Mathematical Analysis and Numerical Methods for Science and Technology: Volume 2 Functional and Variational Methods
Traducere de I.N. Sneddon Autor Robert Dautray Contribuţii de M. Artola, M. Authier Autor Jacques Louis Lions Contribuţii de P. Benilan, M. Cessenat, J.-M. Combes, H. Lanchon, B. Mercier, C. Wild, C. Zuilyen Limba Engleză Paperback – 22 noi 1999
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Specificații
ISBN-13: 9783540660989
ISBN-10: 3540660984
Pagini: 608
Ilustrații: 1
Dimensiuni: 155 x 235 x 32 mm
Greutate: 0.84 kg
Ediția:1st. ed. 1988. 2nd printing 1999
Editura: Springer Berlin, Heidelberg
Colecția Springer
Locul publicării:Berlin, Heidelberg, Germany
ISBN-10: 3540660984
Pagini: 608
Ilustrații: 1
Dimensiuni: 155 x 235 x 32 mm
Greutate: 0.84 kg
Ediția:1st. ed. 1988. 2nd printing 1999
Editura: Springer Berlin, Heidelberg
Colecția Springer
Locul publicării:Berlin, Heidelberg, Germany
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Public țintă
ResearchDescriere
These
6
volumes
-
the
result
of
a
10
year
collaboration
between
the
authors,
two
of
France's
leading
scientists
and
both
distinguished
international
figures
-
compile
the
mathematical
knowledge
required
by
researchers
in
mechanics,
physics,
engineering,
chemistry
and
other
branches
of
application
of
mathematics
for
the
theoretical
and
numerical
resolution
of
physical
models
on
computers.
Since
the
publication
in
1924
of
the
"Methoden
der
mathematischen
Physik"
by
Courant
and
Hilbert,
there
has
been
no
other
comprehensive
and
up-to-date
publication
presenting
the
mathematical
tools
needed
in
applications
of
mathematics
in
directly
implementable
form.
The
advent
of
large
computers
has
in
the
meantime
revolutionised
methods
of
computation
and
made
this
gap
in
the
literature
intolerable:
the
objective
of
the
present
work
is
to
fill
just
this
gap.
Many
phenomena
in
physical
mathematics
may
be
modeled
by
a
system
of
partial
differential
equations
in
distributed
systems:
a
model
here
means
a
set
of
equations,
which
together
with
given
boundary
data
and,
if
the
phenomenon
is
evolving
in
time,
initial
data,
defines
the
system.
The
advent
of
high-speed
computers
has
made
it
possible
for
the
first
time
to
calculate
values
from
models
accurately
and
rapidly.
Researchers
and
engineers
thus
have
a
crucial
means
of
using
numerical
results
to
modify
and
adapt
arguments
and
experiments
along
the
way.
Every
facet
of
technical
and
industrial
activity
has
been
affected
by
these
developments.
Modeling
by
distributed
systems
now
also
supports
work
in
many
areas
of
physics
(plasmas,
new
materials,
astrophysics,
geophysics),
chemistry
and
mechanics
and
is
finding
increasing
use
in
the
life
sciences.
Cuprins
III.
Functional
Transformations.-
A.
Some
Transformations
Useful
in
Applications.-
§1.
Fourier
Series
and
Dirichlet’s
Problem.-
1.
Fourier
Series.-
1.1.
Convergence
in
L2
(T).-
1.2.
Pointwise
Convergence
on
T.-
2.
Distributions
on
T
and
Periodic
Distributions.-
2.1.
Comparison
of
D’(T)
with
the
Distributions
on
?.-
2.2.
Principal
Properties
of
D’(T).-
3.
Fourier
Series
of
Distributions.-
4.
Fourier
Series
and
Fourier
Transforms.-
5.
Convergence
in
the
Sense
of
Césaro.-
6.
Solution
of
Dirichlet’s
Problem
with
the
Help
of
Fourier
Series.-
6.1.
Dirichlet’s
Problem
in
a
Disk.-
6.2.
Dirichlet’s
Problem
in
a
Rectangle.-
§2.
The
Mellin
Transform.-
1.
Generalities.-
2.
Definition
of
the
Mellin
Transform.-
3.
Properties
of
the
Mellin
Transform.-
4.
Inverse
Mellin
Transform.-
5.
