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Variational and Topological Methods in Bifurcation Theory (Winter Semester 2017/18)

Lecture: Monday 15:45-17:15 SR 3.68
Problem class: Thursday 15:45-17:15 SR 3.68
Lecturer PD Dr. Rainer Mandel
Office hours: by appointment
Room -1.019 Kollegiengebäude Mathematik (20.30)
Email: rainer.mandel@kit.edu
Problem classes Dr. Dominic Scheider
Office hours: on appointment
Room -1.021 Kollegiengebäude Mathematik (20.30)
Email: dominic.scheider@kit.edu


In the first part of the lecture we will see how variational methods can be used in order to prove bifurcation results. Applied to elliptic boundary value problems (PDEs or ODEs) these results yield the existence of small nontrivial solutions. In the second part of the lecture the Brouwer degree as well as the Leray-Schauder degree are introduced. These (topological) concepts are used in order to prove the so-called Krasnoselski-Rabinowitz theorem on global bifurcation.

Problem Class

Oct 26 - Some Calculus on Banach Spaces.
(Frechet differentiability, examples; the Implicit Function Theorem with an application)

Nov 09 - (i) Reflexive spaces and weak convergence,
(ii) another step of the proof of Theorem 2, and finally
(iii) an example by Böhme about (non-)existence of branches of solutions.

Nov 30 - (i) On the definition of the Brouwer degree,
(ii) an approximation result for continuous functions (Proposition 9 (i)),
(iii) an explicit calculation of the Brouwer degree in one dimension,
(iv) and another chance to present Böhme's example.

Dec 18 - Compactness.

Jan 11 - Bifurcation: prototypical case and a standard example.
Inverting the Dirichlet Laplacian.

Feb 01 - Global structure of bifurcating branches:
We review the problem and complete the proof of Proposition 32.