Introduction to PDE, AIMS (Senegal) March 14-April1, 20016

Introduction to Partial Differential Equations

Almut Burchard (University of Toronto)

Outline: This is a basic introduction to partial differential equations as they arise in physics, geometry and optimization. It is meant to be accessible to beginners with little or no prior knowledge of the topic, while offering students of all levels an opportunity to enlarge their bag of tools and sharpen their problem-solving skills. To give just a taste of the breadth of the field, we will discuss a variety of ideas and techniques, for different types of equations. Non-smooth solutions are a recurring theme: Why do we need them, what do they mean for the underlying physical or geometric problem, what challenges do they present, and what methods are available for studying them?
Sources: L. C. Evans Chapters 2 & 3, F. John, W. Strauss

Tentative schedule:

("*" denotes possible alternative topics)
Week 1 (March 14-18)
M  PDE as physical models. Well-posed problems
T   The transport equation: method of characteristics
W   Burger's equation: shocks and entropy solutions
R   Second-order linear equations in two variables: non-characteristic data and characteristic curves
F   The wave equation in one space dimension: energy, causality
*   The Cauchy-Riemann system
*   The Cauchy-Kovalevskaya existence theorem
Week 2 (March 21-25)
M   Poisson's equation: fundamental solution
T   Harmonic functions: mean value property, maximum principle
W   Harnack's inequality and regularity. Green's functions
R   Variational methods: the Dirichlet problem
*   Min-max characterization of eigenvalues. Weyl's law
F   The diffusion equation: fundamental solution, maximum principle
*   Distributions and weak solutions. Sobolev spaces
Week 3 (March 28-April 1)
M   The wave equation in odd dimensions: method of spherical means
T   The wave equation in even dimensions: Hadamard's method of descent
*   Schrödinger's equation
W   Hilbert space methods: orthonormal bases and Fourier series
*   Spherical harmonics
R   The Fourier integral transform
F   Evolution equations: Duhamel's formula and Picard iteration