Applications
of
the
Mellin
Transform.-
6.
Table
of
Some
Mellin
Transforms.-
§3.
The
Hankel
Transform.-
1.
Generalities.-
2.
Introduction
to
Bessel
Functions.-
3.
Definition
of
the
Hankel
Transform.-
4.
The
Inversion
Formula.-
5.
Properties
of
the
Hankel
Transform.-
6.
Application
of
the
Hankel
Transform
to
Partial
Differential
Equations.-
6.1.
Dirichlet’s
Problem
for
Laplace’s
Equation
in?+3.
The
Case
of
Axial
Symmetry.-
6.2.
Boundary
Value
Problem
for
the
Biharmonic
Equation
in
?+3,
with
Axial
Symmetry.-
7.
Table
of
Some
Hankel
Transforms.-
Review
of
Chapter
III
A.-
B.
Discrete
Fourier
Transforms
and
Fast
Fourier
Transforms.-
§1.
Introduction.-
§2.
Acceleration
of
the
Product
of
a
Matrix
by
a
Vector.-
§3.
The
Fast
Fourier
Transform
of
Cooley
and
Tukey.-
§4.
The
Fast
Fourier
Transform
of
Good-Winograd.-
§5.
Reduction
of
the
Number
of
Multiplications.-
1.
Relation
Between
the
Discrete
Fourier
Transform
and
the
Problem
of
Cyclic
Convolution.-
2.
Complexity
of
the
Product
of
Two
Polynomials.-
3.
Application
to
the
Cyclic
Convolution
of
Order
2.-
4.
Application
to
the
Cyclic
Convolution
of
Order
3.-
5.
Application
to
the
Cyclic
Convolution
of
Order
6.-
§6.
Fast
Fourier
Transform
in
Two
Dimensions.-
§7.
Some
Applications
of
the
Fast
Fourier
Transform.-
1.
Solution
of
Boundary
Value
Problems.-
2.
Regularisation
and
Smoothing
of
Functions.-
3.
Practical
Calculation
of
the
Fourier
Transform
of
a
Signal.-
4.
Determination
of
the
Spectrum
of
Certain
Finite
Difference
Operators
and
Fast
Solvers
for
the
Laplacian.-
Review
of
Chapter
III
B.-
IV.
Sobolev
Spaces.-
§1.
Spaces
H1(?),
Hm(?).-
§2.
The
Space
Hs(?n).-
1.
Definition
and
First
Properties.-
2.
The
Topological
Dual
of
Hs(?n).-
3.
The
Equation
(-?
+
k2)u
=
f
in
?n,
k
?
?\z0{.-
§3.
Sobolev’s
Embedding
Theorem.-
§4.
Density
and
Trace
Theorems
for
the
Spaces
Hm(?),
(m
?
?
*
=
?\z0{).-
1.
A
Density
Theorem.-
2.
A
Trace
Theorem
for
H1
(?+n).-
3.
Traces
of
the
Spaces
Hm(?+n)
and
Hm(?).-
4.
Properties
of
m-Extension.-
§5.
The
Spaces
H-m(?)
for
all
m
?
?.-
§6.
Compactness.-
§7.
Some
Inequalities
in
Sobolev
Spaces.-
1.
Poincaré’s
Inequality
for
H01(?)
(resp.
H0m(?)).-
2.
Poincaré’s
Inequality
for
H1(?).-
3.
Convexity
Inequalities
for
Hm(?).-
§8.
Supplementary
Remarks.-
1.
Sobolev
Spaces
Wm,
p(?).-
1.1.
Definitions.-
1.2.
Sobolev
Injections.-
1.3.
Trace
Theorems
for
the
Spaces
Wm,
p(?).-
2.
Sobolev
Spaces
with
Weights.-
2.1.
Unbounded
Open
Sets.-
2.2.
Polygonal
Open
Sets.-
Review
of
Chapter
IV.-
Appendix:
The
Spaces
Hs(?)
with
?
the
“Regular”
Boundary
of
an
Open
Set
?
in
?n.-
V.
Linear
Differential
Operators.-
§1.
Generalities
on
Linear
Differential
Operators.-
1.
Characterisation
of
Linear
Differential
Operators.-
2.
Various
Definitions.-
2.1.
Leibniz’s
Formula.-
2.2.
Transpose
of
a
Linear
Differential
Operator.-
2.3.
