Lectures on: Bifurcation Theory and Applications to Elliptic Boundary-Value Problems

Lectures on: Bifurcation Theory and Applications to Elliptic Boundary-Value Problems

Professor Charles Stuart

In the standard results used to treat elliptic boundary-value problems, the connection between the nonlinear problem and its linearisation is made by requiring Fréchet differentiability at the trivial solutions. For some elliptic problems this fails but differentiability in the sense of Hadamard still holds. This will be explored by treating the following topics.

Lecture 1: 17 November - Presentation

• Review of the basic notions concerning bifurcation and asymptotic linearity.

• Review of differentiability in the sense of Gˆateaux, Fréchet, Hadamard.

• Examples which are Hadamard but not Fréchet differentiable.  The Dirichlet problem for a degenerate elliptic equation on a bounded domain. The stationary nonlinear Schrödinger equation on R

Lecture 2: 24 November - Presentation

• Bifurcation from isolated eigenvalues of finite multiplicity of the linearisation.

• Pseudo-inverses and parametrices for paths of Fredholm operators of index zero.

• Detecting a change of orientation along such a path.

• Lyapunov-Schmidt reduction.

Lecture 3: 1 December - Presentation

• Sufficient conditions for bifurcation from points that are not isolated eigenvalues of the linearisation.

• Odd potential operators.

• Defining min-max critical values using sets of finite genus.

• Formulating some necessary conditions for bifurcation.