1360 CHAPTER 41. ELLIPTIC REGULARITY

Ri

Wi

Ω⋂

Wi

spt(ψ i)

Ri(Ω⋂

Wi)

0

b

u′ ∈U ′

Gi

0Rn−1

y Φ−1i (spt(ψ i))

Vi

Φ−1i (Ω

⋂Wi)

Ui

Therefore, by Lemma 38.3.3 on Page 1312, it follows that for t ∈ [m,m+1),

∗i ∈L

(Ht (Wi∩Ω) ,Ht (Vi)

).

Assume

ai j (x)viv j ≥ δ |v|2 . (41.2.32)

Lemma 41.2.2 Let W be one of the sets described in the above definition and let m ≥ 1.Let W1 ⊆W1 ⊆W where W1 is an open set. Suppose also that

u ∈ H1 (Ω) ,

αrs ∈ C0,1 (

Ω),

f ∈ L2 (Ω) ,

hk ∈ H1 (Ω) ,

and that for all v ∈ H1 (Ω∩W ) such that spt(v)⊆Ω∩W,∫Ω

ai j (x)u,i (x)v, j (x)dx+∫

Ω

hk (x)v,k (x)dx =∫

Ω

f (x)v(x)dx. (41.2.33)