This book is a reader-friendly, relatively short introduction to the modern theory of linear partial differential equations. An effort has been made to present complete proofs in an accessible and self-contained form.
The first three chapters are on elementary distribution theory and Sobolev spaces with many examples and applications to equations with constant coefficients. The following chapters study the Cauchy problem for parabolic and hyperbolic equations, boundary value problems for elliptic equations, heat trace asymptotics, and scattering theory. The book also covers microlocal analysis, including the theory of pseudodifferential and Fourier integral operators, and the propagation of singularities for operators of real principal type. Among the more advanced topics are the global theory of Fourier integral operators and the geometric optics construction in the large, the Atiyah-Singer index theorem in Rn, and the oblique derivative problem.
Readership: Graduate students and research mathematicians interested in partial differential equations.
Author(s): Gregory Eskin
Series: Graduate Studies in Mathematics 123
Publisher: American Mathematical Society
Year: 2011
Language: English
Pages: C+xviii+410+B
Cover
S Title
Lectures on Linear Partial Differential Equations, GSM 123
Copyright
© 2011 by the American Mathematical Society
ISBN 978-0-8218-5284-2
QA372.E78 2011 515'.3533-dc22
LCCN 2010048243
Dedicated to Michael Eskin
Contents
Preface
Acknowledgments
Chapter I Theory of Distributions
Introduction to Chapters I, II, III
1. Spaces of infinitely differentiable functions
1.1. Properties of the convolution
1.2. Approximation by Col-functions.
1.3. Proof of Proposition 1.1.
1.4. Proof of property b) of the convolution
2. Definition of a distribution
2.1. Examples of distributions
2.2. Regular functionals.
2.3. Distributions in a domain
3. Operations with distributions
3.1. Derivative of a distribution
3.2. Multiplication of a distribution by a C°°-function
3.3. Change of variables for distributions.
4. Convergence of distributions
4.1. Delta-like sequences
5. Regularizations of nonintegrable functions
5.1. Regularization in R^1.
5.2. Regularization in R^n.
6. Supports of distributions
6.1. General form of a distribution with support at 0.
6.2. Distributions with compact supports
7. The convolution of distributions
7.1. Convolution of f in D' and $\phi$ in Co
7.2. Convolution of f in D' and g in E'.
7.3. Direct product of distributions
7.4. Partial hypoellipticity
8. Problems
Chapter II Fourier Transforms
9. Tempered distributions
9.1. General form of a tempered distribution.
10. Fourier transforms of tempered distributions
10.1. Fourier transforms of functions in S.
10.2. Fourier transform of tempered distributi
10.3. Generalization of Liouville's theorem
11. Fourier transforms of distributions with compact supports
12. Fourier transforms of convolutions
13. Sobolev spaces
13.1. Density of Co (R^n) in Hs (R^n).
13.2. Multiplication by a(x) in S.
13.3. Sobolev's embedding theorem
13.4. An equivalent norm for noninteger
13.5. Restrictions to hyperplanes (traces)
13.6. Duality of Sobolev spaces.
13.7. Invariance of Hs(R^n) under changes of variables
14. Singular supports and wave front sets of distributions
14.1. Products of distributions
14.2. Restrictions of distributions to a surface
15. Problems
Chapter III Applications of Distributions to Partial Differential Equations
16. Partial differential equations with constant coefficients
16.1. The heat equation
16.2. The Schrodinger equation
16.3. The wave equation
16.4. Fundamental solutions for the wave equations
16.5. The Laplace equation
16.6. The reduced wave equation
16.7. Faddeev's fundamental solutions for (-\Delta - k^2).
17. Existence of a fundamental solution
