泛函作业（1）

(k + l)f = kf + lf.
Ïds is a linear space
3
3. f (1)W k,p
ê˜mµW k,p
{ f : D α f ∈ LP , 0
|α |
k}
is a linear space. f Dα f
0 |α | k P,
(2)½Ânorm:
W k,p
f
W k,p ∗ |α|=k
fn − fm
C (R)
m, n > N ž§
+ sup
x=y
|(fn (x) − fm (x)) − (fn (y ) − fm (y ))| < ε. |x − y |α |(fn (x) − fm (x)) − (fn (y ) − fm (y ))| < ε. (1) |x − y |α 1
Ké?¿
?˜…Ü " ε > 0, ∃N § n, m > N žkµ
fn − fm
W k,p
<ε
.
=
Dα (fn − fm )
0 |α | k P
< ε.
Ïd
4
Dα (fn − fm )
|α|=αi k
P
<ε
αi = αž§ KŒ ( ((
kn ˆ l
1 ∂fn |α| ∂fm |α| p ) −( ) ) dx) p → 0 ∂xi ∂xi
x=y
f : α − Lipschitz continuous. ∧α (Rn ) f
∧α
f
C (R)
|f (x) − f (y )| |x − y |α
(L )
y²µ∧α is a Banach space. y: £1¤ c · f
(2)triangle inequality: f1 + f2
∧α ∧α
= |c| · f
∧α .
= f ≤ f1 ≤ f1 ≤ f1
C
+ sup
x=y
|(f1 + f2 )(x) − (f1 + f2 )(y )| |x − y |α
C
C
+ f2 + f2 + f2
+ sup(
x= y
|(f1 (x) − (f1 (y )| |(f2 (x) − (f2 (y )| + ) |x − y |α |x − y |α
x= y
|(f (x) − fn (x) − f (y ) + fn (y )| |fn (x) − fn (y )| + sup . |x − y |α |x − y |α x=y |(fn (x) − fn (y )| < ∞. |x − y |α
dufn ∈ ∧( Rn ),
sup
x= y
d£1¤
sup
x=y
|f (x) − fn (x) − f (y ) + fn (y )| < ε. |x − y |α |(f (x) − f (y )| < ∞. |x − y |α
¤±
sup
x=y
⇒ f ∈ ∧α (Rn ). ∴ ∧α is a Banach space.
n , if 2. ∀α, β ∈ Z+
Dα f
P.
y²µW k,p is Banach space (Soblev Space). y: 1)k y f W k,p Ú f W k,p ∗ Ñ´‰ê" i)šK5´w, § … f W k,p = 0ž§ f = 0 a.e.
ii) af
W k,p
=
0 |α | k
Dα af
P
=a
0 |α| k
(fn − f )
W k,p
→ 0, f ∈ W k,p .
Dα (fn − fm )
0 |α | k P
< ε.
¥
m → ∞.
Kk
Dα (fn − f )
0 |α | k P
< ε.
=
fn − f
W k,p
→ 0
d
Dα (fn − f )
0 |α | k P
<ε
5
•µ
Dα f
P
= Dα (f − fn + fn )
x∈ k n x∈ k n x∈ k n
Ïdαf ∈ s é?¿ f, g, h ∈ S, k, c ∈ R:
(f + g ) + h = f + (g + h)
1∗f =f
0 + f = f, f − f = 0
f +g =g+f
k (f + g ) = kf + kg
kl(f ) = k (lf )
sup |fn (x) − fm (x)| + sup
x∈Rn x= y
Kk
sup |fn (x) − fm (x)| < ε
x∈Rn
={fn (x)}´˜—Âñ . f(x)´ÙÂñ¼ê§ =
n→∞
lim fn (x) = f (x).
eyµf ∈ ∧α (Rn ), … fn − f 3£1¤¥, -m → ∞, Œ
∧α (Rn )
→ 0.
x∈Rn
sup |fn (x) − f (x)| + sup
x= y
|(fn (x) − f (x)) − (fn (y ) − f (y ))| < ε. |x − y |α
= fn (x) − f (x)
sup
x= y
∧α (Rn )
→ 0.
|(f (x) − fn (x) − f (y ) + fn (y ) + fn (x) − fn (y )| |(f (x) − f (y )| = sup α |x − y | |x − y |α x= y ≤ sup
P
≤ Dα (f −
=
Dα (f − fn )
P
→ ∞,
Dα fn
P
´k.þ"
Dα f f ∈ W k,p
P
< ∞,
6
Dα f
P
=a f
W k,p
iii) f +g
W k,p
=
0 |α | k
Dα f + g ( Dα f
0 |α | k P
P
≤ =
0 |α | k
+ Dα g
P)
Dα f
P
+
0 |α | k
Dα g
P
ÓnŒy W k,p ´‰ê§ 2)y²W k,p "
f fn ´W k,p ¥
f
W k,p ∗ ´‰ê"
泛函作业1泛函一作业泛函作业1作业一泛函作业泛函分析密度泛函泛函极值什么是泛函
•¼©ÛŠ’ £2¤
1. X = Rn , f ∈ C (X ) (1
α > 0), if sup
x=y
|f (x) − f (y )| <∞ |x − y |α {f ∈ C (Rn ), (L ) holds.} + sup
1) sup |xα Dβ (f + g )| = sup |xα Dβ f (x) + xα Dβ g (x)|
x∈kn x∈kn
≤ sup |xα Dβ f (x)| + sup |xα Dβ g (x)| < ∞, ∴ f + g ∈ s
x∈ k n x∈ k n
2) sup |xα Dβ αf (x)| = sup |αxα Dβ f (x)| = |α| sup |xα Dβ f (x)|
2
∞
d(f, g )
k=0
(f − g ) 1 ρ∗ . · k 1 + ρ∗ 2k k
(S , d) − complete metric space.y²µ S is a linear space?
yµ ?¿ f, g ∈ S ,Œ•f = supx∈kn |xα Dβ f (x)| < ∞, overlineg = sup |xα Dβ g (x)| < ∞
C
C
+ sup
x= y
|(f1 (x) − (f1 (y )| |(f2 (x) − (f2 (y )| + sup |x − y |α |x − y |α x=y
∧α
∧α
d£1¤ £2¤linearspace + · ∧α ⇒ (∧α , · ) is norm space. ey µ {fn (x)}´∧α (Rn )¥ …Ü § =é∀ε > 0, ∃N, k fn − fm ∧α (Rn ) < ε. =
sup |xα | · Dβ f < ∞, if is called a Schwartz f unction
x∈Rn
(S )
S (R n )
{f : f satisf ies(S )}§ Pραβ = |xα | · Dβ f ,
∗ ραβ can be arranged in an order ρ∗ 1 , ρ2 ... ½Âµ
Kkµ
∂fn ∂xi )|α| − ( ∂fm |α| ) →0 ∂xi a.e.
⇒
(
∂fn |α| ∂fm |α| ) −( ) → 0 ∂xi ∂xi ∂ |fn − fm | → 0 ∂xi ∂ (fn − fm ) → 0 ∂xi n → ∞.
⇒
⇒ il1 nž§ Œ
§ fn , fm 'uz‡xi êªu0. fn − fm → 0"ùÒ`²fn ´˜—ëY § K•3f ¦ µ lim fn = f , ±ey²µ