Helms L.L. Potential Theory
Подождите немного. Документ загружается.
Index 437
Newtonian potential, 127, 316
nonnegative definite, 343
nontangential limit, 98
norm, 5, 6
normed linear space, 6
nucleus, 175
oblique derivative problem, 333
outer capacity, 165
outer unit normal, 8
partition of unity, 132
paved space, 175
paving, 175
peaks, 229
Perron lower solution, 77
Perron subsolution, 357, 420
Perron supersolution, 357, 420
Perron upper solution, 77
Perron-Wiener method, 75
Picard, 20
Poincar´e, 85, 115, 360
point at infinity, 197
Poisson Integral, 25, 34, 201
Poisson Integral for half-space, 33
PoissonIntegralFormula,17
Poisson’s equation, 316
polar cap, 89
polar representation, 365
polar set, 88, 150, 206
pole, 10
positive definite, 343, 407
potential, 122
projection, 258
Proof of Theorem 2.6.28, 232
Property B, 166
PWB method, unbounded regions, 204
q.e., 150
quasi everywhere, 150
reciprocity law, 135
reduction, 116, 159
reflection, 38
reflection principle, 326
regular, 420
regular Borel measure, 5
regular boundary point, 81, 211
regular region, 81
regular,(L,M)-, 420
regularized reduction, 159
resolutive, 207
resolutive boundary function, 78
resolutive,(L,M)-, 420
Riesz Decomposition Theorem, 142
Riesz measure, 142
Riesz potential, 304
Riesz Representation Theorem, 142
right-directed, 56
saturated, 75, 205
Schauder interior estimates, 338
Schwarz, 26, 155, 213
Schwarz’s Reflection Principle, 326
seminorm, 6
Sibony, 186
Sierpinski, 177
signed measure, 5
spherical chip, 288
spherical chip, admissible, 396
spherical coordinates, 1
spherical harmonic, 52
spine, Lebesgue, 128
Stolz domain, 98
straighten boundary, 405
stratifiable, 296
strictly elliptic, 338
strong barrier, 82
strong convergence, 253
strong maximum principle, 346, 409
strongly subadditive, 170
strongly subadditive, countably, 170
subfunction, 354, 411
subharmonic, 60, 198
subnewtonian kernel, 304
subsequence selection principle, 269
subsolution, 356
super-mean-valued, 60
superfunction, 354, 411
superharmonic, 60, 62, 63, 198
superharmonic, alternative definition, 62
superharmonic, yet another definition,
63
supersolution, 356, 414
Suslin scheme, 175
sweeping, 189
sweeping out, 115
swept measure, 189
symmetric derivate, 89
thin, 225, 237
Tonelli, 23
total, 135
total variation, 5
u.s.c., 3
uniform equicontinuity, 4
uniformly H¨older continuous, 309
438 Index
upper class, 76
upper inner normal derivate, 345
upper semicontinuous, 3
vague convergence, 253
Vasilesco, 155
vertex, 259
Vitali covering, 92
w*-topology, 5
weak convergence, 253
weak maximum principle, 344
wedge conditions, 424
Wiener, 81, 164
Wiener’s test, 261
Zaremba, 86
Zaremba’s cone condition, 86, 265, 367
zero-one law, 194
Notation
A(f : x, δ), 60
A(u : y, ρ), 14
A
0
, 313
A
f
, 224
A(x), 408
A(F), 175
