This course builds on Math 16:640:517 Partial Differential Equations I, although that more basic course is not an absolute prerequisite. The course provides an introduction to existence and regularity theory for solutions to boundary value problems for linear second-order elliptic (and parabolic) partial differential equations. The primary goal of the course is to provide students with a solid understanding of the Schauder and L^p (Calderon-Zygmund) a priori estimates, as those are vitally important for the study of nonlinear partial differential equations arising in all areas of pure and applied mathematics. We shall quickly review Holder and Sobolev spaces, the Sobolev and Rellich embedding theorems, maximum principles, and L^2 existence and regularity theory for elliptic equations normally covered in Math 16:640:517. Depending on student interest and available time, we shall also provide brief introductions to one or more of the following topics: s ystems of elliptic partial differential equations, harmonic analysis, partial differential equations on Riemannian manifolds, pseudo-differential operators, singular integral operators, and viscosity solutions. In addition to the primary textbook, "Elliptic partial differential equations of second order" by Gilbarg and Trudinger, supplementary textbooks that students may find useful include: (1) "Lectures on elliptic and parabolic equations in Hölder (Sobolev) spaces" by N. V. Krylov, (2) "Functional analysis" by H. Brezis, (3) "Sobolev spaces" by R. A. Adams and J. J. Fournier, (4) "Second order parabolic differential equations" by G. M. Lieberman, (5) "Invariance theory, the heat equation, and the Atiyah-Singer index theorem" by P. B. Gilkey, (6) "Some nonlinear problems in Riemannian geometry", by T. Aubin, (7) "Lectures on elliptic boundary value problems" by S. Agmon.
"Elliptic partial differential equations of second order" by D. Gilbarg and N. S. Trudinger
Math 16:640:517 or permission of instructor