Partial differential equations:
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Boston [u.a.]
Birkhäuser
2010
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| Ausgabe: | 2. ed. |
| Schriftenreihe: | Cornerstones
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| Schlagworte: | |
| Online-Zugang: | Inhaltstext Inhaltsverzeichnis |
| Beschreibung: | XX, 389 S. graph. Darst. |
| ISBN: | 9780817645519 0817645519 ebook 9780817645526 |
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| 100 | 1 | |a DiBenedetto, Emmanuele |d 1947-2021 |e Verfasser |0 (DE-588)139140034 |4 aut | |
| 245 | 1 | 0 | |a Partial differential equations |c Emmanuele DiBenedetto |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Boston [u.a.] |b Birkhäuser |c 2010 | |
| 300 | |a XX, 389 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Cornerstones | |
| 650 | 4 | |a Differential equations, Partial | |
| 650 | 0 | 7 | |a Partielle Differentialgleichung |0 (DE-588)4044779-0 |2 gnd |9 rswk-swf |
| 655 | 7 | |0 (DE-588)4123623-3 |a Lehrbuch |2 gnd-content | |
| 689 | 0 | 0 | |a Partielle Differentialgleichung |0 (DE-588)4044779-0 |D s |
| 689 | 0 | |5 DE-604 | |
| 856 | 4 | 2 | |q text/html |u http://deposit.dnb.de/cgi-bin/dokserv?id=2857128&prov=M&dok_var=1&dok_ext=htm |3 Inhaltstext |
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Datensatz im Suchindex
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Contents
Preface
to the Second Edition
. xvii
Preface to the First Edition
. xix
0
Preliminaries
. 1
1
Green's Theorem
. 1
1.1
Differential Operators and
Adjoints
. 2
2
The Continuity Equation
. 3
3
The Heat Equation and the Laplace Equation
. 5
3.1
Variable Coefficients
. 5
4
A Model for the Vibrating String
. 6
5
Small Vibrations of a Membrane
. 8
6
Transmission of Sound Waves
. 11
7
The Navier-Stokes System
. 13
8
The
Euler
Equations
. 13
9
Isentropic Potential Flows
. 14
9.1
Steady Potential Isentropic Flows
. 15
10
Partial Differential Equations
. 16
1
Quasi-Linear Equations and the Cauchy-Kowalewski
Theorem
. 17
1
Quasi-Linear Second-Order Equations in Two Variables
. 17
2
Characteristics and Singularities
. 19
2.1
Coefficients Independent of ux and uy
. 20
3
Quasi-Linear Second-Order Equations
. 21
3.1
Constant Coefficients
. 23
3.2
Variable Coefficients
. 23
4
Quasi-Linear Equations of Order
m
> 1. 24
4.1
Characteristic Surfaces
. 25
5
Analytic Data and the Cauchy-Kowalewski Theorem
. 26
5.1
Reduction to Normal Form
([19]). 26
Contents
6
Proof of the Cauchy-Kowalewski Theorem
. 27
6.1
Estimating the Derivatives of
u
at the Origin
. 28
7
Auxiliary Inequalities
. 29
8
Auxiliary Estimations at the Origin
. 31
9
Proof of the Cauchy-Kowalewski Theorem (Concluded)
. 32
9.1
Proof of Lemma
6.1 . 33
Problems and Complements
. 33
lc Quasi-Linear Second-Order Equations in Two Variables
. 33
5c Analytic Data and the Cauchy-Kowalewski Theorem
. 34
6c Proof of the Cauchy-Kowalewski Theorem
. 34
8c The Generalized Leibniz Rule
. 34
9c Proof of the Cauchy-Kowalewski Theorem (Concluded)
. 35
The Laplace Equation
. 37
1
Preliminaries
. 37
1.1
The Dirichlet and Neumann Problems
. 38
1.2
The Cauchy Problem
. 39
1.3
Well-Posedness and a Counterexample
of
Hadamard
. 39
1.4
Radial Solutions
. 40
2
The Green and Stokes Identities
. 41
2.1
The Stokes Identities
. 41
3
Green's Function and the Dirichlet Problem for a Ball
. 43
3.1
Green's Function for a Ball
. 45
4
Sub-Harmonic Functions and the Mean Value Property
. 47
4.1
The Maximum Principle
. 50
4.2
Structure of Sub-Harmonic Functions
. 50
5
Estimating Harmonic Functions and Their Derivatives
. 52
5.1
The Harnack Inequality and the Liouville Theorem
. 52
5.2
Analyticity of Harmonic Functions
. 53
6
The Dirichlet Problem
. 55
7
About the Exterior Sphere Condition
. 58
7.1
The Case
N = 2
and dE Piecewise Smooth
. 59
7.2
A Counterexample of Lebesgue for
N — 3 ([101]). 59
8
The
Poisson
Integral for the Half-Space
. 60
9 Schauder
Estimates of Newtonian Potentials
. 62
10
Potential Estimates in LP(E)
. 65
11
Local Solutions
. 68
11.1
Local Weak Solutions
. 69
12
Inhomogeneous Problems
. 70
12.1
On the Notion of Green's Function
. 70
12.2
Inhomogeneous Problems
. 71
12.3
The Case
ƒ
є
С™
(E).
72
12.4
The Case
ƒ
Є
CT>(Ë)
. 72
Contents
vii
Problems
and Complements
. 73
le
Preliminaries
. 73
1.1c Newtonian Potentials on Ellipsoids
. 73
1.2c
Invariance
Properties
. 74
2c The Green and Stokes Identities
. 74
3c Green's Function and the Dirichlet Problem
for the Ball
. 74
3.1c Separation of Variables
. 75
4c Sub-Harmonic Functions and the Mean Value Property
. 76
4.1c Reflection and Harmonic Extension
. 77
4.2c The Weak Maximum Principle
. 77
4.3c Sub-Harmonic Functions
. 78
5c Estimating Harmonic Functions
. 79
5.1c Harnack-Type Estimates
. 80
5.2c Ill-Posed Problems: An Example of
Hadamard
. 80
5.3c Removable Singularities
. 81
7c About the Exterior Sphere Condition
. 82
8c Problems in Unbounded Domains
. 83
8.1c The Dirichlet Problem Exterior to a Ball
. 83
9c
Schauder
Estimates up to the Boundary
([135, 136]). 84
10c Potential Estimates in LP(E)
. 84
10.1c Integrability of Riesz Potentials
. 85
10.2c Second Derivatives of Potentials
. 85
Boundary Value Problems by Double-Layer Potentials
. 87
1
The Double-Layer Potential
. 87
2
On the Integral Defining the Double-Layer Potential
. 89
3
The Jump Condition of W(dE,
жо;
υ)
Across
дЕ.
