MATE 6677 Partial Differential Equations

Time and place: MJ 9:00--10:15, in M-118. On Friday Dec 17, we meet in M-118 at 7:30pm. The final exam is on Jan 15 at 11:00; the second exam is ready for you to collect.

Prerequisites: According to university policy, officially, none. In effect, the student is expected to have a good background in Advanced calculus and basic topology, an introductory course in Ordinary differential equations, and having taken, or taking concurrently, a course in Real analysis. For an introductory course emphasizing solution methods in a classical setting, the student is encouraged to take or audit MATE 6675.

Topics: We will mainly follow the plan of Folland:; Preliminaries: notation and definitions, some function spaces, convolution, Fourier transform, distributions.; Local existence theory: basic concepts, real first-order equations, the general Cauchy problem, Cauchy-Kovaleskaya theorem.; The Laplace operator: basic properties of harmonic functions, fundamental solution, Dirichlet and Neumann problems, Green's function, Dirichlet problem in a ball and half-space.; The heat operator: heat kernel, heat equation in bounded domains.; The wave operator: Cauchy problem, solutions in a half-space, inhomogeneous equation, the wave equation in bounded domains, Radon transform.; The L^2 theory of derivatives: Sobolev spaces on R^n, local regularity of elliptic operators, Sobolev spaces on bounded domains.; Some topics from integral equations or L^2 methods for elliptic boundary-value problems.

References: "Introduction to partial differential equations" by G. Folland, Princeton University Press, 1976.; Secondary sources:; "Partial differential equations" by L.C. Evans, AMS, 2000.; "Partial differential equations" by J. Rauch, Springer-Verlag, 1991.

Last modified: Thu Jan 13 15:29:54 AST 2011