Brief Syllabus for MATH 674 (AMSC 674): Partial Differential Equations II

  • Prerequisite:
  • Practically, NO prerequisite. Typically, MATH/AMSC 673, or equivalent, or permission of the instructor.
    The course is largely self-contained. Some prior familiarity of students with
    functional analysis would be helpful but NOT required.
    Ask for instructor's permission if you are in doubt.
    Handouts with reviews will be given in class.

  • Focus:
  • Concepts and analytic techniques that permeate the rigorous theory of Partial Differential Equations (PDEs).
    Particular emphasis is placed on modern concepts and techniques
    of functional analysis for the solution of linear (primarily) and some nonlinear PDE.
    Two major theorems to be taught are the Lax-Milgram theorem and the Fredholm alternative,
    along with their applications in understanding solutions of a class of PDE.
    Also: Emphasis will be on WEAK SOLUTIONS of elliptic, parabolic and hyperbolic PDE,
    which form a broad class of solutions needed, e.g., in the use of numerical methods.
    If time permits, a special topic to be addressed will be the theory of nonlinear dispersive PDE.

  • Tentative list of topics:

  • Introduction (some background). Banach and Hilbert spaces. Compact operators and applications.
    Sobolev spaces and weak derivatives. Sobolev and Poincare's inequalities; compactness.
    Related Fourier transform methods. The space $H^{-1}$; gradient flows.

    2nd-order elliptic PDE. The concept of weak solutions. Existence: Lax-Milgram theorem; Fredholm alternative;
    energy estimates. Regularity. Eigenvalues and eigenfunctions.
    Applications: Steady-state heat flow; evaporation-condensation in materials science; fluid mechanics.
    A classic problem: ``can one hear the shape of a drum'' ?

    Linear evolution equations. 2nd-order parabolic PDE:
    Existence of weak solutions; regularity; maximum principles.
    Applications: diffusions; Fokker-Planck equations and connections to stochastic processes.

    2nd-order hyperbolic PDE: Existence of weak solutions;
    regularity; speed of propagation.
    Applications: wave equations; scattering and diffraction of waves.

    Introduction to semigroup theory. Elementary properties; contractions.
    Applications to parabolic and hyperbolic PDE.
    Applications: Stochastic processes with diffusion.

    Special topic: Theory of nonlinear dispersive PDE.
    Introduction to Strichartz estimates.
    The nonlinear Schroedinger equation (nonlinear optics, Bose-Einstein condensation); blowup of solutions.
    The KdV equation (water waves).