91
4
More on the Jump Condition Across dE
. 93
5
The Dirichlet Problem by Integral Equations ([111])
. 94
6
The Neumann Problem by Integral Equations ([111])
. 95
7
The Green Function for the Neumann Problem
. 97
7.1
Finding
£(·; ■). 98
8
Eigenvalue Problems for the Laplacian
. 99
8.1
Compact Kernels Generated by Green's Function
.100
9
Compactness of AF in U>{E) for
1 <
ρ
<
oc
.100
10
Compactness of
ΑΦ
in LP{E) for
1 <
ρ
<
oo
.102
11
Compactness of
ΑΦ
in L°°{E)
.102
Problems and Complements
.104
2c On the Integral Defining the Double-Layer Potential
.104
5c The Dirichlet Problem by Integral Equations
.105
6c The Neumann Problem by Integral Equations
.106
viii Contents
7c
Green's Function for the Neumann Problem
.106
7.1c Constructing
£(■; ·)
for a Ball in R2 and K3
.106
8c Eigenvalue Problems
.107
4
Integral Equations and Eigenvalue Problems
.109
1
Kernels in L2(E)
.109
1.1
Examples of Kernels in L2(E)
.110
2
Integral Equations in L2(E)
.
Ill
2.1
Existence of Solutions for Small
|A|
.
Ill
3
Separable Kernels
.112
3.1
Solving the Homogeneous Equations
.113
3.2
Solving the Inhomogeneous Equation
.113
4
Small Perturbations of Separable Kernels
.114
4.1
Existence and Uniqueness of Solutions
.115
5
Almost Separable Kernels and Compactness
.116
5.1
Solving Integral Equations
for Almost Separable Kernels
.117
5.2
Potential Kernels Are Almost Separable
.117
6
Applications to the Neumann Problem
.118
7
The Eigenvalue Problem
.119
8
Finding a First Eigenvalue and Its Eigenfunctions
.121
9
The Sequence of Eigenvalues
.122
9.1
An Alternative Construction Procedure of the
Sequence of Eigenvalues
.123
10
Questions of Completeness and the ffilbert-Schmidt Theorem
. 124
10.1
The Case of
K(x;
■)
є
L2(E) Uniformly in
ж
.125
11
The Eigenvalue Problem for the Laplacean
.126
11.1
An Expansion of Green's Function
.127
Problems and Complements
.128
2c Integral Equations
.128
2.1c Integral Equations of the First Kind
.128
2.2c Abel Equations
([2, 3]).128
2.3c Solving Abel Integral Equations
.129
2.4c The Cycloid
([3]).130
2.5c Volterra Integral Equations
([158, 159]) .130
3c Separable Kernels
.131
3.1c
Hammerstein
Integral Equations
([64]).131
6c Applications to the Neumann Problem
.132
9c The Sequence of Eigenvalues
.132
10c Questions of Completeness
.132
10.1c Periodic Functions in
RN
.133
10.2c The
Poisson
Equation with Periodic Boundary
Conditions
.134
lie The Eigenvalue Problem for the Lapiacian
.134
Contents
ix
The Heat Equation
.135
1
Preliminaries
.135
1.1
The Dirichlet Problem
.136
1.2
The Neumann Problem
.136
1.3
The Characteristic Cauchy Problem
.136
2
The Cauchy Problem by Similarity Solutions
.136
2.1
The Backward Cauchy Problem
.140
3
The Maximum Principle and Uniqueness (Bounded Domains)
. 140
3.1
A Priori Estimates
.141
3.2
Ill-Posed Problems
.141
3.3
Uniqueness (Bounded Domains)
.142
4
The Maximum Principle in RN
.142
4.1
A Priori Estimates
.144
4.2
About the Growth Conditions
(4.3)
and
(4.4).145
5
Uniqueness of Solutions to the Cauchy Problem
.145
5.1
A Counterexample of Tychonov
([155]).145
6
Initial Data in
L¡OC(RN)
.147
6.1
Initial Data in the Sense of
L¡OC(RN)
.149
7
Remarks on the Cauchy Problem
.149
7.1
About Regularity
.149
7.2
Instability of the Backward Problem
.150
8
Estimates Near
t
= 0 .151
9
The Inhomogeneous Cauchy Problem
.152
10
Problems in Bounded Domains
.154
10.1
The Strong Solution
.155
10.2
The Weak Solution and Energy Inequalities
.156
11
Energy and Logarithmic Convexity
.157
11.1
Uniqueness for Some Ill-Posed Problems
.158
12
Local Solutions
.158
12.1
Variable Cylinders
.162
12.2
The Case \a\
= 0.162
13
The Harnack Inequality
.163
13.1
Compactly Supported Sub-Solutions
.164
13.2
Proof of Theorem
13.1.165
14
Positive Solutions in
5χ
.167
14.1
Non-Negative Solutions
.169
Problems and Complements
.171
2c Similarity Methods
.171
2.1c The Heat Kernel Has Unit Mass
.171
2.2c The Porous Media Equation
.172
2.3c The p-Laplacean Equation
.172
2.4c The Error Function
.173
2.5c The
Appell
Transformation
([7]).173
2.6c The Heat Kernel by Fourier Transform
.173
Contents
2.7c
Rapidly Decreasing Functions
.174
2.8c The Fourier Transform of the Heat Kernel
.174
2.9c The Inversion Formula
.175
3c The Maximum Principle in Bounded Domains
.176
3.1c The Blow-Up Phenomenon for Super-Linear Equations
. 177
3.2c The Maximum Principle for General Parabolic
Equations
.178
4c The Maximum Principle in Rw
.178
4.1c A Counterexample of the Tychonov Type
.180
7c Remarks on the Cauchy Problem
.180
12c On the Local Behavior of Solutions
.180
The Wave Equation
.183
1
The One-Dimensional Wave Equation
.183
1.1
A Property of Solutions
.184
2
The Cauchy Problem
.185
3
Inhomogeneous Problems
.186
4
A Boundary Value Problem (Vibrating String)
.188
4.1
Separation of Variables
.189
4.2
Odd Reflection
.190
4.3
Energy and Uniqueness
.190
4.4
Inhomogeneous Problems
.191
5
The Initial Value Problem in
N
Dimensions
.191
5.1
Spherical Means
.192
5.2
The Darboux Formula
.192
5.3
An Equivalent Formulation of the Cauchy Problem
. 193
6
The Cauchy Problem in
Ш3
.193
7
The Cauchy Problem in
Ж2
.196
8
The Inhomogeneous Cauchy Problem
.198
9
The Cauchy Problem for Inhomogeneous Surfaces
.199
9.1
Reduction to Homogeneous Data on
t
=
Φ.