18. Hypoelliptic equations
18.1. Characterization of hypoelliptic polynomials
18.2. Examples of hypoelliptic operators
19. The radiation conditions
19.1. The Helmholtz equation in R^3.
19.2. Radiation conditions
19.3. The stationary phase lemma
19.4. Radiation conditions for n > 2.
19.5. The limiting amplitude principle
20. Single and double layer potentials
20.1. Limiting values of double layers potentials
20.2. Limiting values of normal derivatives of single layer potentials
21. Problems
Chapter IV Second Order Elliptic Equations in Bounded Domains
Introduction to Chapter IV
22. Sobolev spaces in domains with smooth boundaries
22.1. The spaces Hs(\Omega) and Hs(\Omega).
22.2. Equivalent norm in Hm(\Omega) .
22.3. The density of Co in Hs(\Omega).
22.4. Restrictions to $\partial$\Omega
22.5. Duality of Sobolev spaces in \Omega
23. Dirichlet problem for second order elliptic PDEs
23.1. The main inequality
23.2. Uniqueness and existence theorem in H1(\Omega).
23.3. Nonhomogeneous Dirichlet problem
24. Regularity of solutions for elliptic equations
24.1. Interior regularity
24.2. Boundary regularity
25. Variational approach. The Neumann problem
25.1. Weak solution of the Neumann problem
25.2. Regularity of weak solution of the Neumann problem
26. Boundary value problems with distribution boundary data
26.1. Partial hypoellipticity property of elliptic equations
26.2. Applications to nonhomogeneous Dirichlet and Neumann problems
27. Variational inequalities
27.1. Minimization of a quadratic functional on a convex set
27.2. Characterization of the minimum point
28. Problems
Chapter V Scattering Theory
Introduction to Chapter V
29. Agmon's estimates
30. Nonhomogeneous Schrodinger equation
30.1. The case of q(x)
30.2. Asymptotic behavior of outgoing solutions (the case of q(x)
30.3. The case of q(x)
31. The uniqueness of outgoing solutions
31.1. Absence of discrete spectrum for k^2 > 0.
31.2. Existence of outgoing fundamental solution (the case of q(x)
32. The limiting absorption principle
33. The scattering problem
33.1. The scattering problem (the case of q(x) =
33.2. Inverse scattering problem (the case of q(x) =
33.3. The scattering problem (the case of q(x)
33.4. Generalized distorted plane waves.
33.5. Generalized scattering amplitude
34. Inverse boundary value problem
34.1. Electrical impedance tomography
35. Equivalence of inverse BVP and inverse scattering
36. Scattering by obstacles
36.1. The case of the Neumann conditions.
36.2. Inverse obstacle problem
37. Inverse scattering at a fixed energy
37.1. Relation between the scattering amplitude and the Faddeev's scattering amplitudes
37.2. Analytic continuation of Tr
37.3. The limiting values of T, and Faddeev's scattering amplitude.
37.4. Final step: The recovery of q(x).
38. Inverse backscattering
38.1. The case of real-valued potentials
39. Problems
Chapter VI Pseudo differential Operators
Introduction to Chapter VI
40. Boundedness and composition of $\psi$do's
40.1. The boundedness theorem
40.2. Composition of $\psi$do's
41. Elliptic operators and parametrices
41.1. Parametrix for a strongly elliptic operator.
41.2. The existence and uniqueness theorem.
41.3. Elliptic regularity.
42. Compactness and the Fredholm property
42.1. Compact operators
42.2. Fredholm operators
42.3. Fredholm elliptic operators in R^n
43. The adjoint of a pseudo differential operator
43.1. A general form of $\psi$do's
43.2. The adjoint operator
43.3. Weyl's $\psi$do's
44. Pseudolocal property and microlocal regularity
44.1. The Schwartz kernel
44.2. Pseudolocal property of $\psi$do's.
44.3. Microlocal regularity
45. Change-of-variables formula for $\psi$do's
46. The Cauchy problem for parabolic equations
46.1. Parabolic $\psi$do's.
46.2. The Cauchy problem with zero initial conditions
