Jost J. Partial Differential Equations
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References
1. L. Bers, M. Schechter, Elliptic equations, in: L. Bers, F. John, M. Schechter:
Partial Differential Equations, pp. 131–299, Interscience, New York, 1964
2. D. Braess, Finite Elemente, Springer, 1997
3. I. Chavel, Eigenvalues in Riemannian Geometry, Academic Press, 1984
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reprinted 1968, Springer.
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New York (the German and English versions do not coincide, but both are
highly recommended)
5. L.C. Evans, Partial Differential Equations, Graduate Studies in Math. 19, AMS,
1998
6. A. Friedman, Partial Differential Equations of Parabolic Type, Prentice Hall,
1964
7. M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear
Elliptic Systems, Princeton Univ. Press, 1983
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Birkh¨auser, 1993.
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order, Springer, 1983.
10. F. John, Partial Differential Equations, Springer, 1982
11. J. Jost, Nonpositive Curvature: Geometric and Analytic Aspects,Birkh¨auser,
Basel, 1997
12. J. Jost, Postmodern Analysis, Springer,
3
2005
13. J. Jost, Dynamical Systems, Springer, 2005
14. J. Jost, X. Li-Jost, Calculus of Variations, Cambridge Univ. Press, 1998
15. A. Kolmogoroff, I. Petrovsky, N. Piscounoff,
´
Etude de l’ ´equation de la diffusion
avec croissance de la quantit´edelamati`ere et son application `a un probl`eme
biologique, Moscow Univ.Bull.Math.1, 1937, 1-25
16. O.A. Ladyzhenskya, V.A. Solonnikov, N.N. Ural’tseva, Linear and Quasilinear
Equations of Parabolic Type, Amer.Math.Soc., 1968
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tions, Nauka, Moskow, 1964 (in Russian); English translation: Academic Press,
New York, 1968, 2nd Russian edition 1973
348 References
18. J. Moser, On Harnack’s theorem for elliptic differential equations, Comm. Pure
Appl. Math. 14 (1961), 577–591
19. J. Murray, Mathematical Biology, Springer, 1989
20. G. Strang, G. Fix, An Analysis of the Finite Element Method, Prentice Hall,
Englewood Cliffs, N.J., 1973
21. J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer, 1983
22. M. Taylor, Partial Differential Equations, Vols. I–III, Springer, 1996
23. K. Yosida, Functional Analysis, Springer, 1978
24. E. Zeidler, Nonlinear Functional Analysis and its Applications, Vols. I-IV,
Springer, 1984
Index of Notation
Ω always is an open subset of R
d
, usually bounded as well.
Ω
⊂⊂ Ω :⇔ The closure
¯
Ω
is compact and contained in Ω.
For ϕ : Ω → R, the support of ϕ (supp ϕ) is defined as the closure of
{x ∈ Ω : ϕ(x) =0}.1
PDE 1
u
x
i
:=
∂u
∂x
i
for i =1,...,d 1
x =(x
1
,...,x
d
)1
Δu :=
d
i=1
u
x
i
x
i
=0 1
R
+