200
9.2
The Problem with Homogeneous Data
.200
10
Solutions in Half-Space. The Reflection Technique
.201
10.1
An Auxiliary Problem
.202
10.2
Homogeneous Data on the
Hyperplane £3 = 0.202
11
A Boundary Value Problem
.203
12
Hyperbolic Equations in Two Variables
.204
13
The Characteristic Goursat Problem
.205
13.1
Proof of Theorem
13.1:
Existence
.205
13.2
Proof of Theorem
13.1:
Uniqueness
.207
13.3
Goursat Problems in Rectangles
.207
14
The Non-Characteristic Cauchy Problem and the Riemann
Function
.208
15
Symmetry of the Riemann Function
.210
Contents xi
Problems
and Complements
.211
2c
The d'Alembert Formula
.211
3c Inhomogeneous Problems.211
3.1c The Duhamel
Principle
([38]).211
4c Solutions
for the Vibrating String
.212
6c Cauchy Problems in E3
.214
6.1c Asymptotic Behavior
.214
6.2c Radial Solutions
.214
6.3c Solving the Cauchy Problem by Fourier Transform
. 216
7c Cauchy Problems in R2 and the Method of Descent
.217
7.1c The Cauchy Problem for
N = 4,5.218
8c
Inhomogeneous
Cauchy Problems
.218
8.1c The Wave Equation for the
N
and (N
+
1)-Laplacian
. 218
8.2c Miscellaneous Problems
.219
10c The Reflection Technique
.221
lie Problems in Bounded Domains
.221
11.1c Uniqueness
.221
11.2c Separation of Variables
.222
12c Hyperbolic Equations in Two Variables
.222
12.1c The General Telegraph Equation
.222
14c Goursat Problems
.223
14.1c The Riemann Function and the Fundamental Solution
of the Heat Equation
.223
Quasi-Linear Equations of First-Order
.225
1
Quasi-Linear Equations
.225
2
The Cauchy Problem
.226
2.1
The Case of Two Independent Variables
.226
2.2
The Case of
N
Independent Variables
.227
3
Solving the Cauchy Problem
.227
3.1
Constant Coefficients
.228
3.2
Solutions in Implicit Form
.229
4
Equations in Divergence Form and Weak Solutions
.230
4.1
Surfaces of Discontinuity
.231
4.2
The Shock Line
.231
5
The Initial Value Problem
.232
5.1
Conservation Laws
.233
6
Conservation Laws in One Space Dimension
.234
6.1
Weak Solutions and Shocks
.235
6.2
Lack of Uniqueness
.236
7 Hopf
Solution of The Burgers Equation
.236
8
Weak Solutions to
(6.4)
When
α(·)
is Strictly Increasing
.238
8.1
Lax VariationaJ Solution
.239
9
Constructing Variational Solutions I
.240
9.1
Proof of Lemma
9.1 .241
Contents
10
Constructing Variational
Solutions
II
.242
11 The Theorems
of Existence and Stability
.244
11.1
Existence of Variational Solutions
.244
11.2
Stability of Variational Solutions
.245
12
Proof of Theorem
11.1.246
12.1
The Representation Formula
(11.4).246
12.2
Initial Datum in the Sense of ^(R)
.247
12.3
Weak Forms of the PDE
.248
13
The Entropy Condition
.248
13.1
Entropy Solutions
.249
13.2
Variational Solutions of
(6.4)
are Entropy Solutions
-----249
13.3
Remarks on the Shock and the Entropy Conditions
. 251
14
The Kruzhkov Uniqueness Theorem
.253
14.1
Proof of the Uniqueness Theorem I
.253
14.2
Proof of the Uniqueness Theorem II
.254
14.3
Stability in Ll(RN)
.256
15
The Maximum Principle for Entropy Solutions
.256
Problems and Complements
.257
3c Solving the Cauchy Problem
.257
6c Explicit Solutions to the Burgers Equation
.259
6.2c
Invariance
of Burgers Equations by Some
Transformation of Variables
.259
6.3c The Generalized Riemann Problem
.260
13c The Entropy Condition
.261
14c The Kruzhkov Uniqueness Theorem
.262
Non-Linear Equations of First-Order
.265
1
Integral Surfaces and Monge's Cones
.265
1.1
Constructing Monge's Cones
.266
1.2
The Symmetric Equation of Monge's Cones
.266
2
Characteristic Curves and Characteristic Strips
.267
2.1
Characteristic Strips
.268
3
The Cauchy Problem
.269
3.1
Identifying the Initial Data p(0, s)
.269
3.2
Constructing the Characteristic Strips
.270
4
Solving the Cauchy Problem
.270
4.1
Verifying
(4.3).271
4.2
A Quasi-Linear Example in E2
.272
5
The Cauchy Problem for the Equation
of Geometrical Optics
.273
5.1
Wave Fronts, Light Rays, Local Solutions,
and Caustics
.274
6
The Initial Value Problem for Hamilton-Jacobi Equations
. 274
7
The Cauchy Problem in Terms of the Lagrangian
.276
Contents xiii
8 The Hopf Variational
Solution .