Order
of
a
Linear
Differential
Operator.-
3.
Linear
Differential
Operator
on
a
Manifold.-
4.
Characteristics.-
4.1.
Concept
of
Characteristics.-
4.2.
Bicharacteristics.-
5.
Operators
with
Analytic
Coefficients.
Theorems
of
Cauchy-Kowalewsky
and
of
Holmgren.-
§2.
Linear
Differential
Operators
with
Constant
Coefficients.-
1.
Study
of
a
l.d.o.
with
Constant
Coefficients
by
the
Fourier
Transform.-
1.1.
Existence
of
a
Solution
of
Pu
=
f
in
the
Space
of
Tempered
Distributions.-
1.2.
Example
1:
The
Laplacian.-
1.3.
Elliptic
and
Strongly
Elliptic
Operators.-
1.4.
Hypo-Elliptic
and
Semi-Elliptic
Operators.-
1.5.
Examples.-
1.6.
Reduction
of
Operators
of
Order
2
in
a
Homogeneous,
Isotropic
“Medium”.-
2.
Elementary
Solutions
of
a
l.d.o.
with
Constant
Coefficients.-
2.1.
Introduction.-
2.2.
Elementary
Solutions
in
S’
Examples.-
2.3.
Elementary
Solution
with
Support
in
a
Salient
Closed
Convex
Cone:
Hyperbolic
Operator.-
3.
Characterisation
of
Hyperbolic
Operators.-
3.1.
Characteristics
of
a
l.d.o.
with
Constant
Coefficients.-
3.2.
Algebraic
Characterisation
of
Hyperbolic
Operators.-
3.3.
Hyperbolic
Operators
of
Order
2.-
4.
Parabolic
Operators.-
§3.
Cauchy
Problem
for
Differential
Operators
with
Constant
Coefficients.-
1.
Cauchy
Problem
and
the
Elementary
Solution
in
D’(?n×
?+).-
2.
Propagation
in
Hyperbolic
Cauchy
Problems.-
3.
Choice
of
a
Functional
Space:
Well-Posed
Cauchy
Problem.-
4.
Well-Posed
Cauchy
Problem
in
S’.-
5.
Parabolic
and
Weakly
Parabolic
Operators.-
6.
Study
of
the
Particular
Case
P
=
?/?t
+
P0.-
6.1.
Analysis
of
One-Dimensional
Case.-
6.2.
Case
in
which
P0
is
Strongly
Elliptic.-
6.3.
Schrödinger
Operator.-
7.
Well-Posed
Cauchy
Problem
in
D’:
Hyperbolic
Operators.-
§4.
Local
Regularity
of
Solutions*.-
1.
Characterisation
of
Hypo-Ellipticity.-
1.1.
Necessary
Condition
for
Hypo-Ellipticity.-
1.2.
Algebraic
Transformation
of
the
Necessary
Condition
for
Hypo-Ellipticity.-
1.3.
The
Principal
Result.-
2.
Analyticity
of
Solutions.-
2.1.
Statement
of
Results.-
2.2.
Estimates
of
Analyticity.-
2.3.
Generalisation:
Gevrey
Classes.-
3.
Comparison
of
Operators.-
4.
Local
Regularity
for
Operators
with
Variable
Coefficients
and
of
Constant
Force.-
5.
Construction
of
an
Elementary
Solution.-
§5.
The
Maximum
Principle
*.-
1.
Prerequisites.-
2.
Parabolic
Maximum
Principle
and
Dissipativity.-
3.
Characterisation
of
Operators
P
Satisfying
Maximum
Principles.-
3.1.
The
Weak
Maximum
Principle.-
3.2.
The
Comparison
Principle.-
3.3.
The
Strong
Maximum
Principle.-
3.4.
The
Principle
of
the
Strong
Parabolic
Maximum.-
Review
of
Chapter
V.-
VI.
Operators
in
Banach
Spaces
and
in
Hilbert
Spaces.-
§1.
Review
of
Functional
Analysis:
Banach
and
Hilbert
Spaces.-
1.
Locally
Convex
Topological
Vector
Spaces.
Normed
Spaces
and
Banach
Spaces.-
2.
Linear
Operators.-
3.
Duality.-
4.
The
Hahn-Banach
Theorem
and
its
Applications.-
4.1.
