Book
Lecture Notes.
Partial Differential Equations
Gerald Teschl
Abstract
This manuscript provides a brief introduction to Partial Differential Equations. The first part is intended as a first introduction and
does neither require any functional analytic tools nor Lebesgue or Sobolev spaces.
The second part deals with advanced techniques from functional analysis.
MSC: 3501
Keywords: Method of characteristics, CauchyKovalevskaya theorem, Separation of variables, Fourier transform, Dispersion, Laplace equation, Harmonic functions, Dirichlet problem, Heat equation, Wave equation
Sobolev spaces, Elliptic regularity, Operator semigroups, Reaction diffusion equations, Calculus of variations, Nonlinear Schrödinger equation, Strichartz estimates
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Table of contents

Preface
 First order PDE
 The method of characteristics
 Semilinear equations
 Quasilinear equations
 Fully nonlinear equations
 Classification and canonical forms
 The CauchyKovalevskaya theorem
 First order systems
 Second order equations
 Separation of variables
 The heat equation for a thin rod
 Outlook: The reaction diffusion equation
 The wave equation for a string
 The wave equation on a rectangle and on a disc
 The Laplace equation on a disc
 The Fourier transform and problems on the line
 Motivation
 The Fourier transform in one dimension
 The heat equation on the line
 The wave equation on the line
 Dispersion
 Symmetry groups
 The Laplace equation
 Harmonic functions
 Subharmonic functions
 The Newton potential and the Poisson equation on R^{n}
 Poisson equation on a domain and Green's function
 The Dirichlet principle
 Solution for a half space and for a ball
 The Perron method for solving the Dirichlet problem
 General elliptic equations
 The heat equation
 The Fourier transform
 The fundamental solution
 The heat equation on a bounded domain and the maximum principle
 Energy methods
 General parabolic equations
 The wave equation
 Solution via the Fourier transform
 Solution in arbitrary dimensions
 Energy methods
 A first look at weak derivatives and L^{2} based Sobolev spaces
 Motivation
 The Fourier transform on L^{2}
 The Sobolev spaces H^{r}(R^{n})
 Evolution problems
 General Sobolev spaces
 Basic properties
 Extension and trace operators
 Embedding theorems
 Elliptic equations
 The Poisson equation
 Elliptic equations
 Elliptic regularity
 The Poisson equation in C(U)
 Operator semigroups
 Single variable calculus in Banach spaces
 Uniformly continuous operator groups
 Strongly continuous semigroups
 Generator theorems
 Applications to parabolic equations
 Applications to hyperbolic equations
 Nonlinear equations
 Semilinear equations
 Reaction diffusion equations
 Calculus of Variations
 Differentiation in Banach spaces
 The direct method
 Constraints
 The nonlinear Schrödinger equation
 Local wellposedness in H^{r} for r>n/2
 Global solutions and blowup in H^{1}
 Strichartz estimates
 Wellposedness in L^{2} and H^{1}
 Standing waves
 Appendix: Radial Sobolev spaces
 Appendix: Calculus facts
 Differentiation
 Integration
 Fourier series
 Appendix: Real and functional analysis
 Differentiable and Hölder continuous functions
 Lebesgue spaces
 Closed operators
 The Bochner integral
Part 1: Classical Partial Differential Equations
Part 2: Advanced Partial Differential Equations
Part 3: Appendices
Glossary of notations
Index