277
8.1 The First Hopf Variational
Formula
.278
8.2
The Second
Hopf Variational
Formula
.278
9
Semigroup Property of
Hopf
Variational Solutions
.279
10
Regularity of
Hopf
Variational Solutions
.280
11 Hopf
Variational Solutions
(8.3)
are Weak Solutions of the
Cauchy Problem
(6.4).281
12
Some Examples
.283
12.1
Example I
.283
12.2
Example II
.284
12.3
Example III
.284
13
Uniqueness
.285
14
More on Uniqueness and Stability
.287
14.1
Stability in
ľp{Rn)
for All
ρ
> 1.287
14.2
Comparison Principle
.288
15
Semi-Concave Solutions of the Cauchy Problem
.288
15.1
Uniqueness of Semi-Concave Solutions
.288
16
A Weak Notion of Semi-Concavity
.289
17
Semi-Concavity of
Hopf
Variational Solutions
.290
17.1
Weak Semi-Concavity of
Hopf
Variational Solutions
Induced by the Initial Datum uo
.290
17.2
Strictly Convex Hamiltonian
.291
18
Uniqueness of Weakly Semi-Concave
Variational
Hopf
Solutions
.293
Linear Elliptic Equations with Measurable Coefficients
. 297
1
Weak Formulations and Weak Derivatives
.297
1.1
Weak Derivatives
.298
2
Embeddings of Wl^{E)
.299
2.1
Compact Embeddings of W^P(E)
.300
3
Multiplicative Embeddings of W^p(E) and
Wl*{E)
.300
3.1
Some Consequences of the Multiplicative Embedding
Inequalities
.301
4
The Homogeneous Dirichlet Problem
.302
5
Solving the Homogeneous Dirichlet Problem
(4.1)
by the
Riesz Representation Theorem
.302
6
Solving the Homogeneous Dirichlet Problem
(4.1)
by
Variational Methods
.303
6.1
The Case
N = 2.304
6.2
Gâteaux
Derivative and The
Euler
Equation of J(·)
. 305
7
Solving the Homogeneous Dirichlet Problem
(4.1)
by
Galerkin Approximations
.305
7.1
On the Selection of an
Orthonormal
System
in
W¿*(E)
.306
xiv Contents
7.2
Conditions on f
and
ƒ
for the Solvability of the
Dirichlet Problem
(4.1) .307
8
Traces on dE of Functions in Wl*{E)
.307
8.1
The Segment Property
.307
8.2
Defining Traces
.308
8.3
Characterizing the Traces on dE of Functions in
W1'P(E)
.309
9
The Inhomogeneous Dirichlet Problem
.309
10
The Neumann Problem
.310
10.1
A Variant of
(10.1).311
11
The Eigenvalue Problem
.312
12
Constructing the Eigenvalues of
(11.1) .313
13
The Sequence of Eigenvalues and Eigenfunctions
.315
14
A Priori L°°(E) Estimates for Solutions of the Dirichlet
Problem
(9.1).317
15
Proof of Propositions
14.1-14.2.318
15.1
An Auxiliary Lemma on Fast Geometric Convergence
. 319
15.2
Proof of Proposition
14.1
for
N > 2.319
15.3
Proof of Proposition
14.1
for
N = 2.320
16
A Priori L°°(E) Estimates for Solutions of the Neumann
Problem
(10.1).320
17
Proof of Propositions
16.1-16.2.322
17.1
Proof of Proposition
16.1
for
N > 2.324
17.2
Proof of Proposition
16.1
for
N = 2.325
18
Miscellaneous Remarks on Further Regularity
.325
Problems and Complements
.326
lc Weak Formulations and Weak Derivatives
.326
1.1c The Chain Rule in W^(E)
.326
2c Embeddings of Wl*(E)
.327
2.1c Proof of
(2.4).327
2.2c Compact Embeddings of W1-P{E)
.328
3c Multiplicative Embeddings of W}'P(E) and Whp{E)
.329
3.1c Proof of Theorem
3.1
for
1 <
ρ
< N.329
3.2c Proof of Theorem
3.1
for
ρ
> N > 1.331
3.3c Proof of Theorem
3.2
for
1 <
ρ
< N
and
E
Convex
_332
5c Solving the Homogeneous Dirichlet Problem
(4.1)
by the
Riesz Representation Theorem
.333
6c Solving the Homogeneous Dirichlet Problem
(4.1)
by
Variational Methods
.334
6.1c More General Variational Problems
.334
6.8c
Gâteaux
Derivatives,
Euler
Equations, and
Quasi-Linear Elliptic Equations
.336
Contents xv
8c
Traces
on
дЕ
of Functions in W1'P(E)
.337
8.1c Extending Functions in W1'P(E)
.337
8.2c The Trace Inequality
.338
8.3c Characterizing the Traces on dE of Functions in
Wl*{E) .339
9c The Inhomogeneous Dirichlet Problem
.341
9.1c The Lebesgue Spike
.341
9.2c Variational Integrals and Quasi-Linear Equations
.341
10c The Neumann Problem
.342
lie The Eigenvalue Problem
.343
12c Constructing the Eigenvalues
.343
13c The Sequence of Eigenvalues and Eigenfunctions
.343
14c A Priori L°°(E) Estimates for Solutions of the Dirichlet
Problem
(9.1).343
15c A Priori L°°(E) Estimates for Solutions of the Neumann
Problem
(10.1).344
15.1c Back to the Quasi-Linear Dirichlet Problem (9.1c)
.344
10
DeGiorgi
Classes
.347
1
Quasi-Linear Equations and
DeGiorgi
Classes
.347
1.1
DeGiorgi
Classes
.349
2
Local Boundedness of Functions in the
DeGiorgi
Classes
.350
2.1
Proof of Theorem
2.1
for 1<
ρ
< N.351
2.2
Proof of Theorem
2.1
for
ρ
= N.352
3
Holder Continuity of Functions in the DG Classes
.353
3.1
On the Proof of Theorem
3.1 .354
4
Estimating the Values of
и
by the Measure of the Set where
и
is Either Near
μ+
or Near
μ~
.354
5
Reducing the Measure of the Set where
и
is Either Near
μ+
or Near
μ~
.355
5.1
The Discrete Isoperimetric Inequality
.356
5.2
Proof of Proposition
5.1.357
6
Proof of Theorem
3.1.358
7
Boundary
DeGiorgi
Classes: Dirichlet Data
.359
7.1
Continuity up to dE of Functions in the Boundary
DG Classes (Dirichlet Data)
.360
8
Boundary
DeGiorgi
Classes: Neumann Data
.361
8.1
Continuity up to dE of Functions in the Boundary
DG Classes (Neumann Data)
.363
9
The Harnack Inequality
.364
9.1
Proof of Theorem
9.1
(Preliminaries)
.364
9.2
Proof of Theorem
9.1.
