1320 CHAPTER 38. SOBOLEV SPACES BASED ON L2
Now (∫∞
0
∣∣∣(uk (·,xn) ,φ j
)H−(
u(·,xn) ,φ j
)H
∣∣∣2 dxn
)1/2
=
(∫∞
0
∣∣∣(uk (·,xn)−u(·,xn) ,φ j
)H
∣∣∣2 dxn
)1/2
≤(∫
∞
0|uk (·,xn)−u(·,xn)|2H
∣∣∣φ j
∣∣∣2H
dxn
)1/2
=∣∣∣φ j
∣∣∣2H
(∫∞
0|uk (·,xn)−u(·,xn)|2H dxn
)1/2
=∣∣∣φ j
∣∣∣2H
(∫∞
0
∫Rn−1
∣∣uk(x′,xn
)−u(x′,xn
)∣∣2 dx′dxn
)1/2
which converges to zero. Therefore, there exists a set of measure zero, N j and a subse-quence, still denoted by k such that if xn /∈ N j, then(
uk (·,xn) ,φ j
)H→(
u(·,xn) ,φ j
)H.
Now by Theorem 38.5.4, γuk→ γu in H = L2(Rn−1
). It only remains to consider the term
of 38.5.20 which involves an integral.∣∣∣∣∫ xn
0
(uk,n (·, t) ,φ j
)H
dt−∫ xn
0
(u,n (·, t) ,φ j
)H
dt∣∣∣∣
≤∫ xn
0
∣∣∣(uk,n (·, t)−u,n (·, t) ,φ j
)H
∣∣∣dt
≤∫ xn
0
∣∣uk,n (·, t)−u,n (·, t)∣∣H
∣∣∣φ j
∣∣∣H
dt
≤(∫ xn
0
∣∣uk,n (·, t)−u,n (·, t)∣∣2H dt
)1/2(∫ xn
0
∣∣∣φ j
∣∣∣2H
dt)1/2
= x1/2n
∣∣∣φ j
∣∣∣H
(∫ xn
0
∫Rn−1
∣∣uk,n(x′, t)−u,n
(x′, t)∣∣2 dx′
)1/2
dt
and this converges to zero as k→∞. Therefore, using a diagonal sequence argument, thereexists a subsequence, still denoted by k and a set of measure zero, N ≡ ∪∞
j=1N j such thatfor x′ /∈ N, you can pass to the limit in 38.5.20 and obtain that for all φ j,(
γu,φ j
)H+∫ xn
0
(u,n (·, t) ,φ j
)H
dt =(
u(·,xn) ,φ j
)H.
By density of{
φ j
}, this equality holds for all φ ∈ L2
(Rn−1
). In particular, the equal-
ity holds for every φ ∈ Cc(Rn−1
). Since u is uniformly continuous on the compact set,
spt(φ)× [0,1] , there exists a sequence, (xn)k → 0 such that the above equality holds for