Linear part:structure of function spaces, Sobolev spaces, embeddings, topologies, linear operators, adjoint and inverse operators, spectra, distributions, semigroup theory, integral equations, well-posedness and the notion of a solution. Nonlinear part: inequalities, Frechet and Gateaux derivatives, fixed point theorems. Applications from mechanics, reaction-diffusion equations, the Navier-Stokes equations, nonlinear Schrödinger equation. Prerequisite: MATH 438 or equivalent.
Application to problems in physics of method of steepest descent, Fourier and Laplace transforms; boundary-value problems, integral equations, and Green's functions. Prerequisites: MATH 311 or 411, and 337, or equivalents.