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Tentative Lecture Plan
- From the outset this is the plan of 2017.
- There will be some changes. I will update the plan as I lecture.
| Week | Topic | Chapter | Pages | Remarks |
|---|---|---|---|---|
| 2 | Introducution The basics: Metric and Banach spaces, duals, \(\ell^p\), Hahn-Banach and consequences | 1 2 | p. 5-8, Theorem A.1 (appendix), Brezis chp 1.1: Corollay 1.2-1.4 | Introduction meeting. 1 lecture. |
| 3 | Conclusion to Hahn-Banach, separable and reflexive spaces, Compactness in metric spaces, weak convergence, relation to strong convergence, Banach-Steinhaus without proof | p. 7-10, 15-16 | ||
| 4 | Weak compactnes, Eberlein-Smuljan's thm, weak * convergence, Helley's thm, Alaoglu's thm, strong compactness for functions: Arzela-Ascoli's thm | | p 11-17 | |
| 5 | Distribution theory Definitions, properties, operations, regular/singular operations(cont.), derivatives of regular distr., the fundamental theorem, convolution | 3 | | Remark: We define convolution both as a function as in Holden and as a distribution (see e.g. Wikipedia). The two definitions are related, one is the "density function" of the other. |
| 6 | Convolutions (cont.), convergence, approximations Primitive in 1D, equations in \(D'\), fundamental solutions | |||
| 7 | Lebesgue Spaces Strong and weak \(L^p\), properties, inequalities. Convolutions, approximation in \(L^p\). Approximation in \(L^p\), compactness in \(L^p\) | 4 | 4.1, Prop 4.4 and Thm 4.6, 4.2, 4.7, 4.3 | Proof of Kolmogorov: We take the classical proof and not the one in the Holden notes, see e.g. Theorem A.5 in Holden-Risebro: Front Tracking for Hyperbolic Conservation Laws. The classical proof is an approximation argument that reduces the proof to an application of Arzela-Ascoli. |
| 8 | Modes of convergence | p 86-89 p 59-63 | ||
| 9 | Modes of convergence (cont.), convergence and compactness in \(L^p,\ p\in(1,\infty)\). Limiting cases: Convergence and compactness in\(L^1\), Dunford-Pettis with proof, uniform integrability, de la Vallee-Poussin. | p 60-65 p 65-70 | My lecture notes on Dunford-Pettis and equiintegrability PDF Note that the proof in the notes of Holden lack the conclusion. |
|
| 10 | Convergence and compactness in\(L^1\) (cont.), Convergence and compactness in \(L^\infty\) Examples. Radon measures, the space \(\mathcal M\), Weak * compactness in \(\mathcal M\) and the subspace \(L^1\) | p 69-75 p 75-78 | This material is partly taken from Folland: Real Analysis chp 7, partly from Holden. Note: Riesz representation theorem, the space \(M\) needs be defined as the space of finite Radon measures. My lecture notes on Radon measure and compactness: PDF. |
|
| 11 | Sobolev Spaces Definitions, smooth approximations Smooth approximations (cont.) straightening the boundary, extensions | 5 | 95-97 97-97, 179 (App B.3) | OBS: Global approx up to boundary - the proof in Holden using straightening is not optimal and only gives \(C^1\) approximate functions. In the lectures I used the proof from Evans: PDEs chapter 5 that avoids straightening and give \(C^\infty\) approximations. My lecture notes on smooth approximation up the boundary and straightening the boundary: PDF. |
| 12 | Extensions, restrictions Restrictions (cont), Sobolev inequalities, Gagliardo-Nirenberg-Sobolev | 98-102 102-104 | ||
| 13 | Sobolev inequalities: Gagliardo, Poincare. H\"older spaces, Morrey's inequality. | 104-107, 111-112. 110-111, 118. | ||
| 14 | General Sobolev inequalities, embedding, compactness in \(W^{1,p}\) Compactness (cont.), Sobolev chain rule, finite differences. | 116-118, 18-19, 112-114 114-116, 126, 128 | Rellich-Kondrachov: In the lectures I give a stronger version of this result than presented in the notes of Holden. I follow the book of Evans, and show compact embedding into Hoelder spaces \(C^{0\,\gamma}\). The proof of the first part of Rellich-Kondarchov's compactness theorem follows from extension, Kolmogorov-Riesz compactness theorem, and interpolation in \(L^p\). This way avoids the long regularization + Arzela-Ascoli argument used in the Holden notes (and in PDE book by Evans). In fact the the regularization + Arzela-Ascoli argument is exactly the argument we used in class to proof Kolmogorov-Riesz in the first place (but the proof in the notes of Holden is slightly different). My lecture notes from this week: General Sobolev inequalities and Strong comactness in \(W^{1,p}\) Strong comactness, chain rule, and difference quotients |
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| 15 | Application: To be decided |