Một số vấn đề trong lý thuyết đa thế vị
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E
F
F
F
F
N
R
C
n
Rn
n
n
C
n
B
n
R
n
C
B2n
∂Bn
∂B2n
B(x, r)
n
R
n
C
x
x
(x, r)
B
∂B(x, r)
x
Vn
V
r
r
2n
r
R
n
C
n
σ
∅
∥x∥
A(Ω)
C(Ω)
Ck(Ω)
x
Ω
Ω
k−
k
C0k(Ω)
Ω
k−
k
Ω
C∞(Ω)
Ω
C0∞(Ω)
Ω
Ω
E0(Ω), E(Ω), F(Ω), N (Ω)
H(Ω)
USC(Ω)
L∞(Ω)
L∞ (Ω)
Lp(Ω)
Lp (Ω)
SH(Ω)
Ω
Ω
Ω
Ω
Ω
Ω
Ω
Ω
PSH(Ω)
PSH−(Ω)
MPSH(Ω)
O
X,z
u∗v
Ω
Ω
z ∈ X.
uv
19
“
“
20
R
n
m−
K⊂Ω
Ω
µ
K
µ
K
µ
µ
K
K
µ
u
u
Ω
B
u(z) = −(− log ∥z∥)1/2
2n
B \{0}
2n
l
α
v = −(− log(|f1|λ1 + ... + |fm|λm ))α
f1, ..., fm
0<α<
1
n
v∈E
f1, ..., fm
F
Ω
F
Ω
u ∈ F(Ω)
lim sup u(z) = 0,
F(Ω)
z→∂Ω
limz→∂Ω u(z) = 0
F
F
“
“
“
“
“
“
Ω
n
R (n ≥ 2)
C2
u : Ω −→ R
Ω
∑j
△u =
n
∂2u ≡0 Ω.
=1
∂xj2
H(Ω)
Ω
u : Ω −→ [−∞, +∞)
u
−∞
U
Ω
u≤φ
¯
φ ∈ H(U) ∩ C(U)
∂U =⇒ u ≤ φ
U
SH(Ω)
u, v ∈ SH(Ω)
max{u, v} ∈ SH(Ω)
αu + βv ∈ SH(Ω)
α, β ≥ 0.
u : Ω −→ [−∞, +∞)
−∞
u ∈ SH(Ω)
B (a, R) ⊂ Ω
∫
u(a) ≤
1
u(x)dσ(x);
σ(∂B(0, 1))Rn−1
∂B(a,R)
B
u
(a, R) ⊂ Ω
∫
1
u(a) ≤ V (B(0, 1))Rn
n
u(x)dVn(x).
B(a,R)
Ω
u ∈ L1loc(Ω)
u ∈ SH(Ω)
Ω
Ω
u ∈ SH(Ω) v ∈ SH(ω)
n
R ω
lim sup v(y) ≤ u(y),
x
y
→
max{u, v}
y ∈ ∂ω ∩ Ω
w=
ω,
u
Ω
\
Ω ω
u, v ∈ SH(Ω)
Ω
u=v
Ω
∆u
u ∈ SH(Ω)U
u
u(x) =
−1 (
) ∫
∂B(0, 1)
g(|x − w|)dν(w) + φ(x),
max{1, n − 2}σ
U
Ω
U
ν = ∆u|U φ ∈ H(U)
g(r) =
− log r
r2−n
g
: (0,+∞) → R
(n = 2)
(n > 2)
u, v ∈ L1(Rn)
∫
(u ∗ v)(x) = u(x − y)v(y)dVn(y).
R
n
.
u∗v
u=v
u ∈ L1 (Rn)
u∗v
χ:R→R
v ∈ L1(Rn)
e−1/t
χ(t) =
(t > 0)
0
(t
≤
0).
ρ(x) = Kχ(1 − ∥x∥2),
ρ : Rn → R
−1
χ(1 − ∥x∥2)dλ(x) ) .