:= {t ∈ R : t>0} 2
∇u 7
B(x, r):=
y ∈ R
d
: |x −y|≤r
8
˚
B(x, r):=
y ∈ R
d
: |x −y| <r
8
Γ (x, y):=Γ (|x − y|):=
1
2π
log |x − y| for d =2
1
d(2−d)ω
d
|x − y|
2−d
for d>2
8
ω
d
8
∂
∂ν
x
9
ν 10
u(x
0
)=S(u, x
0
,r):=
1
dω
d
r
d−1
∂B(x
0
,r)
u(x)do(x)16
u(x
0
)=K(u, x
0
,r):=
1
ω
d
r
d
B(x
0
,r)
u(x)dx 16
(t):=
c
d
exp
1
t
2
−1
if 0 ≤ t<1,
0 otherwise,
17
T
+
(v):=
y ∈ Ω : ∃p ∈ R
d
∀x ∈ Ω : v(x) ≤ v(y)+p ·(x −y)
38
τ
v
(y):=
p ∈ R
d
: ∀x ∈ Ω : v(x) ≤ v(y)+p · (x − y)
38
L
d
39
diam(Ω)44
350 Index of Notation
R
d
h
51
¯
Ω
h
:= Ω ∩ R
d
h
51
Γ
h
52
Ω
h
52
u
i
(x):=
1
h
u(x
1
,...,x
i−1
,x
i
+ h, x
i+1
,...,x
d
) − u(x
1
,...,x
d
)
u
¯ı
(x):=
1
h
u(x
1
,...,x
d
) − u(x
1
,...,x
i−1
,x
i
− h, x
i+1
,...,x
d
)
52
Λ(x, y, t, t
0
):=
1
(4π|t−t
0
|)
d
2
exp
|x−y|
2
4(t
0
−t)
80
K(x, y, t)=Λ(x, y, t, 0) =
1
(4πt)
d
2
e
−
|x−y|
2
4t
87
Γ (x)=
∞
0
e
−t
t
x−1
dt for x>097
p(x, y, t)=
1
(4πt)
d
2
e
−
|x−y|
2
4t
127
P
t
: C
0
b
(R
d
) → C
0
b
(R
d
) 127
P
Ω,g,t
f(x) 128
T
t
: B → B 129
D(A) 130
J
λ
v :=
∞
0
λe
−λs
T
s
vds for λ>0 130
D
t
T
t
132
R(λ, A):=(λ Id −A)
−1
133
P (t, x; s, E) 145
C
∞
0
(A):={ϕ ∈ C
∞
(A) : the closure of {x : ϕ(x) =0} is compact
and contained in A} 157
D(u):=
Ω
|∇u(x)|
2
dx 158
C
k
0
(Ω):={f ∈ C
k
(Ω) : the closure of {x : f(x) =0} is a compact
subset of Ω} (k =1, 2,...) 160
v = D
i
u 160
W
1,2
(Ω) 161
(u, v)
W
1,2
(Ω)
:=
Ω
u · v +
d
i=1
Ω
D
i
u · D
i
v 161
u
W
1,2
(Ω)
:= (u, u)
1
2
W
1,2
(Ω)
161
H
1,2
(Ω) 161
H
1,2
0
(Ω) 161
(V
μ
f)(x):=
Ω
|x − y|
d(μ−1)
f(y)dy 167
α := (α
1
,...,α
d
) 193
Index of Notation 351
D
α
ϕ :=
∂
∂x
1
α
1
···
∂
∂x
d
α
d
ϕ for ϕ ∈ C
|α|
(Ω) 193
D
α
u 193
W
k,p
(Ω):={u ∈ L
p
(Ω):D
α
u exists and is contained in L
p
(Ω)
for all |α|≤k} 193
u
W
k,p
(Ω)
:=
|α|≤k
Ω
|D
α
u|
p
1
p
193
H
k,p
(Ω) 193
H
k,p
0
(Ω) 193
·
p
= ·
L
p
(Ω)
193
Du 193
D
2
u 193
(V
μ
f)(x):=
Ω
|x − y|
d(μ−1)
f(y)dy 196
−
Ω
v(x)dx :=
1
|Ω|
Ω
v(x)dx 198
|Ω| 198
u
B
:=
1
|B|
B
u(y)dy 200
|B| 200
osc
Ω∩B(z,r)
u := sup
x,y∈B(z,r)∩Ω
|u(x) − u(y)| 203
f ∈ C
α
(Ω) 204
u
C
α
(Ω)
:= u
C
0
(Ω)
+sup
x,y∈Ω
|u(x)−u(y)|
|x−y|
α
204
C
0,1
(Ω) 204
Δ
h
i
u(x):=
u(x+he
i
)−u(x)
h
208
supp ϕ 209
domain of class C
k
218
C
l,1
(Ω) 224
f,g :=
Ω
f(x)g(x)dx 230
C
α
(Ω) 255
C
k,α
(Ω) 255
|f|
C
α
(Ω)
:= sup
x,y∈Ω
|f(x)−f(y)|
|x−y|
α
255
f
C
k,α
(Ω)
255
· 309
(·, ·) 309
352 Index of Notation
L
p
(Ω):=
u : Ω → R measurable,
with u
p
:= u
L
p
(Ω)
:=
0
Ω
|u|
p
dx
1
1
p
< ∞
311
L
∞
(Ω):=
u : Ω → R measurable, u
L
∞
(Ω)
:= sup |u| < ∞
311
·
p
312
(u, v)
L
2
(Ω)
:=
Ω
u(x)v(x)dx 312
u
h
(x):=
1
h
d
R
d
x−y
h
u(y)dy 313
Index
α-H¨older continuous, 230
alternating method of H.A. Schwarz,
66
Arzela–Ascoli, 20
Banach fixed point theorem, 337
Banach space, 337
barrier, 65, 74
bilinear form, 201
– coercive, 201, 338
– elliptic, 201, 338
boundary point
– nonregular, 75
