The main book to be used was * Partial Differential
Equations,* Springer-Verlag, by Fritz John (1910--1994). More
precisely, I proposed that the curriculum comprise the following parts
of this book:

Chapter 1: all of it;

Chapter 2: sections 1--5;

Chapter 3: sections 1, 2, 4, 6;

Chapter 4: all of it;

Chapter 5: sections 1--2;

Chapter 7: section 1;

Chapter 8: Hans Lewy's example.

The course was held in English because of the participation of some exchange students.

Some exercise problems were distributed, to be solved during the
course. The problems were distributed on paper in the classroom but
are also available on the web:

*Sheet No. 1* available as a DVI file and
as a PostScript file.

*Sheet No. 2* available as a DVI
file and as a PostScript file.

*Sheet No. 3* available as a DVI
file and as a PostScript file.

*Sheet No. 4* available as a DVI
file and as a PostScript file.

*Sheet No. 5* available as a DVI
file and as a PostScript file.

*Sheet No. 6* available as a DVI
file and as a PostScript file.

*Sheet No. 7* available as a DVI
file and as a PostScript file.

*Sheet No. 8* available as a DVI
file and as a PostScript file.

*Sheet No. 9* available as a DVI
file and as a PostScript file.

There is a * Word list * available as a DVI file and as a PostScript file. It contains translations
into Swedish, Esperanto, and French of terms used during the course.

The students were invited to lecture on specified topics. This was intended to make them more familiar with some of the interesting themes in the great theory of PDE that cannot be covered in detail in the course. Giving such a lecture counted as part of the examination. A list of possible topics was distributed on paper on January 26. The list is also available on the web as a DVI file and as a PostScript file. The list of lectures actually delivered by the students is also available.

The course started on January 19 and ended on March 13.

The program was as follows:

*January 19: * A survey of ordinary differential equations.

*January 21: * A survey of first order partial differential equations.
Characteristic curves. Excercise problems on the
existence solutions in different domains: 1.1, 1.2.

*January 23: * Exercise problems on characteristics, complete
description of solutions, blow-up of solutions: 1.3, 1.4, 2.1, 2.2, 2.4.

*January 26: * Every quasilinear equation corresponds to a linear
equation in one more variable. Exercise problems 3.1, 3.5

*January 28: * Waves as solutions to first order equations.
General nonlinear equations of the first order.
Envelopes of solutions.

*January 30: * Characteristic strips. Exercise problem 4.1 and
some other examples where both envelopes and characteristic strips can
be used for solving. Which method is best?

* February 2:* Solving 4.2 and 4.3 using the method of
characteristic strips. Second order equations: introduction. Cauchy's
problem.

* February 4:* Second order equations, cont'd.
Characteristics. Normal form. Well-posed and ill-posed Cauchy
problems. Finding the Taylor series of a solution. Exercise problem
5.1.

* February 6: * Well-posed and ill-posed problems for different
types of equations. Exercise problem 5.3.

* February 9: * Characteristic hypersurfaces (any dimension, any
order of the equation).

* February 10: * Domain of dependence, range of influence. The
wave equation with initial and boundary condition. Reflection of
waves. Discontinuities in a solution. The hodograph method (the
Legendre transformation).

* February 12: * Hyperbolic systems. Real analytic functions.

* February 16: * Real analytic functions. The Cauchy--Kovalevsky
theorem.

* February 18: * Lecture by Bengt Eliasson on * The equations of
fluid mechanics * (6). Proof of the Cauchy--Kovalevsky theorem. Green's
formula.

* February 20: * Lecture by Markus Jonsson on * Differential forms*.
(24). Fundamental solutions for the Laplacian.

* February 23: * Lecture by Peter Borg on * Soft solutions*.
(15). Green's function.

* February 25: * Lecture by Lars Axelson: * Why is the world
three-dimensional? * (36). Lecture by Martin Nilsson on * Wavelet
transforms * (10). Hypoellipticity of the Laplacian.

* March 2: * Lecture by Lawrence McCandless: * The Schrödinger
equation and its relations to the Korteweg--de Vries equation * (39).
Dirichlet's problem for the Laplacian. The Poisson kernel. Hilbert
space methods for the Laplacian.

* March 3: * Lecture by Karl Håkansson and Johan Edlund: *
Solitons and the Korteweg--de Vries equation * (1, 2, 3). Hilbert space
methods for the Laplacian (cont'd). Mean values of functions.

* March 4: * Darboux's equation for the mean values of functions.
Solving the wave equation in any number of variables. Explicit
solution in three variables. Hadamard's method of descent. Duhamel's
principle.

* March 6: * Lecture by Lars Berggren: * The Laplace
transformation * (11). Exercise problems 5.6 and 8.1. The energy
integral. Mixed problems for the wave equation. Eigenfunctions for
the Laplacian. Higher order hyperbolic equations.

* March 9: * Exercise problem 4.4. Gårding's hyperbolicity
condition. Solution using the Fourier transformation of initial-value
problems with a higher-order hyperbolic equation.

* March 13: * Lecture by Per Andersson: * Representation
formulas for solutions to the equation Laplacian(u) = u*. Lecture
by Roger Agenstam and Daniel: * Geometric interpretation of the
Hamilton--Jacoby equation*. Parabolic equations. Fundamental
solutions to the heat equation. The maximum principle for solutions to
the heat equation. Regularity of solutions to the heat
equation. Uniqueness if the solution is bounded from below.

The written examination took place on March 16, 1998. There were oral examinations afterwards. All results have now been reported to the university's computer system UPPDOK. Those who did not succeed can try again in June or August---the exact days are not known to me.

*Christer*

Last change 1998 04 18. Christer Kiselman, kiselman@math.uu.se

URL: http://www.math.uu.se/~cok