∫
K=
(
B(0,1)
∞
ρ = B(0, 1)
∫n ρ(x)dVn(x) = 1.
n
ρ ∈ C (R )
R
ϵ>0
1
ρϵ(x) =
Ω
(x
ϵ
ϵn
R
).
n
u ∈ SH(Ω)
ϵ>0
(x, ∂Ω) > ϵ} = ∅,
Ωϵ := {x ∈ Ω :
u ∗ ρϵ ∈ C∞ ∩ SH(Ωϵ)
ϵ
ρ
u ∗ ρϵ
lim u ∗ρ (x) = u(x)
ϵ→0
ϵ
x∈Ω
Ω
u ∈ C2(Ω)
u ∈ SH(Ω)
∆u ≥ 0
∆u ≥ 0
u ∈ SH(Ω)
∫Ω u(x)∆φ(x)dVn(x) ≥ 0
φ ∈ C0∞(Ω)
≥
∆v
v ∈ Lloc1(Ω) ∗
u = lim(v ρ )
→
Ω
0 Ω
ϵ 0
Ω
v
X
F
ϵ
Xφ
F
δ>0
{E }
i i=1,2,...
δ
X = ∪i∞=1Eid(Ei) ≤
⊂ F
E
d(E)
d E
sup x
)=
(
x,y
y ;
E|−
|
∈
δ>0
0<δ≤∞
φ(E) ≤ δd(E) ≤ δ
E∈F
U⊂X
∪
∑
ψδ(U) = inf
{
i
φ(Ei) :
U⊂
}
Ei, d(Ei) ≤ δ, Ei ∈ F ,
i
ψ(U) = lim ψδ(U).
δ↓0
ψ
F
ψ
X
φ(U) = d(U)k
ψ
0≤k<∞
F={U⊂X}
00=1
k−
d(∅)k = 0
U ⊂ Rn
{
Hδk(U) = inf
∑
}
∪
i
d(Ei)k : U ⊂
i
Ei, d(Ei) ≤ δ .
k
Hk(U)
U
Hk(U) = lim Hδk(U).
δ↓0
k = 0 H0
•
•
k=m
m
R
•
1≤m
n
U
m−
m
U
Hn =
k=n
2n
Vn.
Vn(B(0, 1))
n
n
H (B(a, R)) = (2R)
a ∈ Rn0 < R < ∞
•
k>n
Hk(Rn) = 0 Hk
k
Rn
k
R
n
Hk
R
U ⊂ Rn a ∈ Rn
0<α<∞
Hk(U + a) = Hk(U)
U + a = {x + a : x ∈ U}
Hk(αU) = αkHk(U)
αU = {αx : x ∈ U}.
Hk
0≤k
Hl(U) > 0
U ⊂ Rn
Hl(U) = 0
Hk(U) = ∞
U ⊂ Rn
U
dim U = sup{k : Hk(U) > 0}
=
sup{k : Hk(U) = ∞}
=
inf{k : Hk(U) < ∞}
=
inf{k : Hk(U) = 0}.
n
dim U
∞
k < dim U
Hk(U) = ∞
k > dim U
Hk(U) = 0
•
dim Rn = n
• dim C = log 2/ log 3
C
0 < Hk(U) < ∞ Hk(U) = ∞
Hk(U) = 0
k = dim U
Hk(U)
k
U Rn
Hk
U
φ(U) = d(U)k
h : [0, +∞) → [0, +∞)
0
h
U
Λh(
)=
lim inf
δ↓0
{
h
0
∑
i
U
U ⊂ Rn
∪
h dE
((
R
U
i))
⊂
:
i
E ,dE
( i) ≤
i
δ.
}
n
k
C0>0
C0−1Rk ≤ Hk(U ∩ B(x, R)) ≤ C0Rk, x
∈ U 0 < R < d(U)
0≤k<∞
k
η x ∈ Rn
∗k
Θ (
η, x
)=
lim sup
r↓0
η
R
η B(x, r)
(
(2r)k
k
)
η,
lim inf η B(x, r) .
x )=
)
(
Θ∗(
r↓0
(2r)k
,
n
Θk(η, x) = Θ∗k(η, x) = Θk∗(η, x)
k−
η x
k−
η
n
η
R U ⊂R
Θ∗k(η, x) ≤ α
x∈U
η(U) ≤ 2kαHk(U).
