Lectures on linear partial differential equations

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Lectures on Linear Partial Differential Equations, GSM 123


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     © 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


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