640:501 or permission of instructor.
This is the first half of a year-long introductory graduate course on PDEs, and should be useful for students with a variety of research interests: physics and mathematical physics, applied analysis, numerical analysis, differential geometry, complex analysis, and, of course, partial differential equations. The beginning weeks of the course aim to develop enough familiarity and experience with the basic phenomena, approaches, and methods in solving initial/boundary value problems in the contexts of the classical prototype linear PDEs of constant coefficients: the Laplace equation, the D'Alembert wave equation, the heat equation and the Schroedinger equation. A variety of tools and methods, such as Fourier series/eigenfunction expansions, Fourier transforms, energy methods, and maximum principles will be introduced. More importantly, appropriate methods are introduced for the purpose of establishing quantitative as well as qualitative characteris tic properties of solutions to each class of equations. It is these properties that we will focus on later in extending our beginning theories to more general situations, such as variable coefficient equations and nonlinear equations.
Next we will discuss some notions and results that are relevant in treating general PDEs: characteristics, non-characteristic Cauchy problems and Cauchy-Kowalevski theorem, Hadamard-Petrovsky wellposedness criteria.
Towards the end of the semester, we will begin some introductory discussion on the extension of the energy methods to variable coefficient wave/heat equations, and/or the Dirichlet principle in the calculus of variations.
The purpose here is to motivate and introduce the notion of weak solutions and Sobolev spaces, which will be more fully developed in the second semester.