boundary point lemma of E. Hopf,
37, 89
boundary regularity, 244, 248
boundary value problem, 6
boundedness, 245
Brownian motion, 173, 174
Caccioppoli inequality, 324
calculus of variations
– direct method, 199
Calderon–Zygmund inequality, 269,
274
Campanato estimates, 326
Cauchy–Riemann equations, 1
chain rule for Sobolev functions,
191
Chapman–Kolmogorov equation, 171,
172
compactness theorem of Rellich,
194, 251, 257
comparison theorem, 45
concave, 40, 304
constructive method, 53
constructive technique, 6
continuous semigroup, 155
contracting, 155
convex, 23, 304
convolution, 90
Courant’s minimax principle, 263
Courant’s nodal set theorem, 267
cutoff function, 236, 271, 272
Darboux equation, 143
delta distribution, 190
difference equation, 53
difference method, 53
difference quotient, 234
– forward and backward, 54
difference scheme, 60
– consistent, 60
– convergent, 60
differential equation
– parabolic, 79
differential operator
– elliptic, 59
– linear elliptic, 33
diffusion process, 2
Dirac delta distribution, 10
Dirichlet boundary condition, 255
Dirichlet integral, 185, 196, 200
– transformation behavior, 199
Dirichlet principle, 183, 196, 241
Dirichlet problem, 14, 15, 25, 26,
36, 46, 55, 66, 92, 183, 241, 297
– weak solution, 245
Dirichlet problem on the ball
– solution, 14
discretely connected, 54
discretization
– heat equation, 113
discretization of the heat equation,
113
distribution, 10, 189
distributional derivative, 189
divergence theorem, 7
Duhamel principle, 105
354 INDEX
edge path, 53
edges, 53
eigenvalue, 128, 132, 135, 255
eigenvalue problem, 259
Einstein field equation, 3
elliptic, 5, 43, 44
elliptic differential operator
– divergence type, 303
elliptic regularity theory, 240
ellipticity, 33, 245, 291
ellipticity condition, 46
energy, 142
energy norm, 147
equilibrium state, 2
estimates of J. Schauder, 291
Euler–Lagrange equations, 198, 200
example of Lebesgue, 75
existence, 6
existence problem, 297
extension of Sobolev functions, 250
exterior sphere condition, 74
first eigenvalue, 266
Fisher equation, 125
fixed point, 337
fundamental estimates of J. Moser,
306
fundamental solution, 269
gamma function, 100
Gierer-Meinhardt system, 131
global bound, 241
global error, 60
global existence, 123
Green function, 11, 14, 25, 57
– for a ball, 13
Green representation formula, 9
Green’s formulae, 7
– first Green’s formula, 7
– second Green’s formula, 7
Hadamard, 6
harmonic, 8, 14, 16, 19, 23, 24, 240
harmonic polynomials, 8
Harnack convergence theorem, 29,
64, 68
Harnack inequality, 28, 314
heat equation, 2, 79, 90, 110, 141,
149, 153
– semidiscrete approximation, 114
– strong maximum principle, 88
heat kernel, 82, 104, 153, 174
Hilbert space, 337
Hille–Yosida theorem, 165
H¨older continuous, 281
H¨older’s inequality, 340
