Page established 2005-07-12. Latest update 2005-11-07.

**Mål (4 poäng)**

Kursen avser att
fördjupa kunskapen om teorin för hyperboliska, paraboliska och
elliptiska partiella differentialekvationer i anslutning till
fysikaliska problem. Huvudtema är välställdhet av olika begynnelse-
och/eller randvärdesproblem, samt egenskaper hos lösningar till
vågekvationen, värmeledningsekvationen och Laplace ekvation.

**Mål (6 poäng)**

Kursen avser
att utveckla teorin för hyperboliska, paraboliska och elliptiska
partiella differentialekvationer i anslutning till fysikaliska
problem. Huvudtema är välställdhet av olika begynnelse- och/eller
randvärdesproblem, samt egenskaper hos lösningar till vågekvationen,
värmeledningsekvationen och Laplace ekvation.

** Goals:** The course aims at developing the theory for
hyperbolic, parabolic, and elliptic partial differential equations in
connection with physical problems. Main themes are well-posedness of
various initial-value or boundary-value problems, as well as
properties of solutions to the wave equation, the heat equation, and
the Laplace equation.

**Kursernas innehåll **

Karakteristikor. Klassificering av andra ordningens ekvationer.
Maximumprincipen. Sobolevrum. Linjära elliptiska ekvationer.
Energimetoder för Cauchyproblem (paraboliska och hyperboliska
ekvationer). Fredholmteori och
egenfunktionsutveckling. Potentialteori.

**Contents:** Characteristics. Classification of
second order equations. The maximum principle. Sobolev spaces. Linear
elliptic equations. Energy methods for the Cauchy problem (parabolic and
hyperbolic equations). Fredholm theory and eigenfunction expansions.
Potential theory.

**Examinationsform (4 poäng)**

Skriftligt prov med problem och teoriuppgifter vid kursens
slut. Muntlig examination kan dessutom förekomma. Inlämningsuppgifter
kan förekomma under kursen.

**Examinationsform (6 poäng)**

Ett skriftligt och i allmänhet ett muntligt prov ges vid kursens
slut. Dessutom förekommer obligatoriska inlämningsuppgifter eller ett
teoriprov som redovisas i skriftlig och/eller muntlig form.

Deltagarna förväntas utföra ett projektarbete.

**Kurslitteratur**

Robert
C. McOwen. * Partial differential equations: methods and
applications.* 2nd Edition, Pearson Education, Inc., 2003.

Alternativt:

Lawrence C. Evans, * Partial Differential
Equations,* American Mathematical Society, Providence, RI (1998),
Graduate Studies in Mathematics, Vol. 19.

Boken av McOwen
rekommenderas för kursen under 2005.

**Material for study**

- There are ten sheets of exercise problems available.
- For those who take the six-point course there is a list of proposals for projects available.
- A note on Volumes, areas, and masses is available. It collects formulas on the volume of balls and areas of spheres as well as on the Riesz mass of subharmonic functions.
- A list of terms in English, Swedish, Esperanto, French, and German is available.

** Möten**

** 1. 2005-08-29. ** The first meeting. Introduction. The
participants' expectations. Goals and structure of the course.
Comparison between derivatives and difference quotients; between
integrals and sums. Review of ordinary differential equations
(existence, uniqueness and non-uniqueness). Their significance for
the study of first-order partial differential equations (McOwen
1.1). Differential equations of different orders (orders one through
four appear in applications). Characteristic curves for a linear
equation. Formulation of the Cauchy problem for a first order
equation. Exercises, sheet 1, distributed.

** 2. 2005-08-30. ** Examples of difficulties when the Cauchy
problem is characteristic. The noncharacteristic Cauchy problem for a
linear equation of the first order: precise formulation and idea of
proof. Overdetermined systems: the system *u _{x}* =

** 3. 2005-08-31. ** The noncharacteristic Cauchy problem for a
linear equation of the first order: proof. Quasilinear equations and
their relation to linear equations in one more variable. (Hadamard's
lemma is needed but was not proved.) Exercises, sheet 3, distributed.
Proposals for projects distributed.

** 4. 2005-09-05. ** Proof of Hadamard's lemma. Examples of
quasilinear equations. Methods for quasilinear equations can be used
also for linear equations, for every linear equation is quasilinear.
Discussion of exercises: 1.1, 2.1.c, 2.1.d. Waves of constant shape
and speed:
*u*(*x,t*) = *f*(*x – ct*). Waves
of variable speed: the inviscid Burger equation
*u _{t}* +

** 5. 2005-09-07. ** General nonlinear partial differential
equations of the first order. Definition of the envelope of a family
of curves; of a family of surfaces. The envelope of a family of
graphs of solutions to a first order equation is also the graph of a
solution. Three examples where this idea can be used: the equation
*u _{y}* + (

** 6. 2005-09-08. ** Strips, characteristic strips, and integral
strips. Examples of solving first-order differential equations using
the method of characteristic strips and the method of envelopes:
exercise problems 4.1 and 4.4. Exercises, sheets 5 and 6,
distributed.

