Time: Mondays, Wednesdays, Fridays, at 2:15pm.
Room: Building 380, room 381T.
Teacher: Y. A.
Rubinstein. Office hours: by appointment, in 382F.
Course plan:
This is the first course in the 256 sequence "Partial Differential
Equations." It is a firstyear 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 inclass presentation of a topic.
Schedule:

March 31
Overview/syllabus/references. Solving linear and semilinear firstorder
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  RankineHugoniot
jump condition.

April 12
Conservation laws  uniqueness of entropy solutions.
HW1.

April 16
HamiltonJacobi equations  HopfLax formula.

April 19
No class.

April 21
Motivation for the HopfLax formula and Hamiltonian formalism.
Properties of the HopfLax solution.

April 23
Properties of the HopfLax solution.
Application to the LaxOleinik entropy solution for conservation laws.

April 26
Hans Lewy's example of an equation with no local solution (Presentation
1).
HW2.

April 30
CauchyKowalevsky theorem.

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

May 7
Weak solutions of the homogeneous MongeAmpere equation (Alexandrov
theory).

May 10
Analytic definition of the MongeAmpere 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 HamiltonJacobi and second order elliptic PDE (Presentation 2).

May 19
Control theory and HamiltonJacobi 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 3space (Presentation 6).