Expansion of
Positivity
.365
9.3
Proof of Theorem
9.1.365
10
Harnack Inequality and Holder Continuity
.367
xvi Contents
11
Local Clustering of the
Positivity
Set of Functions
in W^l{E)
.368
12
A Proof of the Harnack Inequality Independent of Holder
Continuity
.370
References
.373
Index
.381 |
| adam_txt |
Contents
Preface
to the Second Edition
. xvii
Preface to the First Edition
. xix
0
Preliminaries
. 1
1
Green's Theorem
. 1
1.1
Differential Operators and
Adjoints
. 2
2
The Continuity Equation
. 3
3
The Heat Equation and the Laplace Equation
. 5
3.1
Variable Coefficients
. 5
4
A Model for the Vibrating String
. 6
5
Small Vibrations of a Membrane
. 8
6
Transmission of Sound Waves
. 11
7
The Navier-Stokes System
. 13
8
The
Euler
Equations
. 13
9
Isentropic Potential Flows
. 14
9.1
Steady Potential Isentropic Flows
. 15
10
Partial Differential Equations
. 16
1
Quasi-Linear Equations and the Cauchy-Kowalewski
Theorem
. 17
1
Quasi-Linear Second-Order Equations in Two Variables
. 17
2
Characteristics and Singularities
. 19
2.1
Coefficients Independent of ux and uy
. 20
3
Quasi-Linear Second-Order Equations
. 21
3.1
Constant Coefficients
. 23
3.2
Variable Coefficients
. 23
4
Quasi-Linear Equations of Order
m
> 1. 24
4.1
Characteristic Surfaces
. 25
5
Analytic Data and the Cauchy-Kowalewski Theorem
. 26
5.1
Reduction to Normal Form
([19]). 26
Contents
6
Proof of the Cauchy-Kowalewski Theorem
. 27
6.1
Estimating the Derivatives of
u
at the Origin
. 28
7
Auxiliary Inequalities
. 29
8
Auxiliary Estimations at the Origin
. 31
9
Proof of the Cauchy-Kowalewski Theorem (Concluded)
. 32
9.1
Proof of Lemma
6.1 . 33
Problems and Complements
. 33
lc Quasi-Linear Second-Order Equations in Two Variables
. 33
5c Analytic Data and the Cauchy-Kowalewski Theorem
. 34
6c Proof of the Cauchy-Kowalewski Theorem
. 34
8c The Generalized Leibniz Rule
. 34
9c Proof of the Cauchy-Kowalewski Theorem (Concluded)
. 35
The Laplace Equation
. 37
1
Preliminaries
. 37
1.1
The Dirichlet and Neumann Problems
. 38
1.2
The Cauchy Problem
. 39
1.3
Well-Posedness and a Counterexample
of
Hadamard
. 39
1.4
Radial Solutions
. 40
2
The Green and Stokes Identities
. 41
2.1
The Stokes Identities
. 41
3
Green's Function and the Dirichlet Problem for a Ball
. 43
3.1
Green's Function for a Ball
. 45
4
Sub-Harmonic Functions and the Mean Value Property
. 47
4.1
The Maximum Principle
. 50
4.2
Structure of Sub-Harmonic Functions
. 50
5
Estimating Harmonic Functions and Their Derivatives
. 52
5.1
The Harnack Inequality and the Liouville Theorem
. 52
5.2
Analyticity of Harmonic Functions
. 53
6
The Dirichlet Problem
. 55
7
About the Exterior Sphere Condition
. 58
7.1
The Case
N = 2
and dE Piecewise Smooth
. 59
7.2
A Counterexample of Lebesgue for
N — 3 ([101]). 59
8
The
Poisson
Integral for the Half-Space
. 60
9 Schauder
Estimates of Newtonian Potentials
. 62
10
Potential Estimates in LP(E)
. 65
11
Local Solutions
. 68
11.1
Local Weak Solutions
. 69
12
Inhomogeneous Problems
. 70
12.1
On the Notion of Green's Function
. 70
12.2
Inhomogeneous Problems
. 71
12.3
The Case
ƒ
є
С™
(E).
72
12.4
The Case
ƒ
Є
CT>(Ë)
. 72
Contents
vii
Problems
and Complements
. 73
le
Preliminaries
. 73
1.1c Newtonian Potentials on Ellipsoids
. 73
1.2c
Invariance
Properties
. 74
2c The Green and Stokes Identities
. 74
3c Green's Function and the Dirichlet Problem
for the Ball
. 74
3.1c Separation of Variables
. 75
4c Sub-Harmonic Functions and the Mean Value Property
. 76
4.1c Reflection and Harmonic Extension
. 77
4.2c The Weak Maximum Principle
. 77
4.3c Sub-Harmonic Functions
. 78
5c Estimating Harmonic Functions
. 79
5.1c Harnack-Type Estimates
. 80
5.2c Ill-Posed Problems: An Example of
Hadamard
. 80
5.3c Removable Singularities
. 81
7c About the Exterior Sphere Condition
. 82
8c Problems in Unbounded Domains
. 83
8.1c The Dirichlet Problem Exterior to a Ball
. 83
9c
Schauder
Estimates up to the Boundary
([135, 136]). 84
10c Potential Estimates in LP(E)
. 84
10.1c Integrability of Riesz Potentials
. 85
10.2c Second Derivatives of Potentials
. 85
Boundary Value Problems by Double-Layer Potentials
. 87
1
The Double-Layer Potential
. 87
2
On the Integral Defining the Double-Layer Potential
. 89
3
The Jump Condition of W(dE,
жо;
υ)
Across
дЕ.
91
4
More on the Jump Condition Across dE
. 93
5
The Dirichlet Problem by Integral Equations ([111])
. 94
6
The Neumann Problem by Integral Equations ([111])
. 95
7
The Green Function for the Neumann Problem
. 97
7.1
Finding
£(·; ■). 98
8
Eigenvalue Problems for the Laplacian
. 99
8.1
Compact Kernels Generated by Green's Function
.100
9
Compactness of AF in U>{E) for
1 <
ρ
<
oc
.100
10
Compactness of
ΑΦ
in LP{E) for
1 <
ρ
<
oo
.102
11
Compactness of
ΑΦ
in L°°{E)
.102
Problems and Complements
.104
2c On the Integral Defining the Double-Layer Potential
.104
5c The Dirichlet Problem by Integral Equations
.105
6c The Neumann Problem by Integral Equations
.106
viii Contents
7c
Green's Function for the Neumann Problem
.106
7.1c Constructing
£(■; ·)
for a Ball in R2 and K3
.106
8c Eigenvalue Problems
.107
4
Integral Equations and Eigenvalue Problems
.109
1
Kernels in L2(E)
.109
1.1
Examples of Kernels in L2(E)
.110
2
Integral Equations in L2(E)
.
Ill
2.1
Existence of Solutions for Small
|A|
.
