viv j ≥ δ |v|2 .

Suppose also that u ∈ H1 (Ω) and∫Ω

ai j (x)u,i (x)v, j (x)dx+∫

Ω

hk (x)v,k (x)dx =∫

Ω

f (x)v(x)dx

for all v ∈ H1 (Ω) . Then u ∈ H2 (Ω) and for some C independent of f ,g, and u,

||u||2H2(Ω) ≤C

(|| f ||2L2(Ω)+ ||u||

2H1(Ω)+∑

k||hk||2H1(Ω)

).

Proof: Let the Wi for i = 1, · · · , l be as described in Definition 41.2.1. Thus ∂Ω ⊆∪l

j=1Wj. Then let C1 ≡ ∂Ω\∪li=2Wi, a closed subset of W1. Let D1 be an open set satisfying

C1 ⊆ D1 ⊆ D1 ⊆W1.