Vorlesung: Elliptic Regularity Theory (SoSe 2019)

Lecture (Vorlesung):
Thu 10--12 (in B 046).

Exercises (Ãœbungen):
There are NO exercises.

Synopsis (Kurzbeschreibung):
This course studies the regularity theory of solutions to linear elliptic partial differential equations (PDEs) of 2nd order. This topic is of its own interest (proving that weak solutions are classical solutions for example) but also gives important tools for the study of semilinear, quasilinear, and fully nonlinear elliptic PDEs. Such equations occur in many fields of mathematics, but in particular in Mathematical Physics and Geometry.
Some keywords: L^2-theory and weak Maximum Principle (DeGiorgi interation), Schauder theory (via mollifiers), L^p-theory, DeGiorgi-Nash-Moser theory (Hölder continuity).
Important analysis-tools (also useful in other contexts) will be developed along the way.

Audience (Hörerkreis):
Master students of Mathematics (WP 17.2, 18.1, 18.2, 44.3, 45.2, 45.3), TMP-Master.

Credits:
3 ECTS.

Prerequisites (Vorkenntnisse):
Knowledge of Sobolev spaces on domains and the theory of weak solutions to elliptic PDEs, as presented in various courses in Analysis (for example PDE1 in WS18/19; see Evans' book below, pp. 255-295 and 313-327).
Some knowledge of Functional Analysis (FA1) (Lax-Milgram and Fredholm Alternative).
A brief handout with the needed facts (without proofs) will be made available here

Language (Sprache):
English. (Die mündliche Prüfung kan auch auf Deutsch gemacht werden).

Exam (Prüfung):
There will be an oral exam (Es wird eine mündliche Prüfung geben).

Content (Inhalt):

0. Introduction and motivation
   

1. L^2 - theory

   1.1 Weak derivatives and the Sobolev spaces W^{k,p}(Omega)
   1.2 Weak solutions of elliptic equations
   1.3 H^2 - regularity of weak solutions
   1.4 A maximum principle for weak solutions (DeGiorgi iteration)
   

2. Schauder theory

   2.1 Hölder spaces and mollifiers
   2.1 Schauder estimates
   

Literature (Literatur):
There will be no lecture notes. Above you will find a short description of the content of the lecture
(to be updated as we go along).
(Es wird kein Skript geben. Hier wird laufend eine Kurzübersicht der Vorlesung erstellt.)

Supplementary literatur (Ergänzende Literatur):

• L. Beck, Elliptic Regularity Theory, Springer (2016).
• Y.-Z. Chen, L.-C. Wu, Second order elliptic equations and elliptic systems, AMS (1998).
• L. C. Evans, Partial Differential Equations, 2nd edition, AMS (2010).
• M. Giaquinta, L. Martinazzi, An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs, 2nd edition, Edizioni della Normale, Pisa (2012).
• D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, Springer (1977).
• Q. Han, F. Lin, Elliptic Partial Differential Equations, 2nd edition, AMS (2011).
• N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Hölder Spaces, AMS (1996).
• N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, AMS (2008).

Office hours (Sprechstunde):
Wednesday 10:15-11:00 (Room B 408) or by appointment via email.


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Letzte Änderung: 23 September 2019 (No more updates)

Thomas Østergaard Sørensen