Ill
3
Separable Kernels
.112
3.1
Solving the Homogeneous Equations
.113
3.2
Solving the Inhomogeneous Equation
.113
4
Small Perturbations of Separable Kernels
.114
4.1
Existence and Uniqueness of Solutions
.115
5
Almost Separable Kernels and Compactness
.116
5.1
Solving Integral Equations
for Almost Separable Kernels
.117
5.2
Potential Kernels Are Almost Separable
.117
6
Applications to the Neumann Problem
.118
7
The Eigenvalue Problem
.119
8
Finding a First Eigenvalue and Its Eigenfunctions
.121
9
The Sequence of Eigenvalues
.122
9.1
An Alternative Construction Procedure of the
Sequence of Eigenvalues
.123
10
Questions of Completeness and the ffilbert-Schmidt Theorem
. 124
10.1
The Case of
K(x;
■)
є
L2(E) Uniformly in
ж
.125
11
The Eigenvalue Problem for the Laplacean
.126
11.1
An Expansion of Green's Function
.127
Problems and Complements
.128
2c Integral Equations
.128
2.1c Integral Equations of the First Kind
.128
2.2c Abel Equations
([2, 3]).128
2.3c Solving Abel Integral Equations
.129
2.4c The Cycloid
([3]).130
2.5c Volterra Integral Equations
([158, 159]) .130
3c Separable Kernels
.131
3.1c
Hammerstein
Integral Equations
([64]).131
6c Applications to the Neumann Problem
.132
9c The Sequence of Eigenvalues
.132
10c Questions of Completeness
.132
10.1c Periodic Functions in
RN
.133
10.2c The
Poisson
Equation with Periodic Boundary
Conditions
.134
lie The Eigenvalue Problem for the Lapiacian
.134
Contents
ix
The Heat Equation
.135
1
Preliminaries
.135
1.1
The Dirichlet Problem
.136
1.2
The Neumann Problem
.136
1.3
The Characteristic Cauchy Problem
.136
2
The Cauchy Problem by Similarity Solutions
.136
2.1
The Backward Cauchy Problem
.140
3
The Maximum Principle and Uniqueness (Bounded Domains)
. 140
3.1
A Priori Estimates
.141
3.2
Ill-Posed Problems
.141
3.3
Uniqueness (Bounded Domains)
.142
4
The Maximum Principle in RN
.142
4.1
A Priori Estimates
.144
4.2
About the Growth Conditions
(4.3)
and
(4.4).145
5
Uniqueness of Solutions to the Cauchy Problem
.145
5.1
A Counterexample of Tychonov
([155]).145
6
Initial Data in
L¡OC(RN)
.147
6.1
Initial Data in the Sense of
L¡OC(RN)
.149
7
Remarks on the Cauchy Problem
.149
7.1
About Regularity
.149
7.2
Instability of the Backward Problem
.150
8
Estimates Near
t
= 0 .151
9
The Inhomogeneous Cauchy Problem
.152
10
Problems in Bounded Domains
.154
10.1
The Strong Solution
.155
10.2
The Weak Solution and Energy Inequalities
.156
11
Energy and Logarithmic Convexity
.157
11.1
Uniqueness for Some Ill-Posed Problems
.158
12
Local Solutions
.158
12.1
Variable Cylinders
.162
12.2
The Case \a\
= 0.162
13
The Harnack Inequality
.163
13.1
Compactly Supported Sub-Solutions
.164
13.2
Proof of Theorem
13.1.165
14
Positive Solutions in
5χ
.167
14.1
Non-Negative Solutions
.169
Problems and Complements
.171
2c Similarity Methods
.171
2.1c The Heat Kernel Has Unit Mass
.171
2.2c The Porous Media Equation
.172
2.3c The p-Laplacean Equation
.172
2.4c The Error Function
.173
2.5c The
Appell
Transformation
([7]).173
2.6c The Heat Kernel by Fourier Transform
.173
Contents
2.7c
Rapidly Decreasing Functions
.174
2.8c The Fourier Transform of the Heat Kernel
.174
2.9c The Inversion Formula
.175
3c The Maximum Principle in Bounded Domains
.176
3.1c The Blow-Up Phenomenon for Super-Linear Equations
. 177
3.2c The Maximum Principle for General Parabolic
Equations
.178
4c The Maximum Principle in Rw
.178
4.1c A Counterexample of the Tychonov Type
.180
7c Remarks on the Cauchy Problem
.180
12c On the Local Behavior of Solutions
.180
The Wave Equation
.183
1
The One-Dimensional Wave Equation
.183
1.1
A Property of Solutions
.184
2
The Cauchy Problem
.185
3
Inhomogeneous Problems
.186
4
A Boundary Value Problem (Vibrating String)
.188
4.1
Separation of Variables
.189
4.2
Odd Reflection
.190
4.3
Energy and Uniqueness
.190
4.4
Inhomogeneous Problems
.191
5
The Initial Value Problem in
N
Dimensions
.191
5.1
Spherical Means
.192
5.2
The Darboux Formula
.192
5.3
An Equivalent Formulation of the Cauchy Problem
. 193
6
The Cauchy Problem in
Ш3
.193
7
The Cauchy Problem in
Ж2
.196
8
The Inhomogeneous Cauchy Problem
.198
9
The Cauchy Problem for Inhomogeneous Surfaces
.199
9.1
Reduction to Homogeneous Data on
t
=
Φ.
