Vorlesung: Semi-linear Elliptic PDEs (WiSe 2020/21)
[UPDATE 12.10.2020:] Due to the present situation, this course will be online
(uploaded videos, lecture notes; all details now on uni2work). It will start 03 November 2020.
For updated information, check back here, on LSF, and on uni2work (where all material will be uploaded).
A brief video introduction can be found here.
To access the course material (videos, lecture notes, Zoom data etc), you need to sign up in uni2work here.
Lecture (Vorlesung):
Tue 08-10 Online (videos). LSF
Exercises (Übungen):
There are NO exercises!
Synopsis (Kurzbeschreibung):
This course studies existence of weak solutions of semi-linear elliptic Partial Differential Equations (PDEs). Existence of solutions of PDEs is not easy to establish, the best strategy is often to first show the existence of solutions in some generalised sense, and then establish regularity (to conclude existence of a classical solution). For equations in divergence form, this leads to the study of weak solutions (and Sobolev spaces) by testing (multiplying and integrating) against smooth functions (as studied for linear elliptic PDEs in the course PDE 2 last semester).
Examples of semi-linear elliptic PDEs are abundant, in particular from Physics, Geometry, and Biology. They in particular describe solitary (or, stationary) waves for nonlinear time-dependent equations from Physics, such as the Klein-Gordon equation and the nonlinear Schrödinger equation (sometimes called 'nonlinear scalar field equations' in these cases). They also appear as stationary states for nonlinear heat equations, or in nonlinear diffusion in population genetics. On the other hand, such equations often appear in problems in Differential Geometry, such as the Yamabe Problem. There are also connections with constant mean curvature and minimal surfaces, as well as to stationary solutions for various geometric flows.
In this course we will study various techniques to prove existence of weak solutions to such equations in bounded domains.
Keywords: Nonlinear functional analysis; Critical Points; variational methods (Minimization Techniques: compact problems; Minimax Methods: Palais-Smale sequences, Mountain Pass Theorem).
(Depending on interest, a sequel treating non-variational methods (monotone operators; fix point methods) might be planned in the following semester.)
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 of linear elliptic PDEs, as normally presented in (some version of) PDE2 will be an advantage. The course will start with a (quick!) review of this material. Students who wish to follow this course, but did not yet follow a course on this material, should (in due time!) contact the Lecturer (Prof. Sørensen) via email to discuss the prerequisites needed. (These are basically the content of Chapters 1.2, 1.4, and 1.7 in the book by Badiale and Serra mentioned below.)
Language (Sprache):
English. (Die mündliche Prüfung kan auch auf Deutsch gemacht werden).
Exam (Prüfung):
There will be an oral exam of 30min (Es wird eine mündliche Prüfung von 30min geben). See uni2work for details.
Literature:
In uni2work you will find a copy of the notes from the lecture (to be updated as we go along).
The lecture will mainly follow the book:
[BS] M. Badiale, E. Serra (2011), Semilinear Elliptic Equations for Beginners, Springer (Universitext), 2011.
(Login with your Campus-account.)
Supplementary literatur (Ergänzende Literatur):
- A. Ambrosetti, D. Arcoya, An Introduction to Nonlinear Functional Analysis and Elliptic Problems, Birkhäuser, 2011.
- A. Ambrosetti, A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge University Press, 2010.
- H. Le Dret, Nonlinear Elliptic Partial Differential Equations, Springer, 2018.
- P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, AMS, 1986.
- L. Boccardo, G. Croce, Elliptic Partial Differential Equations, De Gruyter, 2013.
- R. Precup, Linear and Semilinear Partial Differential Equations, De Gruyter, 2013.
Office hours (Sprechstunde):
Via Zoom; see uni2work.
-----------------------------------
Letzte Änderung: 02 March 2021 (No more updates).
Thomas Østergaard Sørensen