Kenneth Kuttler

1248 CHAPTER 35. WEAK DERIVATIVES

h(B(x,r))⊆ h(x)+Dh(x)B(0,r (1+ ε)), (35.6.14)

(h(

B(x,r)))≤ mn (Dh(x)B(0,r (1+ ε))) (35.6.15)

If x ∈Ω\ (S∪N) is also a point of density of Ω, then

limr→0

mn (h(B(x,r)∩Ω))

mn (h(B(x,r)))= 1. (35.6.16)

If x ∈Ω\N, then

|detDh(x)|= limr→0

mn (h(B(x,r)))mn (B(x,r))

a.e. (35.6.17)

Proof: Since Dh(x)−1 exists,

h(x+v) = h(x)+Dh(x)v+o(|v|) (35.6.18)

= h(x)+Dh(x)

v+

=o(|v|)︷︸︸︷Dh(x)−1 o(|v|)

 (35.6.19)

Consequently, when r is small enough, 35.6.14 holds. Therefore, 35.6.15 holds. From35.6.19, and the assumption that Dh(x)−1 exists,

Dh(x)−1 h(x+v)−Dh(x)−1 h(x)−v =o(|v|). (35.6.20)

LettingF(v) = Dh(x)−1 h(x+v)−Dh(x)−1 h(x),

apply Lemma 35.6.9 in 35.6.20 to conclude that for r small enough, whenever |v|< r,

Dh(x)−1 h(x+v)−Dh(x)−1 h(x)⊇ B(0,(1− ε)r).

Therefore,h(

B(x,r))⊇ h(x)+Dh(x)B(0,(1− ε)r)

which impliesmn

(h(

B(x,r)))≥ mn (Dh(x)B(0,r (1− ε)))

which shows 35.6.13.Now suppose that x is a point of density of Ω as well as being a point where Dh(x)−1

and Dh(x) exist. Then whenever r is small enough,

1− ε <mn (h(B(x,r)∩Ω))

mn (h(B(x,r)))≤ 1

and so

1− ε <mn(h(B(x,r)∩ΩC

))mn (h(B(x,r)))

+mn (h(B(x,r)∩Ω))

mn (h(B(x,r)))

≤mn(h(B(x,r)∩ΩC

))mn (h(B(x,r)))

+1.

1248CHAPTER 35. WEAK DERIVATIVESh(B(x,r)) Ch(x) + Dh(x) B(0,r(1+8)),mn (h (B()) < mp, (Dh (x) B(0,r(1+)))Ifx € Q\ (SUN) is also a point of density of Q, thenIfx € Q\N, thenMy (h (B(x, r) NQ))r30 my (h(B(X,r))) IL(h(B(x,r)))\det Dh (x)| = lim “2PMr>0 =m, (B(x,r))Proof: Since Dh (x)! exists,h(x+v)= h(x)+Dh(x)v+o(|v))=0(|y|)————= h(x)+Dh(x) | v+Dh(x)~'o0({v))(35.6.14)(35.6.15)(35.6.16)(35.6.17)(35.6.18)(35.6.19)Consequently, when r is small enough, 35.6.14 holds. Therefore, 35.6.15 holds. From35.6.19, and the assumption that Dh (x)! exists,Dh (x) 'h(x+v)—Dh(x) 'h(x) —v=o(|v|).LettingF (v) = Dh(x)'h(x+v) —Dh(x)~'h(x),apply Lemma 35.6.9 in 35.6.20 to conclude that for r small enough, whenever |v| < r,Dh(x)~'h(x+v) —Dh(x)~'h(x) > B(0,(1—e)r).Therefore,h (8) > h(x) + Dh(x) B(0,(1—e)r)which impliesm, (h(B(xr)) ) > mn (D(x) B (0,r (1 ~e)))which shows 35.6.13.(35.6.20)Now suppose that x is a point of density of Q as well as being a point where Dh (x)!and Dh (x) exist. Then whenever r is small enough,and sol-é€ <IAmn (h(B(x,r) NQ))mn (h(B(x,r)))l-e< <1mn (bh (B(x,r)MQ°)) my (h(B(x,r) NQ))m, (h(B(x,r))) m, (h(B(X,r)))mn ( (B(x,r NQ°))m,(h(BQxr))