Problems
of
Approximation.-
4.2.
Problems
of
Existence.-
4.3.
Problems
of
Separation
of
Convex
Sets.-
5.
Bidual,
Reflexivity,
Weak
Convergence,
Weak
Compactness.-
5.1.
Bidual.-
5.2.
Reflexivity.-
5.3.
Weak
Convergence.-
5.4.
Weak
Compactness.-
5.5.
Weak-Star
Convergence.-
6.
Hilbert
Spaces.-
6.1.
Definitions.-
6.2.
Projection
on
a
Closed
Convex
Set.-
6.3.
Orthonormal
Bases.-
6.4.
The
Riesz
Representation
Theorem.
Reflexivity.-
7.
Ideas
About
Functions
of
a
Real
or
Complex
Variable
with
Values
in
a
Banach
Space.-
7.1.
Weak
Topology.-
7.2.
Weak
Differentiability.-
7.3.
Weak
Holomorphy.-
§2.
Linear
Operators
in
Banach
Spaces.-
1.
Generalities
on
Linear
Operators.-
1.1.
Domain,
Kernel
and
Image
of
a
Linear
Operator.-
1.2.
Nullity
and
Deficiency
Indices.-
1.3.
Basic
Properties
of
Linear
Operators.-
2.
Spaces
of
Bounded
Operators.-
2.1.
Introduction.-
2.2.
Various
Concepts
of
Convergence
of
Operators.-
2.3.
Composition
and
Inverse
of
Bounded
Operators.-
2.4.
Transpose
of
a
Bounded
Operator.-
2.5.
Some
Classes
of
Bounded
Operators.-
2.6.
Some
Ideas
on
Functions
of
a
Real
or
Complex
Variable
with
Operator
Values;
Families
of
Operators.-
3.
Closed
Operators.-
3.1.
Definition
and
Examples.-
3.2.
Basic
Properties.-
3.3.
The
Set
?(X,
Y)
of
Closed
Operators
from
X
into
Y.-
3.4.
Transpose
of
a
Closed
Operator.-
3.5.
Operators
with
Closed
Image.-
§3.
Linear
Operators
in
Hilbert
Spaces.-
1.
Bounded
Operators
in
Hilbert
Spaces.-
1.1.
Adjoint
Sesquilinear
Form.-
1.2.
Hermitian
Operators.-
1.3.
Orthogonal
Projectors.-
1.4.
Isometries
and
Unitary
Operators.-
1.5.
Hilbert-Schmidt
Operators.-
2.
Unbounded
Operators
in
Hilbert
Spaces.-
2.1.
Adjoint
of
an
Unbounded
Operator.-
2.2.
Symmetric
Operators.-
2.3.
The
Cayley
Transform.-
2.4.
Normal
Operators.-
2.5.
Sesquilinear
Forms
and
Unbounded
Operators.-
Review
of
Chapter
VI.-
VII.
Linear
Variational
Problems.
Regularity.-
§1.
Elliptic
Variational
Theory.-
1.
The
Lax-Milgram
Theorem.-
2.
First
Examples.-
2.1.
Example
1.
Dirichlet
Problem.-
2.2.
Example
2.
Neumann
Problem.-
3.
Extensions
in
the
Case
in
which
V
and
H
are
Spaces
of
Distributions
or
of
Functions.-
4.
Sesquilinear
Forms
Associated
with
Elliptic
Operators
of
Order
Two.-
5.
Sesquilinear
Forms
Associated
with
Elliptic
Operators
of
Order
2m.-
6.
Miscellaneous
Remarks.-
7.
Application
to
the
Solution
of
General
Elliptic
Problems
(of
Dirichlet
Type).-
§2.
Examples
of
Second
Order
Elliptic
Problems.-
1.
Generalities.-
2.
Examples
of
Variational
Problems.-
2.1.
Mixed
Problem.-
2.2.
Non-Local
Boundary
Conditions.-
3.
Problems
Relative
to
Integro-Differential
Forms
on
?
+
?.-
3.1.
Problem
of
the
Oblique
Derivative.-
3.2.
Robin’s
Problem.-
4.
Transmission
Problem.-
5.
Miscellaneous
Remarks.-
6.
Application:
Stationary
Multigroup
Equation
for
the
Diffusion
of
Neutrons.-
7.