Θ∗k(η, x) ≥ α
x∈U
η(U) ≥ αHk(U).
n
Hk
0<α<∞
{
H=
h : (0, +∞) → (0, +∞) : ∃ M > 0, c > 4
cϵ
∫0
h(r)
h(ϵ)
dr ≤ M
rn−1
}.
,ϵ
ϵn−2
rk
h1, h2 ∈ H
k>n−2
a
F
h1 + h2 ∈ H
: (0, +∞) → (0, +∞)
1
c> 4
∃
lim sup
+
ϵ→0
h
h(r)/rn−1
(∗)
cϵ
a · h1 ∈ H
∫cϵ
0
rk| log r|
h
F (r)dr
<∞ .
F(ϵ)
H
∗
F (r) =
µ
f
X
∫
∫
+∞
µ({x ∈ X : f(x) ≥ t})dt.
f dµ =
X
0
Ω
Kh ∈ H
A,B>0
u
Ω x0
µ
n
R (n ≥ 2) K
ϵ0>0
(
)
ϵ<ϵ0
µ K ∩ B(x0, ϵ) ≥ Ah(ϵ)
(
)
x∈K
µ K ∩ B(x, ϵ) ≤ Bh(ϵ)
1
lim
ϵ→0µK
ϵ<ϵ0
u(x)dµ(x) = u(x
∩ B(x0 , ϵ)
∫
)
(
).
0
ϵ1 < ϵ0
x0 = 0
K∩B(x0,ϵ)
c
cϵ
∫0
h(r)
h(ϵ)
rn−1
ϵn−2
γ>1p>1
ϵ≤ϵ1
,
dr ≤ M
c = 2γ(1 + p)
Kϵ = K ∩ B(0, ϵ).
1
lim
ϵ→0
u(0) = −∞
lim sup
ϵ→0
µ(Kϵ)
u(x)dµ(x) = u(0).
∫
Kϵ
1
µ(Kϵ)
u(x)dµ(x)
∫
lim sup sup u(y)
≤
ϵ→0
y∈Kϵ
Kϵ
1
lim
ϵ→0
µ(Kϵ)
u(x)dµ(x) = u(0).
∫
Kϵ
u(0) =
≤
.
−∞
u(0) > −∞
∫
B(0, 1) b Ω
u(x) = max 1, n
B(0, 1)
−1
g( x − w
− 2 } σ ∂B(0, 1)
(
{
)
|
B(0,1)
)dν(w) + φ(x),
|
(
g ν = ∆u|
B(0,1)
lim
ϵ→0
φ
u x dµ x
∫
1
( )
µ(Kϵ)
(
)=
∫
Kϵ
lim
ϵ→0
−1
− 2 σ ∂B(0, 1)
∫
(
}
max 1, n
{
1
fϵ(w) = µ(Kϵ)
x
0
u(0) = max 1, n
f (w)dν(w) + φ(0),
ϵ
)
B(0,1)
g(|x − w|)dµ(x).
Kϵ
∫
−1
g( w
2 σ ∂B(0, 1)
{
)dν(w) + φ(0).
||
(
−}
)
B(0,1)
lim
ϵ→0
∫ g( w
f (w)dν(w) =
∫
)dν(w).
