Document

L
|L(u)−L(v)|=|L(u−v)| ≤ Cku−vkV.
{ϕk}V ϕ ∈V
k→ ∞
|L(ϕk)−L(ϕ)|=|L(ϕk−ϕ)| ≤ Ckϕk−ϕkV→0k→ ∞.
L L
vkV|L(vk)|/kvkkV≥k
wk=vk/(kkvkkV)∈V.
|L(wk)| ≥ 1
kwkkV= 1/k ⇒wk→0,
L L(wk)→0
L V
kLkV0:= sup
06=v∈V
|L(v)|
kvkV
.
k·kV0
V V
V0
f: [a, b]→R
f0f∈L1
loc(a, b)f0
Tf0
hTf0, ϕi=Zb
a
f0(x)ϕ(x)dx, ∀ϕ∈ D(a, b),
= [fϕ]b
a−Zb
a
f(x)ϕ0(x)dx
=−Zb
a
f(x)ϕ0(x)dx =−hTf, ϕ0i.
T(n)n T
hT(n), ϕi:= (−1)nhT, ϕ(n)i.
Ω⊂RNTα
¿∂|α|T
∂xα, ϕÀ= (−1)|α|¿T, ∂|α|ϕ ∂xα,À∀ϕ∈ D(Ω).

f: [a, b]→R
hT0
f, ϕi=−Zb
a
fϕ0dx =hTf0, ϕi,∀ϕ∈ D(Ω) ⇒T0
f=Tf0.
f∈L1
loc(Ω)
Tf
g∈L1
loc(Ω)
¿∂|α|Tf
∂xα, ϕÀ=ZΩ
g(x)ϕ(x)dx∀ϕ∈ D(Ω).
α f
α f g
g=Dαf
Dαf
N= 1 Ω = (−1,1) f(x) = 1 − |x|
D1f
D1f
g(x) := ½1x < 0,
−1x > 0,
hT0
f, ϕi=−Z1
−1
f(x)ϕ0(x)dx, ∀ϕ∈ D(−1,1)
[−1,0] [0,1]
−Z1
−1
f(x)ϕ0(x)dx =−Z0
−1
f(x)ϕ0(x)dx −Z1
0
f(x)ϕ0(x)dx,
=−fϕ ¯¯0
−1+Z0
−1
1×ϕ(x)dx −fϕ ¯¯1
0+Z1
0
(−1) ×ϕ(x)dx,
=Z1
−1
g(x)ϕ(x)dx −f(0−)ϕ(0−) + f(0+)ϕ(0+),
=Z1
−1
g(x)ϕ(x)dx =hTg, ϕi
hT00
f, ϕi= (−1)2Z1
−1
f(x)ϕ00(x)dx =−Z0
−1
ϕ0(x)dx + [fϕ0]0
−1+Z1
0
ϕ0(x)dx + [fϕ]1
0,
=f(0−)ϕ0(0−)−f(0+)ϕ0(0+) + ϕ(1) −ϕ(0) −[ϕ(0) −ϕ(−1)],
=−2ϕ(0) = h−2δ0, ϕi,

T00
f−2δ0
f x 6= 0
f
Lp
k f ∈L1
loc(Ω)
Dαf|α| ≤ k
kfkWk
p(Ω) := 
X
|α|≤kkDαfkp
Lp(Ω)
1/p
,
1≤p < ∞p=∞
kfkWk
∞(Ω) := max
|α|≤kkDαfkL∞(Ω).
Wk
p(Ω)
Wk
p(Ω) := nf∈Lp(Ω) : kfkWk
p(Ω) <∞.o
•Wk
2(Ω) = {v∈L2(Ω) : Dαv∈L2(Ω) ∀|α| ≤ k.}
Wk
2(Ω) Hk(Ω)
•H1(Ω) = nv∈L2(Ω) : ∂v
∂xi∈L2(Ω), i = 1,2,3, . . . , No
∂v/∂xi
•H1(Ω) ⊃H1
0(Ω) := {v∈H1(Ω) : v= 0 ∂Ω}
(·,·)1,ΩH1(Ω)
(u, v)1,Ω:= ZΩÃuv +
N
X
i=1
∂u
∂xi
∂v
∂xi!dx,
kuk1,Ω= ((u, u)1,Ω)1/2=ÃZΩÃu2+
N
X
i=1 µ∂u
∂xi¶2!dx!1/2
.