Application:
Statical
Problems
of
Elasticity.-
7.1.
Introduction.-
7.2.
Variational
Formulation.-
7.3.
Korn’s
Inequality.-
7.4.
Application
to
Problem
(2.39).-
7.5.
Inhomogeneous
Problem.-
8.
Statical
Problems
of
the
Flexure
of
Plates.-
§3.
Regularity
of
the
Solutions
of
Variational
Problems.-
1.
Introduction.-
2.
Interior
Regularity.-
3.
Global
Regularity
of
the
Solutions
of
Dirichlet
and
Neumann
Problems
for
Elliptic
Operators
of
Order
2.-
4.
Miscellaneous
Results
on
Global
Regularity.-
5.
Green’s
Functions.-
5.1.
Case
of
the
Laplacian
in
a
Bounded
Open
Set
?
with
Dirichlet
Condition.-
5.2.
Some
Other
Particular
Examples.-
5.3.
Green’s
Functions
in
a
More
General
Setting.-
Review
of
Chapter
VII.-
Appendix.
“Distributions”.-
§1.
Definition
and
Basic
Properties
of
Distributions.-
1.
The
Space
D(?).-
1.1.
Definition.-
1.2.
Elementary
Properties
of
the
Convolution
Product
of
Two
Functions.-
1.3.
A
Procedure
for
the
Construction
of
Functions
of
D(?).-
1.4.
The
Notion
of
Convergence
in
D(?).-
1.5.
Some
Inclusion
and
Density
Properties.-
2.
The
Space
D’(?)
of
Distributions
on
?.-
2.1.
Definition
of
Distributions
and
the
Concept
of
Convergence
in
D’(?).-
2.2.
First
Examples
of
Distributions:
Measures
on
?.-
2.3.
Differentiation
of
Distributions.
Examples.-
2.4.
Support
of
a
Distribution.
Distributions
with
Compact
Support.-
3.
Some
Elementary
Operations
on
Distributions.-
3.1.
Product
by
a
Function
of
Class
??.-
3.2.
Primitives
of
a
Distribution
on
a
Interval
of
?.-
3.3.
Tensor
Product
of
Two
Distributions.-
3.4.
Direct
Image
and
Inverse
Image
of
a
Function
and
of
a
Distribution
by
a
Function
of
Class
??.-
4.
Some
Examples.-
4.1.
Primitives
of
the
Dirac
Measure.-
4.2.
A
Division
Problem
(Case
n
=
1).-
4.3.
Derivative
of
a
Function
of
?n
Discontinuous
on
a
Surface.-
4.4.
Distributions
Defined
by
Inverse
Image
from
Distributions
on
the
Real
Line.-
§2.
Convolution
of
Distributions.-
1.
Convolution
of
a
Distribution
on
?n
and
a
Function
of
D(?n).-
2.
Convolution
of
Two
Distributions
of
Which
One
(at
Least)
is
with
Compact
Support.-
3.
Distributions
with
Convolutive
Supports.-
4.
Convolution
Algebras.-
§3.
Fourier
Transforms.-
1.
Fourier
Transform
of
L1-Functions.-
2.
The
Space
L(?n).-
3.
Fourier
Transform
in
L2.-
4.
Fourier
Transforms
of
Tempered
Distributions.-
5.
Fourier
Transform
of
Distributions
with
Compact
Support.-
6.
Examples
of
the
Calculation
of
Fourier
Transforms.-
7.
Partial
Fourier
Transform.-
8.
Fourier
Transform
and
Automorphisms
of
?n:
Homogeneous
Distributions.-
8.1.
Fourier
Transform
and
Automorphisms
of
?n.-
8.2.
Homogeneous
Distributions.-
9.
Fourier
Transform
and
Convolution.
Spaces
OM(?n)
and
O’C(?n).-
9.1.
The
Space
OM(?n)
(
=
OM).-
9.2.
The
Space
O’C.-
10.
Fourier
Transform
of
Tempered
Measures.-
11.
Distribution
of
Positive
Type.
Bochner’s
Theorem.-
11.1.
Functions
of
Positive
Type.-
11.2.
Distributions
of
Positive
Type.-
12.
Schwartz’s
Theorem
of
Kernels.-
13.
Some
Distributions
and
Their
Fourier
Transforms.-
Table
of
Notations.-
of
Volumes
1,
3–6.