Huygens principle, 146
hyperbolic, 5
infinitesimal generator, 156
inhomogeneous Neumann bound-
ary conditions, 254
initial boundary value problem, 84,
103, 106, 120
initial value problem, 91, 140, 144,
146, 154
integration by parts, 186
isolated singularity, 24
iteration argument, 247
Korteweg–de Vries equation, 2
Laplace equation, 1, 9, 53, 55, 92
– discrete, 55
– discretized, 114
– fundamental solution, 9
– weak solution, 197
Laplace operator, 1, 33, 175
– eigenvalues, 255
– rotational symmetry, 9
– transformation behavior, 199
Lax–Milgram theorem, 204
linear, 8
linear equation, 4
Liouville theorem, 27
Lipschitz continuous, 230, 281
local error, 60
local existence, 120
Markov process, 172
INDEX 355
– spatially homogeneous, 173
Markov property, 171
maximum principle, 21, 24, 44, 46,
67, 85, 93, 106
– discrete, 55
– of Alexandrov and Bakelman, 40
– strong, 23, 62
– – of weak subsolutions, 317
– strong, E. Hopf, 37
– weak, 23, 34
Maxwell equation, 3
mean, 224, 226
mean value formula, 16
mean value inequality, 20, 21
mean value property, 17, 19, 114
methods of Campanato, 324
minimal surface equation, 3
minimizing sequence, 196
mollification, 18, 236, 341
mollifier, 341
Monge–Amp`ere equation, 2, 43, 44
Morrey’s Dirichlet growth theorem,
231
Moser iteration scheme, 311
Moser–Harnack inequality, 306, 315
natural boundary condition, 253
Navier–Stokes equation, 3
Neumann boundary condition, 30,
123, 127, 253, 255
Neumann boundary value problem,
12
Newton potential, 269, 290
nonlinear, 46, 334
nonlinear equation, 4
nonlinear parabolic equation, 119
numerical scheme, 6
parabolic, 5
partial differential equation, 1
pattern formation, 130
periodic boundary condition, 13
Perron Method, 62
Picard-Lindel¨of theorem, 119
plate equation, 4
Poincar´e inequality, 192, 197, 228,
257, 262
Poisson equation, 1, 24–26, 28, 201,
297
– discrete, 57
– gradient estimate for solutions,
26
– uniqueness of solutions, 24
– weak solution, 197, 205, 236
Poisson representation formula, 14
Poisson’s formula, 16
propagation of waves, 2
quasilinear equation, 4
Rayleigh–Ritz scheme, 264
reaction-diffusion equation, 120, 123
reaction-diffusion system, 126, 130
reduced boundary, 79
regular point, 65
regularity issues, 110
regularity result, 236
regularity theorem of de Giorgi,
319
regularity theory, 198
– L
p
-regularity theory, 274
replacement lemma, 190
representation formula, 14, 141
resolvent, 159, 165
resolvent equation, 160
Riesz representation theorem, 235,
338
scalar product, 337
Schauder estimates, 291
Schnakenberg reaction, 131
Schr¨odinger equation, 4
Schwarz inequality, 338
semidiscrete approximation of the
heat equation, 114
semigroup, 155, 156, 175
– continuous, 155, 164
– contracting, 155, 158, 173
semigroup property, 172
semilinear equation, 5