f∈H1(Ω) f
L2(Ω) H1(Ω) ⊂L2(Ω)
Hm+1(Ω) ⊂Hm(Ω) ⊂. . . H1(Ω) ⊂L2(Ω).Ω
f∈L2(Ω) AΩ
ZA|f|dx≤µZA
f2dx¶1/2µZA 1dx¶1/2 = (volA)1/2kfk0,A <∞.
L2(Ω) ⊂L1
loc(Ω).
Ω = SN
i=1 Ωi
ΩiΩiTΩj=∅i6=j
uh∈Uh
n Uh⊂H1(Ω)
uh
Ωiuh∈H1(Ω)
1uh
uh
ZΩ¡uh¢2dx=X
iZΩi³uh¯¯Ωi´2dx<∞.
uh
gj∈L2(Ω)
ZΩ
gjϕ dx=−ZΩ uh∂ϕ
∂xj
dx,∀ϕ∈ D(Ω).
gj
gj|Ωi=∂uh
∂xj¯¯¯¯Ωi
gj∈L2(Ω)
ZΩ
g2
jdx=X
iZΩiÃ∂uh
∂xj¯¯¯¯Ωi!2
dx<∞.
ZΩi
gj|Ωiϕ dx=ZΩi
∂uh
∂xj¯¯¯¯Ωi
ϕ dx=Z∂Ωi uh¯¯Ωiϕnij ds −ZΩi uh¯¯Ωi
∂ϕ
∂xj
dx,

nij jni∂Ωi
Ωi
Ωi
ZΩ
gjϕ dx=
N
X
i=1 Z∂Ωi
uh¯¯Ωiϕnij ds −ZΩ uh∂ϕ
∂xj
dx.
Γik Ωi,Ωk
ZΓik
uh¯¯Ωiϕnij +uh¯¯Ωkϕnkj ds =ZΓik uh¯¯Ωiϕ(nij +nkj )ds = 0,
uhΓik uh
nij =−nkj Γik
Z∂Ωi
uh¯¯Ωiϕnij ds,
∂Ωi⊂∂Ωϕ= 0 ∂Ω
Ω = (a, b)⊂RH1(Ω)
H1(Ω) = ©u:u∈C0(Ω), D1(u)∈L2(Ω)ª.
Wk
p(Ω)
{vj} k·kWk
p(Ω)
Lp∀|α|< k {Dαvj}
k · kLp(Ω) ∃vα∈Lp(Ω)
Dαvj→vαj→ ∞
Dαv∀|α|< k Dαv=vα
wj→w Lp(Ω) ∀ϕ∈ D(Ω)
ZΩ
wj(x)ϕ(x)dx→ZΩ w(x)ϕ(x)dx.
1/p + 1/q = 1
kwjϕ−wϕkL1(Ω) ≤ kwj−wkLp(Ω)kϕkLq(Ω) →0j→ ∞
Dαv=vα
ZΩ
vαϕ(x)dx= (−1)|α|ZΩ v∂|α|ϕ
∂xα,∀ϕ∈ D(Ω).
Dαvj
ZΩ
vαϕ dx= lim
j→∞ ZΩ
Dαvjϕ dx,
= (−1)|α|lim
j→∞ ZΩ
vj
∂|α|ϕ
∂xαdx= (−1)|α|ZΩ
v∂|α|ϕ
∂xαdx.
Wk
2(Ω) = Hk(Ω)
H1
0(Ω)