Date 
Material covered 
Remarks 
Aug 25

Introduction, notation, differential operators.


Aug 27

Laplace's and Poisson's equations. Fundamental solution.
Properties of harmonic functions. Green's function.


Sep 1

The heat equation. Fundamental solution. Smoothing and infinite
propagation speed. Duhamel's principle.


Sep 3

The wave equation. D'Alembert's, Kirchhoff's, and Poisson's
formulas. Solutions for n even and odd. Domains of dependence and
influence.


Sep 8

Weak derivatives and Sobolev spaces: basic
properties.


Sep 10

Approximation by smooth functions: interior and uptothe
boundary approximation; the segment condition. Extensions.


Sep 15

The cone condition. The Sobolev embedding theorem. Sobolev's
inequality. 

Sep 17

More on Sobolev embedding: embedding into Holder spaces.


Sep 22

Canceled. To be rescheduled at a later time.


Sep 24

Canceled. To be rescheduled at a later time. 

Sep 29

Compact embedding. Traces. Fractional and negative order Sobolev
spaces.


Oct 1

Dualities. More on negative Sobolev
spaces.


Oct 6

On necessary and sufficient conditions for existence of
solutions to linear PDEs. A PDE that is not locally solvable at
the origin (Egorov's counterexample).


Oct 8

Canceled. To be rescheduled at a later time. 

Oct 13

Linear elliptic equations; existence and uniqueness of weak
solutions.


Oct 15 
Elliptic regularity.


Oct 20 
Maximum principles. Semilinear elliptic equations: the method
of sub and supersolutions. Applications: the uniformization
theorem for Riemann surfaces.


Oct 22 
Nonlinear elliptic equations. The continuity method and implicit
function theorem methods.


Oct 27 
Canceled. To be rescheduled at a later time. 

Oct 29 
Linear hyperbolic equations. Gronwall's lemma. Energy estimates
for firstorder symmetric hyperbolic systems.


Nov 3 
Existence and uniqueness for linear firstorder symmetric
hyperbolic systems.


Nov 5 
Existence and uniqueness for linear second order hyperbolic
systems (linear wave equations). 

Nov 10 
Local existence and uniqueness for quasilinear wave equations.
Preliminaries. Uniqueness.


Nov 12 
Local existence and uniqueness for quasilinear wave equations.
Energy estimates. 

Nov 17 
Local existence and uniqueness for quasilinear wave equations.
Existence. 

Nov 19 
Canceled. To be rescheduled at a later time. 

Nov 23  Nov 27 
Thanksgiving holidays, no classes.


Dec 1 
The role of the characteristics. Hyperbolicity.


Dec 3 
Einstein's equations: the basic setup.


Dec 7 (makeup class) 
The Einstein constraint equations. The meaning of the Cauchy
problem for Einstein's equations. Wave coordinates.


Dec 8 (makeup class) 
Local existence for the Cauchy problem for Einstein's equations.
Uniqueness: the maximal globally hyperbolic development of the
data.


Dec 9 (makeup class) 
The characteristic geometry of quasilinear wave equations.
Motivation: global existence and decay of solutions.


Dec 10 (makeup class) 
Null frames and eikonal functions.


Dec 11 (makeup class) 
The nullstructure equations.