200
9.2
The Problem with Homogeneous Data
.200
10
Solutions in Half-Space. The Reflection Technique
.201
10.1
An Auxiliary Problem
.202
10.2
Homogeneous Data on the
Hyperplane £3 = 0.202
11
A Boundary Value Problem
.203
12
Hyperbolic Equations in Two Variables
.204
13
The Characteristic Goursat Problem
.205
13.1
Proof of Theorem
13.1:
Existence
.205
13.2
Proof of Theorem
13.1:
Uniqueness
.207
13.3
Goursat Problems in Rectangles
.207
14
The Non-Characteristic Cauchy Problem and the Riemann
Function
.208
15
Symmetry of the Riemann Function
.210
Contents xi
Problems
and Complements
.211
2c
The d'Alembert Formula
.211
3c Inhomogeneous Problems.211
3.1c The Duhamel
Principle
([38]).211
4c Solutions
for the Vibrating String
.212
6c Cauchy Problems in E3
.214
6.1c Asymptotic Behavior
.214
6.2c Radial Solutions
.214
6.3c Solving the Cauchy Problem by Fourier Transform
. 216
7c Cauchy Problems in R2 and the Method of Descent
.217
7.1c The Cauchy Problem for
N = 4,5.218
8c
Inhomogeneous
Cauchy Problems
.218
8.1c The Wave Equation for the
N
and (N
+
1)-Laplacian
. 218
8.2c Miscellaneous Problems
.219
10c The Reflection Technique
.221
lie Problems in Bounded Domains
.221
11.1c Uniqueness
.221
11.2c Separation of Variables
.222
12c Hyperbolic Equations in Two Variables
.222
12.1c The General Telegraph Equation
.222
14c Goursat Problems
.223
14.1c The Riemann Function and the Fundamental Solution
of the Heat Equation
.223
Quasi-Linear Equations of First-Order
.225
1
Quasi-Linear Equations
.225
2
The Cauchy Problem
.226
2.1
The Case of Two Independent Variables
.226
2.2
The Case of
N
Independent Variables
.227
3
Solving the Cauchy Problem
.227
3.1
Constant Coefficients
.228
3.2
Solutions in Implicit Form
.229
4
Equations in Divergence Form and Weak Solutions
.230
4.1
Surfaces of Discontinuity
.231
4.2
The Shock Line
.231
5
The Initial Value Problem
.232
5.1
Conservation Laws
.233
6
Conservation Laws in One Space Dimension
.234
6.1
Weak Solutions and Shocks
.235
6.2
Lack of Uniqueness
.236
7 Hopf
Solution of The Burgers Equation
.236
8
Weak Solutions to
(6.4)
When
α(·)
is Strictly Increasing
.238
8.1
Lax VariationaJ Solution
.239
9
Constructing Variational Solutions I
.240
9.1
Proof of Lemma
9.1 .241
Contents
10
Constructing Variational
Solutions
II
.242
11 The Theorems
of Existence and Stability
.244
11.1
Existence of Variational Solutions
.244
11.2
Stability of Variational Solutions
.245
12
Proof of Theorem
11.1.246
12.1
The Representation Formula
(11.4).246
12.2
Initial Datum in the Sense of ^(R)
.247
12.3
Weak Forms of the PDE
.248
13
The Entropy Condition
.248
13.1
Entropy Solutions
.249
13.2
Variational Solutions of
(6.4)
are Entropy Solutions
-----249
13.3
Remarks on the Shock and the Entropy Conditions
. 251
14
The Kruzhkov Uniqueness Theorem
.253
14.1
Proof of the Uniqueness Theorem I
.253
14.2
Proof of the Uniqueness Theorem II
.254
14.3
Stability in Ll(RN)
.256
15
The Maximum Principle for Entropy Solutions
.256
Problems and Complements
.257
3c Solving the Cauchy Problem
.257
6c Explicit Solutions to the Burgers Equation
.259
6.2c
Invariance
of Burgers Equations by Some
Transformation of Variables
.259
6.3c The Generalized Riemann Problem
.260
13c The Entropy Condition
.261
14c The Kruzhkov Uniqueness Theorem
.262
Non-Linear Equations of First-Order
.265
1
Integral Surfaces and Monge's Cones
.265
1.1
Constructing Monge's Cones
.266
1.2
The Symmetric Equation of Monge's Cones
.266
2
Characteristic Curves and Characteristic Strips
.267
2.1
Characteristic Strips
.268
3
The Cauchy Problem
.269
3.1
Identifying the Initial Data p(0, s)
.269
3.2
Constructing the Characteristic Strips
.270
4
Solving the Cauchy Problem
.270
4.1
Verifying
(4.3).271
4.2
A Quasi-Linear Example in E2
.272
5
The Cauchy Problem for the Equation
of Geometrical Optics
.273
5.1
Wave Fronts, Light Rays, Local Solutions,
and Caustics
.274
6
The Initial Value Problem for Hamilton-Jacobi Equations
. 274
7
The Cauchy Problem in Terms of the Lagrangian
.276
Contents xiii
8 The Hopf Variational
Solution .
277
8.1 The First Hopf Variational
Formula
.278
8.2
The Second
Hopf Variational
Formula
.278
9
Semigroup Property of
Hopf
Variational Solutions
.279
10
Regularity of
Hopf
Variational Solutions
.280
11 Hopf
Variational Solutions
(8.3)
are Weak Solutions of the
Cauchy Problem
(6.4).281
12
Some Examples
.283
12.1
Example I
.283
12.2
Example II
.284
12.3
Example III
.284
13
Uniqueness
.285
14
More on Uniqueness and Stability
.287
14.1
Stability in
ľp{Rn)
for All
ρ
> 1.287
14.2
Comparison Principle
.288
15
Semi-Concave Solutions of the Cauchy Problem
.288
15.1
Uniqueness of Semi-Concave Solutions
.288
16
A Weak Notion of Semi-Concavity
.289
17
Semi-Concavity of
Hopf
Variational Solutions
.290
17.1
Weak Semi-Concavity of
Hopf
Variational Solutions
Induced by the Initial Datum uo
.290
17.2
Strictly Convex Hamiltonian
.291
18
Uniqueness of Weakly Semi-Concave
Variational
Hopf
Solutions
.293
Linear Elliptic Equations with Measurable Coefficients
. 297
1
Weak Formulations and Weak Derivatives
.297
1.1
Weak Derivatives
.298
2
Embeddings of Wl^{E)
.299
2.1
Compact Embeddings of W^P(E)
.300
3
Multiplicative Embeddings of W^p(E) and
Wl*{E)
.300
3.1
Some Consequences of the Multiplicative Embedding
Inequalities
.301
4
The Homogeneous Dirichlet Problem
.302
5
Solving the Homogeneous Dirichlet Problem
(4.1)
by the
Riesz Representation Theorem
.302
6
Solving the Homogeneous Dirichlet Problem
(4.1)
by
Variational Methods
.303
6.1
The Case
N = 2.304
6.2
Gâteaux
Derivative and The
Euler
Equation of J(·)
. 305
7
Solving the Homogeneous Dirichlet Problem
(4.1)
by
Galerkin Approximations
.305
7.1
On the Selection of an
Orthonormal
System
in
W¿*(E)
.306
xiv Contents
7.2
Conditions on f
and
ƒ
for the Solvability of the
Dirichlet Problem
(4.1) .307
8
Traces on dE of Functions in Wl*{E)
.307
8.1
The Segment Property
.307
8.2
Defining Traces
.308
8.3
Characterizing the Traces on dE of Functions in
W1'P(E)