46.3. The Cauchy problem with nonzero initial conditions
47. The heat kernel
47.1. Solving the Cauchy problem by Fourier-Laplace transform
47.2. Asymptotics of the heat kernel as t--> +0.
48. The Cauchy problem for strictly hyperbolic equations
48.1. The main estimate.
48.2. Uniqueness and parabolic regularization
48.3. The Cauchy problem on a finite time interval
48.4. Strictly hyperbolic equations of second order.
49. Domain of dependence
50. Propagation of singularities
50.1. The null-bicharacteristics
50.2. Operators of real principal type
50.3. Propagation of singularities for operators of real principal type.
50.4. Propagation of singularities in the case of a hyperbolic Cauchy problem
51. Problems
Chapter VII Elliptic Boundary Value Problems and Parametrices
Introduction to Chapter VII
52. Pseudo differential operators on a manifold
52.1. Manifolds and vector bundles
52.2. Definition of a pseudo differential operator on a manifold
53. Boundary value problems in the half-space
53.1. Factorization of an elliptic symbol
53.2. Explicit solution of the boundary value problem
54. Elliptic boundary value problems in a bounded domain
54.1. The method of "freezing" coefficients
54.2. The Fredholm property
54.3. Invariant form of the ellipticity of boundary conditions
54.4. Boundary value problems for elliptic systems of differential equations
55. Parametrices for elliptic boundary value problems
55.1. Plus-operators and minus-operators
55.2. Construction of the parametrix in the half-space
55.3. Parametrix in a bounded domain
56. The heat trace asymptotics
56.1. The existence and the estimates of the resolvent
56.2. The parametrix construction
56.3. The heat trace for the Dirichlet Laplacian
56.4. The heat trace for the Neumann Laplacian
56.5. The heat trace for the elliptic operator of an arbitrary order
57. Parametrix for the Dirichlet-to-Neumann operator
57.1. Construction of the parametrix
57.2. Determination of the metric on the boundary
58. Spectral theory of elliptic operators
58.1. The nonselfadjoint case.
58.2. Trace class operators
58.3. The selfadjoint case
58.4. The case of a compact manifold.
59. The index of elliptic operators in R^n
59.1. Properties of Fredholm operators.
59.2. Index of an elliptic $\psi$do.
59.3. Fredholm elliptic $\psi$do's in R^n
59.4. Elements of K-theory.
59.5. Proof of the index theorem.
60. Problems
Chapter VIII Fourier Integral Operators
Introduction to Chapter VIII
61. Boundedness of Fourier integral operators (FIO's)
61.1. The definition of a FIO.
61.2. The boundedness of FIO's.
61.3. Canonical transformations
62. Operations with Fourier integral operators
62.1. The stationary phase lemma
62.2. Composition of a Odo and a FIO.
62.3. Elliptic FIO's
62.4. Egorov's theorem
63. The wave front set of Fourier integral operators
64. Parametrix for the hyperbolic Cauchy problem
64.1. Asymptotic expansion
64.2. Solution of the eikonal equation
64.3. Solution of the transport equation
64.4. Propagation of singularities
65. Global Fourier integral operators
65.1. Lagrangian manifolds
65.2. FIO's with nondegenerate phase functions
65.3. Local coordinates for a graph of a canonical transformation
65.4. Definition of a global FIO.
65.5. Construction of a global FIO given a global canonical transformation
65.6. Composition of global FIO's
65.7. Conjugation by a global FIO and the boundedness theorem
66. Geometric optics at large
66.1. Generating functions and the Legendre transforms
66.2. Asymptotic solutions
66.3. The Maslov index.
67. Oblique derivative problem
67.1. Reduction to the boundary
67.2. Formulation of the oblique derivative problem
67.3. Model problem
67.4. First order differential equations with symbols depending on x'.
67.5. The boundary value problem on $\partial$\Omega
68. Problems
Bibliography
Index
Back Cover
