Partial Differential Equations. I
MATH/AMSC 673, Fall 2006
Course Information
Course Description (Preliminary)
Analysis of boundary value problems for Laplace's
equation. Initial value problems for the heat and wave equations.
Fundamental solutions, maximum principles, energy methods. First order nonlinear
PDEs. Conservation laws. Characteristics, shock formation, weak solutions.
Distributions, Fourier transform.

Introduction
Preliminaries
The divergence theorem; convolutions (approximate identities);
Fourier transform (series); distributions; The Laplacian operator: invariance under rotations and radial
representations
Classifications
Linear, quasilinear, nonlinear equations
Cauchy initial data, boundary conditions
Existence and wellposed problems: CauchyKovalevski: Lewy's counterexample
Constant coefficients: local solvability; Canonical
forms; Classification of 2nd order equations
Linear and nonlinear examples: The transport equation; The wave, heat and Laplace equations;
The nonlinear transport and HamiltonJacobi equations

Elliptic Equations
Properties of Harmonic functions
The fundamental solution; Perron's method; Wyel lemma and regularity
The Meanvalue property (MVP);
The maximum principle;
Regularity: Distributions and mollifiers; gain of regularity; analyticity; Harnack's inequalities
Assignment #1 [
pdf file ] with answers [ pdf file]
Dirichlet and Neumann problems
Representation of solutions by Green's function:
Half space Dirichlet problem  Poisson kernel;
Dirichlet problem in ball  spherical harmonics;
Potential theory (integral equations)
Lecture Notes: Dirichlet problem in a ball
[ pdf file ]
Representation of solutions by superposition:
separation of variables rectangular and circular geometries
Lecture Notes: Fourier and Chebyshev expansions
[ pdf file ]
Dirichlet principle  the weak and variational formulations:
Elliptic equations of 2nd order (*)
The maximum principle; Harnack inequality; Gain of regularity

Parabolic Equations
Where do they come from?
Diffusion; 2D averaging; random walk; Fourier law
The heat equation (Cauchy problem)
Derivation:
The 1D Gaussian: Fourier Transform; Detour: mollifiers
The multiD Gaussian: Fourier Transform; Rotation invariance
Irreversibility
The energy method
The heat equation (bounded domains)
Superposition: Separation of variables
The maximum principle
The energy method
Parabolic systems (*)
Dissipation
Fourier method
Energy decay (the energy method)
Assignment #2 [
pdf file ]
with answers [ pdf file ]

Hyperbolic Equations
Where do they come from?
Transport phenomena; speed of propagation
The linear transport equation (Cauchy problem)
Characteristics; The energy method; A maximum principle?
The wave equation
Solution by spherical means
Fourier methods
The energy method
Hyperbolic systems (*)
Dissipation
Fourier method
Energy decay (the energy method)

Nonlinear PDEs
Conservation laws
Characteristics; Shocks and entropy condition
Burgers' equation; existence and uniqueness
Lecture Notes: Nonlinear conservation laws
[ pdf file ]
HamiltonJacobi equations
Geometrical optics; Characteristics; Legendre transform;
Assignment #3 [
pdf file ]

Extensions
Weak vs. strong solutions
Sobolev spaces
Weak vs. strong derivatives
Sobolev embeddings and related inequalities
Compactness and existence of solutions
Textbooks
Lawrence C. Evans, Partial Differential Equations (Graduate Studies in Mathematics, V. 19), AMS
Robert McOwen, Partial Differential Equations, Methods and Applications
Gerald Folland Introduction to PDEs, Princeton University Press
TEXTBOOKS  mostly ELLIPTIC
D. Gilbarg and N. Trudinger, Elliptic PDEs of 2nd order, SpringerVerlag
Classical reference for elliptic PDEs with detailed proofs  more than required for our introductory course
Eitan Tadmor


