Welcome! Here is the course catalogue.
Time and Place of the lectures: Thursdays, 8:15-10:00, in HG D1.1.
Exam format: 20-minute oral exam.
In this course we will cover some of the most popular techniques to show existence of solutions to nonlinear problems. We will explore
This section contains the (handwritten) lecture notes of the course, which will be uploaded after each class, along with a summary of that class's content. Any comments are welcome!
| Date | Summary |
|---|---|
| 19.09 | Banach spaces, linear maps between Banach spaces, compact maps, differentiability in Banach spaces, weak and weak-* topology. |
| 26.09 | Banach-Alaoglu's theorem, Hölder spaces, weak derivatives, Sobolev spaces, traces, GNS inequality, Poincaré and Poincaré-Wirtinger inequality, Morrey's theorem. |
| 04.10 | Rellich-Kondrachov theorem, PDEs in divergence and non-divergence form, ellipticity, weak solutions, weak formulation, Lax-Milgram theorem, Fredholm alternative, interior and boundary regularity, Euler-Lagrange equations. |
| 10.10 | Brachistochrone and isoperimetric examples, Weierstrass counterexample, the Dirichlet energy, Fréchet-differentiability of the functional associated to \(-\Delta u+g(u)=0\), weak continuity and weak lower-semicontinuity with examples, the direct method theorem. |
| 17.10 | The Direct method in Banach spaces, coercivity, Fermat's theorem, applications (\(n\)-body problem, generalised pendulum, elliptic equation with boundary conditions). |
| 24.10 | Implicit function theorem and Lagrange multiplier theorem in Banach spaces, the first eigenvalue of the laplacian, Rayleigh quotient, simplicity of the first eigenvalue and sign of the first eigenfunction (proof not examinable), optimal Poincaré inequality, orthogonality between eigenfunctions, unboundedness of the spectrum, Courant-Fisher minmax principle, application to nonlinear PDE. |
| 31.10 | No class. |
| 07.11 | Critical levels, deformation lemma in \(\mathbb{R}^n\), the Palais-Smale condition, pseudo-gradient, pseudo-gradient vector field, compactness of critical sets with levels in a compact set. |
| 14.11 | Deformation lemma in Banach spaces, minmax principle, minimisation using (PS), Mountain Pass theorem (statement). |
| 21.11 | Mountain pass theorem, two proofs and application, intrduction to topological degree. |
| 28.11 | Topological degree and properties, Theorem: min-max with degree, Leray-Schauder degree, Schauder fixed point theore, example. |
| 05.12 | Linking, examples, minmax with linked sets, genus of a symmetric sets and its properties. |
| 12.12 | Theorem (genus of the sphere), Borsuk-Ulam, Existence of \(n\) critical points for even functionals on spheres (finite dimensional case with proof, infinite dimensional case without. Mountain pass with symmetry. |
| 19.12 | Conclusion of Mountain pass with symmetry proof, bifurcation theory and the Lyapunov-Schmidt reduction. |
Lecture notes