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):
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.
Here
is a longer list of books.
Office hours (Sprechstunde):
Via Zoom; see uni2work.
-----------------------------------
Letzte Änderung: 02 March 2021 (
No more updates
).
Thomas Østergaard Sørensen