Kenneth Kuttler

1050 CHAPTER 29. THE AREA FORMULA

and so using the dominated convergence theorem in 29.11.68, it equals

limε→0

∫Ω

detDh(x)∫

f (y)φ ε (h(x)−y)dydx

= limε→0

∫Ω

detDh(x)∫

f (h(x)−y)φ ε (y)dydx

= limε→0

∫Ω

detDh(x)∫

B(0,1)f (h(x)− εu)φ (u)dudx

Now ∣∣∣∣∫Ω

detDh(x)∫

B(0,1)f (h(x)− εu)φ (u)dudx

−∫

Ω

detDh(x) f (h(x))dx∣∣∣∣≤∣∣∣∣∫B(0,1)

∫Ω

|detDh(x)| | f (h(x)− εu)− f (h(x))|dxφ (u)du∣∣∣∣

which needs to converge to 0 as ε → 0. However, from the area formula, Theorem 29.5.3applied to the inside integral, the above equals∫

B(0,1)

∫h(Ω)| f (y− εu)− f (y)|dyφ (u)du≤

∫B(0,1)

|| fεu− f ||L1(Rn) φ (u)du

which converges to 0 by continuity of translation in L1 (Rn). Thus as in the lemma,∫f (y)d (y,Ω,h)dy = lim

ε→0

∫f (y)

∫Ω

φ ε (h(x)−y)detDh(x)dxdy

= limε→0

∫Ω

detDh(x)∫

B(0,1)f (h(x)− εu)φ (u)dudx

=∫

Ω

detDh(x) f (h(x))dx

and this proves the corollary.Note that in this corollary h is one to one.

29.12 The Case Of W 1,p

There is a very interesting application of the degree to integration [52]. Recall Lemma23.1.11. I want to generalize this to the case where h :Rn → Rn has the property that itsweak partial derivatives and h are in Lp (Rn;Rn) , p > n. This is denoted by saying

h ∈W 1,p (Rn;Rn) .

In the following proposition, let φ ε be a symmetric nonnegative mollifier,

φ ε (x)≡1εn φ

(xε

),sptφ ⊆ B(0,1) .

1050 CHAPTER 29. THE AREA FORMULAand so using the dominated convergence theorem in 29.11.68, it equalslim det Dh (x (x) | f(y) @, (a(x) —y) dydx€0= lim [ det (x) | (h(x) -y) 0¢ (y) aya= lim / detDh(x) f (h(x) — €u) ¢ (u) dudxe0JQ B(0,1)Now(x) | F(tn(x)—eu)9 (uw) dudsB(0,1)<- [ detDh (x) f (ln(x)) dxJQLoy fet m9) OCs) — eu) — F(x) v9 (ucB(0,1) JQwhich needs to converge to 0 as € — 0. However, from the area formula, Theorem 29.5.3applied to the inside integral, the above equalsDoo ee LO Fao) dus [|New Flargan 6(w) a(0,1) B(0,1)which converges to 0 by continuity of translation in L! (IR”). Thus as in the lemma,[F0aQy.2.n)dy = tim [ £09) fe (h(x) — y)douDh (x) dxdy€0= lim [ dewh(x) hy yf LC) eu) 9 (u) du— I det Dh (x) f (h(x)) dxand this proves the corollary.Note that in this corollary h is one to one.29.12 The Case Of W!?There is a very interesting application of the degree to integration [52]. Recall Lemma23.1.11. I want to generalize this to the case where h :IR” — R” has the property that itsweak partial derivatives and h are in L? (IR";IR”) , p > n. This is denoted by sayingheWw!?(R";R").In the following proposition, let @, be a symmetric nonnegative mollifier,6e(x) = 50 (*).spt9 CB (0.1),