| |
ϵ
B(0,1)
B(0,1)
fϵ(w) → g(|w|), ϵ → 0,
ν
B(0, 1)
ν
∫
g
u(0) ≤
{
≤
(0, 1)
g(w )dν(z) + φ(0)
−1
max 1, n
2 σ ∂B(0, 1)
−} (
)
−1
(
|
|
B(0,δ)
g(δ)ν (0, δ) + φ(0),
)
max{1, n − 2}σ ∂B(0, 1)
(B
)
0<δ<1
ν
)
φ ∈ H B(0, 1)
(B(0, δ)) ≤ max{1, n − 2}σ(∂B(0, 1))
−u(0) + φ(0)
, g(δ)
δ ↘ 0 0<δ<1
δ→0
ν(B(0, δ)) ↘ 0
ν({0}) = 0
fϵ(w) → g(|w|),
w ∈ B(0, 1) {0}
C1,C2 > 0
fϵ(w) ≤ C1g(|w|) + C2,
w ∈ B(0, 1) {0}
ϵ<ϵ1
|w| > pϵ
|w| ≤ pϵ
|w| > pϵ |w|
x
w
| −
w
x
w
ϵ>
w
|≥| |−| |≥| |−
f
p−1
| |− p
1
x ∈ Kϵ
w .
p | |
g
ϵ
(0, 1)
f (w) ≤
=
∫
g(|w| − |x|)dµ(x)
µ(Kϵ)
1 ∫ (p − 1 )
K
µ(K ϵ) g
p |w| dµ(x) ϵ
ϵ
≤
Kϵ
g
=
(
(p − 1
p 1
)
g
p−1
p |
w
)
p |w| .
(p
=
|
−
gw +g
)
(| |)2
n
−
(p−1 )
||
(n = 2)
.
p
g( w ) (n > 2)
|w | ≤ pϵ
fϵ
fϵ(w) =
=
α(ϵ)
0
∫
)
: g(|x − w|) ≥ t} dt
µ {x ∈ Kϵ
µ(Kϵ)
+∞
1 ∫
µ(Kϵ)
(
(
+∞
1
(
(
Kϵ ∩
µ
(
B
))
w, g−1(t) dt.
0
1 ∫
µ(Kϵ)
α(ϵ) = g(|w| + ϵ)
0
fϵ(w) =
α(ϵ)
µ Kϵ∩B
)
)
w, g (t) dt +
−1
1
α(ϵ)
∫
+∞ (
µ
(
Kϵ ∩B w, g−1(t)
)
)
dt.
µ(Kϵ)
(
)
−1(t)
B(0, ϵ) ⊂ B w, g
(
))
µ(Kϵ ∩ B w, g (t) = µ(Kϵ),
−1
0
≤ t ≤ α(ϵ).
(
)
−1
K ∩ B w, γg (t) =
(
) )
−1
µ(Kϵ ∩ B w, g (t) =0.
−1
∅
−1
()()() Kϵ ∩ B w, g (t) ⊂ K ∩ B w, γg (t) ⊂ K ∩ B w0,
2γg−1(t) ,
(
)
w0 ∈ K ∩B w, γg (t)
t ≥ α(ϵ)
−1
ϵ<ϵ1,
2γg−1(t) ≤ 2γg−1(α(ϵ)) = 2γ(|w| + ϵ) ≤ 2γ(1 + p)ϵ < cϵ1 < ϵ0,
(
−1
µ Kϵ ∩ B w, g (t)
(
) ≤ Bh 2γg−1(t) ,
)
(
t ≥ α(ϵ).
)
fϵ
α(ϵ)
ϵ<ϵ1
+∞
1
B
fϵ(w) ≤ µ(Kϵ) ∫ µ(Kϵ)dt + µ(Kϵ) ∫ h(2γg−1(t))dt
0
α(ϵ)
2γ(|w|+ϵ)
= α(ϵ) + (2γ)
n−2
max{1, n − 2}B
∫0
µ(Kϵ)
||
(
≤
g(|w|) +
(2γ)
ϵ
2γ(p+1)ϵ
∫
Ah(ϵ)
)
0
n−2
max{1, n − 2}BM , Aϵn−2
h
ϵ<ϵ1
f (w) ≤
rn−1
+ (2γ)n−2 max{1, n − 2}B
≤ g w +ϵ
h(r) dr
r := 2γg−1(t)
g( w ) + (2γp)n−2 max{1, n − 2}BM .
||
A|w|n−2
h(r) dr
rn−1