.309
9
The Inhomogeneous Dirichlet Problem
.309
10
The Neumann Problem
.310
10.1
A Variant of
(10.1).311
11
The Eigenvalue Problem
.312
12
Constructing the Eigenvalues of
(11.1) .313
13
The Sequence of Eigenvalues and Eigenfunctions
.315
14
A Priori L°°(E) Estimates for Solutions of the Dirichlet
Problem
(9.1).317
15
Proof of Propositions
14.1-14.2.318
15.1
An Auxiliary Lemma on Fast Geometric Convergence
. 319
15.2
Proof of Proposition
14.1
for
N > 2.319
15.3
Proof of Proposition
14.1
for
N = 2.320
16
A Priori L°°(E) Estimates for Solutions of the Neumann
Problem
(10.1).320
17
Proof of Propositions
16.1-16.2.322
17.1
Proof of Proposition
16.1
for
N > 2.324
17.2
Proof of Proposition
16.1
for
N = 2.325
18
Miscellaneous Remarks on Further Regularity
.325
Problems and Complements
.326
lc Weak Formulations and Weak Derivatives
.326
1.1c The Chain Rule in W^(E)
.326
2c Embeddings of Wl*(E)
.327
2.1c Proof of
(2.4).327
2.2c Compact Embeddings of W1-P{E)
.328
3c Multiplicative Embeddings of W}'P(E) and Whp{E)
.329
3.1c Proof of Theorem
3.1
for
1 <
ρ
< N.329
3.2c Proof of Theorem
3.1
for
ρ
> N > 1.331
3.3c Proof of Theorem
3.2
for
1 <
ρ
< N
and
E
Convex
_332
5c Solving the Homogeneous Dirichlet Problem
(4.1)
by the
Riesz Representation Theorem
.333
6c Solving the Homogeneous Dirichlet Problem
(4.1)
by
Variational Methods
.334
6.1c More General Variational Problems
.334
6.8c
Gâteaux
Derivatives,
Euler
Equations, and
Quasi-Linear Elliptic Equations
.336
Contents xv
8c
Traces
on
дЕ
of Functions in W1'P(E)
.337
8.1c Extending Functions in W1'P(E)
.337
8.2c The Trace Inequality
.338
8.3c Characterizing the Traces on dE of Functions in
Wl*{E) .339
9c The Inhomogeneous Dirichlet Problem
.341
9.1c The Lebesgue Spike
.341
9.2c Variational Integrals and Quasi-Linear Equations
.341
10c The Neumann Problem
.342
lie The Eigenvalue Problem
.343
12c Constructing the Eigenvalues
.343
13c The Sequence of Eigenvalues and Eigenfunctions
.343
14c A Priori L°°(E) Estimates for Solutions of the Dirichlet
Problem
(9.1).343
15c A Priori L°°(E) Estimates for Solutions of the Neumann
Problem
(10.1).344
15.1c Back to the Quasi-Linear Dirichlet Problem (9.1c)
.344
10
DeGiorgi
Classes
.347
1
Quasi-Linear Equations and
DeGiorgi
Classes
.347
1.1
DeGiorgi
Classes
.349
2
Local Boundedness of Functions in the
DeGiorgi
Classes
.350
2.1
Proof of Theorem
2.1
for 1<
ρ
< N.351
2.2
Proof of Theorem
2.1
for
ρ
= N.352
3
Holder Continuity of Functions in the DG Classes
.353
3.1
On the Proof of Theorem
3.1 .354
4
Estimating the Values of
и
by the Measure of the Set where
и
is Either Near
μ+
or Near
μ~
.354
5
Reducing the Measure of the Set where
и
is Either Near
μ+
or Near
μ~
.355
5.1
The Discrete Isoperimetric Inequality
.356
5.2
Proof of Proposition
5.1.357
6
Proof of Theorem
3.1.358
7
Boundary
DeGiorgi
Classes: Dirichlet Data
.359
7.1
Continuity up to dE of Functions in the Boundary
DG Classes (Dirichlet Data)
.360
8
Boundary
DeGiorgi
Classes: Neumann Data
.361
8.1
Continuity up to dE of Functions in the Boundary
DG Classes (Neumann Data)
.363
9
The Harnack Inequality
.364
9.1
Proof of Theorem
9.1
(Preliminaries)
.364
9.2
Proof of Theorem
9.1.
Expansion of
Positivity
.365
9.3
Proof of Theorem
9.1.365
10
Harnack Inequality and Holder Continuity
.367
xvi Contents
11
Local Clustering of the
Positivity
Set of Functions
in W^l{E)
.368
12
A Proof of the Harnack Inequality Independent of Holder
Continuity
.370
References
.373
Index
.381 |
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| author | DiBenedetto, Emmanuele 1947-2021 |
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| ctrlnum | (OCoLC)171111213 (DE-599)BVBBV022470541 |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.35 |
| dewey-search | 515.35 |
| dewey-sort | 3515.35 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV022470541 |
| illustrated | Illustrated |
| index_date | 2024-07-02T17:44:27Z |
| indexdate | 2024-07-20T09:18:09Z |
| institution | BVB |
| isbn | 9780817645519 0817645519 ebook 9780817645526 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015678032 |
| oclc_num | 171111213 |
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| physical | XX, 389 S. graph. Darst. |
| publishDate | 2010 |
| publishDateSearch | 2010 |
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| publisher | Birkhäuser |
| record_format | marc |
| series2 | Cornerstones |
| spelling | DiBenedetto, Emmanuele 1947-2021 Verfasser (DE-588)139140034 aut Partial differential equations Emmanuele DiBenedetto 2. ed. Boston [u.a.] Birkhäuser 2010 XX, 389 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Cornerstones Differential equations, Partial Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Partielle Differentialgleichung (DE-588)4044779-0 s DE-604 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=2857128&prov=M&dok_var=1&dok_ext=htm Inhaltstext Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015678032&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | DiBenedetto, Emmanuele 1947-2021 Partial differential equations Differential equations, Partial Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| subject_GND | (DE-588)4044779-0 (DE-588)4123623-3 |
| title | Partial differential equations |
| title_auth | Partial differential equations |
| title_exact_search | Partial differential equations |
| title_exact_search_txtP | Partial differential equations |
| title_full | Partial differential equations Emmanuele DiBenedetto |
| title_fullStr | Partial differential equations Emmanuele DiBenedetto |
| title_full_unstemmed | Partial differential equations Emmanuele DiBenedetto |
| title_short | Partial differential equations |
| title_sort | partial differential equations |
| topic | Differential equations, Partial Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| topic_facet | Differential equations, Partial Partielle Differentialgleichung Lehrbuch |
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