Kenneth Kuttler

1290 CHAPTER 37. BASIC THEORY OF SOBOLEV SPACES

where en−1i is the unit vector in Rn−1 having all zeros except for a 1 in the ith position. Now

by Rademacher’s theorem, Dg(ŷ) exists for a.e. ŷ and so except for a set of measure zero,the expression, o

(g(ŷ−hen−1

)−g(ŷ)

)is o(h) and also for ŷ not in the exceptional set,

g(ŷ−hen−1

i)−g(ŷ) =−hDg(ŷ)en−1

i +o(h) .

Therefore, since the integrand in 37.2.28 has compact support and because of the Lips-chitz continuity of all the functions, the dominated convergence theorem may be applied toobtain ∫

Uw(ŷ,yn)φ ,i (y)dy =∫

Uφ (y)

[−D1u(ŷ,2g(ŷ)− yn)

(en−1

i)+2D2u(ŷ,2g(ŷ)− yn)

(Dg(ŷ)en−1

i)]

=∫

Uφ (y)

[− ∂u

∂yi(ŷ,2g(ŷ)− yn)+2

∂u∂yn

(ŷ,2g(ŷ)− yn)∂g(ŷ)

∂yi

]dy

and so

w,i (y) =∂u∂yi

(ŷ,2g(ŷ)− yn)−2∂u∂yn

(ŷ,2g(ŷ)− yn)∂g(ŷ)

∂yi(37.2.29)

whenever i < n which is what you would expect from a formal application of the chain rule.Next suppose i = n. ∫

Uw(ŷ,yn)φ ,n (y)dy

= limh→0−∫

u(ŷ,2g(ŷ)− (yn +h))−u(ŷ,2g(ŷ)− yn)

hφ (y)dy

= limh→0

∫U

D2u(ŷ,2g(ŷ)− yn)h+o(h)h

φ (y)dy

=∫

∂u∂yn

(ŷ,2g(ŷ)− yn)φ (y)dy

showing that

w,n (y) =−∂u∂yn

(ŷ,2g(ŷ)− yn) (37.2.30)

which is also expected.From the definnition, for y ∈ Rn \U ≡ {(ŷ,yn) : yn ≤ g(ŷ)} it follows w,i = u,i and on

U,w,i is given by 37.2.29 and 37.2.30. Consider ||w,i||pLp(U)for i < n. From 37.2.29

||w,i||pLp(U)=∫

∣∣∣∣ ∂u∂yi

(ŷ,2g(ŷ)− yn)−2∂u∂yn

(ŷ,2g(ŷ)− yn)∂g(ŷ)

∂yi

∣∣∣∣p dy

≤ 2p−1∫

∣∣∣∣ ∂u∂yi

(ŷ,2g(ŷ)− yn)

∣∣∣∣p+2p

∣∣∣∣ ∂u∂yn

(ŷ,2g(ŷ)− yn)

∣∣∣∣p Lip(g)p dy

1290 CHAPTER 37. BASIC THEORY OF SOBOLEV SPACESwhere e”! is the unit vector in R”~! having all zeros except for a 1 in the i” position. Nowby Rademacher’s theorem, Dg (y) exists for a.e. Y and so except for a set of measure zero,the expression, o (g (y — he? ') —g(y)) is o(A) and also for y not in the exceptional set,g (¥— hei") — 8 (¥) = —hDg (y)e? | +0(h).Therefore, since the integrand in 37.2.28 has compact support and because of the Lips-chitz continuity of all the functions, the dominated convergence theorem may be applied toobtainI w (Fyn) 0; (y)dy =[ 0 (y) [Diu (¥,2¢(¥) — yn) (€?-!) + 2Dau (¥, 28 (F) — yn) (De He? !)] dy= [ow )|-5* -(¥,28 (¥) — yn) +254 2469) yn) SO dyand soOU. Oui. 02(y)wily) = Oi (y,28(¥) —yn) — a, (Y,2g(¥) —yn) Oy (37.2.29)whenever i < n which is what you would expect from a formal application of the chain rule.Next suppose i = n.[ Gyno ,(9)ayJU= fy — [ #5289) = Ou +1) ul 2069) —hoo Jy h| Dou(Y,28(¥) —yn)h+o(h)hJn) 6 (yay= limh>0 JU= I 5p, (828 8) Yn) 6 (W) dy@ (y)dyshowing that—OU 7 8wa (Y) = > (¥.28(¥) —yn) (37.2.30)Ynwhich is also expected.From the definnition, for y € R"\U = {(Y,yn) : yn < g(y)} it follows w;, =u; and onU,w, is given by 37.2.29 and 37.2.30. Consider ||w, illpow) for i <n. From 37.2.29Ou. OU x dg(y) |?||P = _ a, _Ilwillrowy I | Oy; (y, 2g (y) Yn) oy, (y,2¢ (y) Yn) Oy;Ou<2 | =< y fayS* §.26(9) —mn)dyP(y,28 (¥) —yn)Pp+2? Lip (g)? dy