** 7. 2005-09-26. ** Second order equations. Survey: the
Laplace and Poisson equations; the wave equation; the heat equation.
The equation *u _{xt}* = 0 with initial conditions on the line

** 8. 2005-09-27. ** Spherical means. Solution of the wave
equation in three space variables using radial functions. Solution of
the wave equation in two space variables using Hadamard's method of
descent. A note on *Volumes, areas, and
masses* is distributed.

** 9. 2005-09-30. ** Huygens' principle. Conservation of energy
in the wave equation. Exponential solutions to partial differential
equations with constant coefficients. Dispersion of waves.
Dissipation of waves. Starting a new chapter: The Laplace equation.
Green's formula.

** 10. 2005-10-03. ** The Dirichlet and Neumann problems for the
Laplace equation. Uniqueness in these problems. The convolution
algebra *C*_{0}, which does not have a unit element. The
Titchmarsh support theorem. A fundamental solution defines a right
inverse to a differential operator. Construction of a radial
fundamental solution to the Laplacian in any dimension. Exercises,
sheet 7, distributed.

** 11. 2005-10-04 10:15–12:00, 2245. ** The smoothness of
the fundamental solution implies C-infinity smoothness of harmonic
functions—even analyticity. The Cauchy problem is unsolvable for
general smooth data. Green's function in a bounded domain. The
Poisson kernel for a ball. Exercises, sheet 8,
distributed.

** 12. 2005-10-05. ** The Poisson kernel for a half space.
Continuous subharmonic functions defined by means of the mean-value
inequality. Perron's method. Proof that the supremum of all
subharmonic minorants is harmonic if it is continuous.

** 13. 2005-10-06 13:15–15:00, 2214. ** Proof that the
supremum of all subharmonic minorants is actually continuous. The
Perron method works for strictly convex domains and some other
domains.

The heat equation in
**R**^{n+1}. Exercises, sheet 9, distributed.

** 14. 2005-10-07. ** Final remarks about the heat equation in
**R**^{n+1}. Non-uniqueness. Weak solutions:
definition.

** 15. 2005-10-10. ** Regularity of solutions to the heat
equation: smooth in all variables, analytic in the space variables.
Weak solutions: examples. Weak solutions to the inhomogeneous heat
equation. The maximumum principle for the heat equation. Uniqueness
in bounded domains resulting from the maximum principle. Uniqueness in
unbounded domains under extra hypotheses. Exercises, sheet 10,
distributed.

** 16. 2005-10-11. ** Existence of solutions to the heat
equation in bounded domains. Hyperbolic equations of higher order.
Reduction to a standard problem, and further reduction to an ordinary
differential equation.

** 17. 2005-10-12. ** Estimates for an ordinary differential
equation. Gårding's condition for hyperbolicity. Ordinary
differential equations can be reduced to systems of the first order.
Similarly for partial differential equations. Hyperbolic and strictly
hyperbolic systems of the first order.

** 18. 2005-10-14. ** Pontus Leitz
presents project 39, * Why is the world is three-dimensional?*
Exercise problems 2.4 and 3.1 discussed.

** 19. 2005-10-18. ** Mattias
Moëll presents project 8, * Equations from combustion
theory.* Exercise problem 3.6. To measure regularity: Hölder
spaces.

** 20. 2005-10-19. ** Ole Andersson
presents project 4, * Minimal surfaces.* Final remarks about
Hölder spaces. Remarks about normed spaces and Banach spaces.

** 21. 2005-10-20. ** Katharina
Kormann and Martin Kronbichler
present projects 6, * Equations from fluid mechanics,* and 7, *
The Navier–Stokes equation and related equations*.
Motivation for introducing Sobolev spaces.

** 22. 2005-10-21. ** Michael Koller
presents project 17, * Similarity analysis.*
Virginie Konlack presents project 38, * Maximum principles
for parabolic equations. *

** 23. 2005-10-28. ** Klas
Pettersson and David Österberg
present projects 27, * Differential forms,* and 28, * Pfaffian
differential equations.* Sobolev spaces. Extra session with
discussions about some of the excercise problems.

** 24. 2005-10-31. ** Stefanie
Mahlberg presents project 9, * Laws of fluid mechanics.*
Sobolev spaces of higer order. Sobolev spaces of periodic functions
and their definition in terms of Fourier coefficients.

In addition to the ten oral presentations mentioned above, Christer Modin has made a written presentation of
project 33, * Compact operators and spectrum.*

** 25. 2005-11-01 08:00—13:00. ** Written examination. Five
persons wrote the four-point-course; eight the six-point-course. All
thirteen have passed. At the next opportunity, 2006-01-18, nobody turned up.

* Christer *

A quotation from the students' evaluation: "Det märks att han tycker om studenter."

Last update 2006-04-08. Back to Kiselman's home page.