Partial Differential Equations (course 256A)

Stanford University

Department of Mathematics

Spring 2010






Time: Mondays, Wednesdays, Fridays, at 2:15pm.
Room: Building 380, room 381-T.

Teacher: Y. A. Rubinstein. Office hours: by appointment, in 382-F.

Course plan:
This is the first course in the 256 sequence "Partial Differential Equations." It is a first-year graduate level course on PDE assuming essentially no previous encounter with the subject aside from familiarity with Laplace's equation as discussed in my Complex Analysis class last Autumn. The main goal of the course will be to gain a familiarity with a range of partial differential equations occuring naturally in mathematical physics, differential geometry and other areas of mathematics and science, as well as with the methods used to analyze them.

Main references:
L. C. Evans, Partial Differential Equations (Second Edition).
L. Simon, lecture notes for PDE 256A, available here.


Additional references:
G. B. Folland, Introduction to PDE (Second Edition).
Further references will be given as we go along.



Assignments:
There will be some homework assignments (not more than five) during the course, and no exams. The only formal requirement for a grade will be an in-class presentation of a topic.

Schedule:

  • March 31
    Overview/syllabus/references. Solving linear and semi-linear first-order equations by integration along curves.

  • April 2
    The method of characteristics - derivation of the characteristic equations.

  • April 5
    The method of characteristics - the compatibility condition and existence of local characteristic curves.

  • April 7
    The method of characteristics - obtaining a local solution.

  • April 9
    First order equations of conservation laws - Rankine-Hugoniot jump condition.

  • April 12
    Conservation laws - uniqueness of entropy solutions.

    HW1.

  • April 16
    Hamilton-Jacobi equations - Hopf-Lax formula.

  • April 19
    No class.

  • April 21
    Motivation for the Hopf-Lax formula and Hamiltonian formalism. Properties of the Hopf-Lax solution.

  • April 23
    Properties of the Hopf-Lax solution. Application to the Lax-Oleinik entropy solution for conservation laws.

  • April 26
    Hans Lewy's example of an equation with no local solution (Presentation 1).

    HW2.

  • April 30
    Cauchy-Kowalevsky theorem.

  • May 3
    Isometric embedding of surfaces in 3-space: the equation for curvature. Constructing flat embedded surfaces as a Dirichlet problem for the homogeneous Monge-Ampere equation. The Monge-Ampere operator.

  • May 7
    Weak solutions of the homogeneous Monge-Ampere equation (Alexandrov theory).

  • May 10
    Analytic definition of the Monge-Ampere operator.

  • May 12
    Regularity results for fully nonlinear second order elliptic equations.

  • May 14
    Schauder estimates for elliptic second order equations.

    HW3.

  • May 17
    Viscosity solutions for Hamilton-Jacobi and second order elliptic PDE (Presentation 2).

  • May 19
    Control theory and Hamilton-Jacobi equations (Presentation 3).

    HW4.

  • May 21
    Systems of conservation laws - introduction and Riemann's problem (Presentation 5).

  • May 24
    Systems of two conservation laws and Riemann invariants for first order hyperbolic systems.

    HW5.

  • May 28
    No class.

  • May 31
    No class.

  • June 2
    Entropy criteria for conservation laws (Presentation 4).

  • June 4
    Some aspects of analysis of isometric embeddings of surfaces in 3-space (Presentation 6).