Partial Differential Equations

Partiella differentialekvationer, 4 poäng, 1TT462

Partiella differentialekvationer D, 6 poäng, 1MA294

Partial Differential Equations, 4 or 6 credit points

Hösten 2005     Fall Semester, 2005

Christer Kiselman

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.

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


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 ux = f, uy = g (one unknown function, two equations) is locally solvable if and only if fy = gx. Streamlines and stream functions of a planar flow. Local existence of a stream function for a divergence-free planar flow. Exercises 1.3 and 1.4 discussed. Exercises, sheet 2, distributed.

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 ut + uux = 0 as a model for breaking surface waves.

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 uy + (ux)2 = 0 with initial condition u(x,0) = 0; exercise problem 4.2; the equation uy = log ux with initial condition u(x,0) = log x. Exercises, sheet 4, distributed.

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 uxt = 0 with initial conditions on the line t = ax, with a zero or nonzero. Good and bad initial value problems. The wave equation in one space variable. The inhomogeneous wave equation: Duhamel's principle.

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 C0, 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 Rn+1. Exercises, sheet 9, distributed.

14. 2005-10-07. Final remarks about the heat equation in Rn+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.


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

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