Brief Syllabus for MATH 673 (AMSC 673): Partial Differential Equations I

  • Prerequisite:
  • Typically, MATH 411 (Advanced Calculus II), or equivalent. The course is largely self-contained; however, prior familiarity of students with analytic proofs will be very helpful. 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), especially PDEs that arise in applications across different disciplines. Emphasis will be placed on the existence, uniqueness and regularity of solutions, as well as special solution techniques (transforms, power series etc).

  • Tentative list of topics:
  • PART I:

    Introduction: Classical and weak solutions; regularity. Major linear PDEs.
    Transport equation: Initial-value problem; nonhomogeneous problem.
    Laplace and Poisson equations: Derivations; boundary value problems; fundamental solution; maximum principle; properties of harmonic functions; Green's function; energy methods.
    Heat equation: Derivations; initial value problems; fundamental solution; properties and estimates; energy methods.
    Wave equation: Derivations; initial value problems; d'Alembert formula; solution by spherical means; nonhomogeneous problem; energy methods.
    Applications in classical and quantum mechanics.

    PART II:

    Nonlinear first-order PDEs: Complete integrals; characteristics; calculus of variations and Hamilton-Jacobi equations; conservations laws; shock formation and entropy condition; weak solutions; the Riemann problem. Applications in gas dynamics, materials science and fluid mechanics.


    Special representations of solutions: Fourier transform; conversion of nonlinear to linear PDEs (Hopf-Cole transform, potential functions, hodograph and Legendre transforms). Applications in fluid mechanics, statistical physics and materials science.

    PART IV:

    Maximum principles for 2nd-order elliptic equations:
    Weak maximum principle; strong maximum principle; Harnack's inequality.