Elliptic Partial Differential Equations and Harmonic Function Theory

(Reading course, 7.5hp, Fall 2015 )

Jonatan Lenells, jlenells@kth.se,         

Henrik Shahgholian, henriksh@kth.se

First meeting: Tuesday 13:15-15:00, September 1, room 3721 Lindstedtsvägen 25, Math building.

Tuesday 13:15-15:00,  room 3721.

The target group of this reading course are students with a common interest to learn (intermediate)  elliptic partial differential equations (PDEs) and harmonic function theory, and several other aspects which are usually not taught in standard courses. The focus will be on various methods, tools, and ideas that are used by mathematicians working with partial differential equations.

Lectures: At the beginning of the semester, each student will have the opportunity to choose a topic (see suggested list of topics below). Each student will then be assigned a two-hour class period at which he/she is asked to present this topic. In connection with the presentation, the student is also asked to write a 3-4 page summary of the topic and to prepare three homework problems, with solutions, for the other students.

The first four lectures of the class will be delivered by the instructors and will consist of an introduction to the subject.

Examination: In order to achieve a passing grade, the student must fulfill the following objectives:

1. Choose and present a topic.
2. Prepare three homework problems, with solutions, within the chosen topic. 3. Successfully solve at least 50% of the homework problems suggested by the other participants throughout the semester.

Lectures: Tuesday 13:15-15:00, (start September 1), room 3733 Lindstedtsvägen 25, Math building.

Prerequisites: Good knowledge of basic Analysis; some introductory PDE course at the undergraduate level.

Target group: PhD students in Mathematics and related disciplines with an in- terest in partial differential equations. Advanced undergraduate students planning to pursue doctoral studies in Mathematics.

List of suggested topics:

1. 1.Maximum/comparison principle, Hopf’s lemma.
   

2. 2.Harnack’s inequality, boundary Harnack.
   

3. 3.Poisson kernel, Harmonic measure.
   

4. 4.Caratheodory’s theorem, Koebe quarter theorem.
   

5. 5.Potential theory, Wiener’s solution of the Dirichlet problem.
   

6. 6.Fundamental & Green’s function, Green’s integral identities.
   

7. 7.Elliptic estimates, Alexandroff’s estimates.
   

8. 8.Barriers, regularity up to the boundary.
   

9. 9.Sobolev spaces: Weak and strong convergence in function spaces, imbedding, Compactness arguments.
   

10. 10.Notion of solutions: Wk,m, Viscosity, Classical in Ck.
    

11. 11.Fractional Sobolev spaces, and Fractional Operators.
    

Suggested literature:

1. 1.Caffarelli, Luis A.; Cabre, Xavier; Fully nonlinear elliptic equations. American Mathematical Society Colloquium Publications, 43. American Mathematical Society, Providence, RI, 1995. vi+104 pp. ISBN: 0-8218-0437-5
   

2. 2.Gilbarg, David; Trudinger, Neil S. Elliptic partial differential equations of second order. Reprint of the 1998 edition. Classics in Mathematics. Springer- Verlag, Berlin, 2001. xiv+517 pp. ISBN: 3-540-41160-7 35-02 (35Jxx)
   

3. 3.Pucci, Patrizia; Serrin, James; The maximum principle. Progress in Nonlinear Differential Equations and their Applications, 73. Birkh ̈auser Verlag, Basel, 2007. x+235 pp. ISBN: 978-3-7643-8144-8
   

4. 4.Garnett, John B.; Marshall, Donald E.; Harmonic measure. New Mathemat- ical Monographs, 2. Cambridge University Press, Cambridge, 2005. xvi+571 pp. ISBN: 978-0-521-47018-6
   

5. 5.Other appropriate literatures/articles.