B
x,ρ
,1
H
(b)
2+α
(Ω; φ), 272
B
x
, 112
B(X), 5
C, 294
C(n, a, b), 312
C(Γ ), 164
C
0
(Ω), 3
C
0
(Ω
−
), 3
C
k
(Ω), 3
C
k
(Ω
−
), 3
C
k
0
(Ω), 3
C
k
(Ω ∪ Σ), 305
C
z,α
,88
C
∗
, 165
C
∗
, 165
C, 166
C
a
, 405
cl
f
A, 224
D
i
,2
D
ij
,2
D
x
i
,3
D
x
i
x
j
,3
D
β
(x)
,3
D
β
,3
D
j
u, 280
D
+
ν
u, 345
D(A), 6
D
s
μ,89
Dμ,92
d, 272
d(Ω), 272
d(x, y), 271
d(F ), 92
d(F
0
), 100
d
x
, 272
*
d(x), 284
*
d
x
, 284
*
d
xy
, 284
*
d(x, y), 284
div v,8
E, 250
E
+
, 242
F
−
,1
∂F,1
cl F ,1
int F ,1
∼ F ,1
F(Γ ), 259
f
∗
,33
f-lim, 224
f-limsup, 224
G
Λ
Ω
, 189
G
Ω
f, 129
G
1/2
, 376
G
Ω
μ, 122
G
Λ
Ω
(x, ·), 189
G, 111
G
Ω
, 110
G
B
,16
G
1/2
, 372
ghm
Ω
h, 116
H
f
,78
H
f
,77
H
f
,77
H
f,g,h
, 420
439
440 Notation
H
−
f,g,h
, 420
H
+
f,g,h
, 420
H
k+α
(Ω), 268
H
(b)
k+α
(Ω), 285
H
k+α,loc
(Ω), 268
H
f,g
, 357
H
+
f,g
, 357
H
−
f,g
, 357
H
∗(b)
k+α
(Ω), 427
h, 372
int
f
A, 224
J
h
,71
K, 305
K
+
, 135
K
B
(x, z), 46
K
z,α
,89
K
+
0
, 135
K
+
(Ω), 135
K(Ω), 164
k(x, y), 303
k(| x − y |), 312
L, 333
L
0
, 335
L(f : x, δ), 60
LS(x, B, g), 354
LS(x, U, φ), 410
L(u : y,ρ), 14
L
x
, 100
L(Ω), 60
L
f
,76
L
Ω
f
,77
L(f,g), 356, 411
L(f,g,h), 414
M, 333
M
0
, 372
M, 338
m, 338
m, M , 336
N
∞
, 175
N (x), 407
N (F, ), 92
n, 175
n(x), 8
O(Ω), 164
PI(c, μ, Ω), 35
PI(f : B), 25
PI(c, f, Ω), 35
PI(f,∼ B
y,ρ
), 201
PI(μ : B), 25
PI(μ : y, ρ), 25
P
m,n
,51
P (d(Ω)), 277
R
, 164
R
Γ
, 164
ˆ
R
Γ
, 164
ˆ
R
, 164
ˆ
R
u
Λ
, 159
R
u
Λ
, 159
R(A), 6
R
n
,1
R
n
∞
, 197
R
n
0
, 313
R
n
±
, 313
ˆ
R
Λ
(u, ·), 189
(r, θ), 1
S(Ω), 60
S
+
(Ω), 60
S
x,ρ
,1
S
z,θ
,98
Uf, 316
U
μ
, 23, 127
U
f
,76
U
Ω
f
,77
U(f, g), 356, 411
U(f, g, h), 414
ˆu,4
u
∞
, 115
u
∞,Λ
, 115
u
B
,u
B
, 70, 355
u
U
, 413
u
∼B
, 206
u(| x − y |),u
y
(x), 10
u
y
(x), 10
V
x,u
, 411
v
U
, 413
w
+
j
, 358
w
−
j
, 358
w
±
j
, 358
x,1
x · y,1
(x
,x
n
), 204
x
, 204
x
∗
, 14, 32
Y
n,m
,52
y
r
,38
β
t
, 373
β!, 2
β
n
, 373
γ
j
, 309
χ
C
, 101
δ
Ω
(x, ·), 189
δ
Λ
Ω
(·,u), 193
δ
Λ
Ω
(μ, ·), 193
δ
Λ
Ω
(x, ·), 189
Notation 441
ε,1
j
, 273
Λ
r
,38
λ
0
, 392
μ, μ
+
,μ
−
, | μ |,5
μ
n
,5
μ
F
, 257
μ
Γ
, 164
μ
F(Γ )
, 259
μ
Ω
x
, 217
μ
x
, 100
ν
n
,2
ν
n
(ρ), 2
ν, 392
Ω,3
Ω
δ
, 272, 309
*
Ω
δ
, 284
Ω
0
, 326
Ω
±
, 326
∂
∞
Ω, 198
Φ
u
Λ
, 158
ψ, 330
σ
n
,2
σ
n
(ρ), 2
| f |
∗(b)
k+α,Ω
, 427
| g |
∗(b)
k+α,Σ
, 427
f
0,Ω
,3
f
k
, 268
| g |
1+α,Σ
, 271
| g; d |
1+α,Σ
, 380
| g |
(1+b)
1+α,Σ
, 287
|u|
k+0,Ω
, 268
[u]
k+α,Ω
, 268
|u|
k+α,Ω
, 268
[u]
k,Ω
, 268
|u|
k,Ω
, 268
|u; φ|
k+α,Ω
, 272
[u; φ]
(b)
k+0,Ω
, 271
|u; φ|
(b)
k+0,Ω
, 271
[u; φ]
(b)
k+α,Ω
, 271
[u; φ]
(b)
k,Ω
, 271
|u; φ|
(b)
k,Ω
, 271
|u; φ|
(b)
k+α,Ω
, 271
[u]
(b)
k+α,Ω
, 285
|u|
(b)
k+α,Ω
, 285
|x|,1
|β|,2
|λ|
2
e
, 250
| μ |
e
, 250
|μ|
2
e
, 241
·
X
,6
Δu,8
Δ
(x)
,8
∧, 284
∂
∞
Ω, 198
∞, 197
[λ
1
,λ
2
]
e
, 250
[μ, ν]
e
, 241
∇,3
θ − lim
x→z
u(x), 98