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ma8105:2019v:lectures [2019-02-13]
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ma8105:2019v:lectures [2019-04-05]
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| ^ 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. | | ^ 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 | | | | | ^ 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. \\ \\ | | ^ 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 \\ \\ p 86-89 | **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. \\ \\ OBS: 3 lectures this week, only one next week. \\ \\ | |
| ^ 8 | Modes of convergence | | p 86-89 \\ \\ p 59-63 | | | ^ 8 | Modes of convergence | | p 59-63 | Only one lecture this week, three last week. \\ \\ | |
| ^ 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 [[http://www.math.ntnu.no/emner/MA8105/2017v/notes/LecNote02.03.2017_DunfordPettis_and_Equiintegrability.pdf|PDF]] \\ \\ Note that the proof in the notes of Holden lack the conclusion. \\ \\ | | ^ 9 | Convergence and compactness in \(L^p,\ p\in(1,\infty)\). \\ \\ Convergence and compactness in\(L^1\), \\ Dunford-Pettis with proof, uniform integrability, de la Vallee-Poussin. \\ \\ Convergence and compactness in \(L^\infty\) \\ Examples. | | p 60-65 \\ \\ p 65-75 \\ \\ | **My lecture notes** on Dunford-Pettis and equiintegrability [[http://www.math.ntnu.no/emner/MA8105/2019v/public/DunfordPettisAndEquiintegrabilityNote2019.pdf|PDF]] \\ \\ Note that the proof of Dunford-Pettis in the notes of Holden lack the conclusion. \\ \\ The discussion on equiintegrability can not be found in Holden, see my notes. | |
| ^ 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: [[http://www.math.ntnu.no/emner/MA8105/2017v/notes/LecNote09.03.2017_RadonMeasuresAndCompactness.pdf|PDF]]. \\ \\ | | ^ 10 | Radon measures, the space \(\mathcal M\),\\ Weak * compactness in \(\mathcal M\) and the subspace \(L^1\) \\ \\ **Sobolev Spaces** \\ Definitions, smooth approximations | \\ \\ \\ \\ \\ \\ \\ 5 | p 75-79 \\ \\ \\ \\ \\ \\ \\ 95-97 \\ \\ | 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: [[http://www.math.ntnu.no/emner/MA8105/2019v/public/LecNote04.03.2019_RadonMeasuresAndCompactness.pdf|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: [[http://www.math.ntnu.no/emner/MA8105/2017v/notes/LecNote16.03.2017_SobolevSmApproStraightening.pdf|PDF]]. \\ \\ | | ^ 11 | Smooth approximations (cont.) \\ straightening the boundary, extensions \\ \\ Extensions, restrictions/trace | | 97-97, 179 (App B.3) \\ \\ \\ \\ \\ \\ 97-99 | **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: [[http://www.math.ntnu.no/emner/MA8105/2019v/public/LecNote11.03.2019_SobolevSmApproStraightening.pdf|PDF]]. \\ \\ | |
| ^ 12 | Extensions, restrictions \\ \\ Restrictions (cont), \\ Sobolev inequalities, Gagliardo-Nirenberg-Sobolev | | 98-102 \\ \\ 102-104 | | ^ 12 | Restrictions/trace (cont), \\ Sobolev inequalities, Gagliardo-Nirenberg-Sobolev \\ \\ Sobolev inequalities: Gagliardo, Poincare. \\ \\ H\"older spaces, Morrey's inequality. \\ \\ | | 98-102 \\ \\ 102-104 \\ \\ \\ 105-107, 111-112 \\ \\ 110-111, 118. | **Obs:** 3 lectures this week. | |
| ^ 13 | Sobolev inequalities: Gagliardo, Poincare. \\ \\ H\"older spaces, Morrey's inequality. \\ \\ | | 104-107, 111-112. \\ \\ 110-111, 118. | | | ^ 13 | General Sobolev inequalities, \\ embedding, compactness in \(W^{1,p}\) \\ \\ | | 116-118, 18-19, 112-114 \\ \\ | **Obs:** 1 only one lecture this week. \\ \\ **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: \\ \\ [[http://www.math.ntnu.no/emner/MA8105/2019v/notes/LecNote25.03.2019_GenSobolevIneq_and_RellichKondrachovCompThm.pdf|General Sobolev inequalities and Strong comactness in \(W^{1,p}\)]] \\ \\ | |
| ^ 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: \\ \\ [[http://www.math.ntnu.no/emner/MA8105/2017v/notes/LecNote03.04.2017_GenSobolevIneq_and_RellichKondrachovCompThm.pdf|General Sobolev inequalities and Strong comactness in \(W^{1,p}\)]] \\ \\ [[http://www.math.ntnu.no/emner/MA8105/2017v/notes/LecNote06.04.2017_Rellich_ChainRule_and_DifferenceQuotients.pdf|Strong comactness, chain rule, and difference quotients]] \\ \\ | | ^ 14 | Compactness (cont.), \\ Sobolev chain rule, finite differences. \\ \\ **Application:** Application: Convergence of finite difference approximation for the Porous Medium Equation (PME). \\ \\ A. About PME (eq'n, background, self-similar solutions, derivation, well-posedness and a priori estimates). \\ \\ B. The explicit monotone finite difference approximation. \\ \\ C. A priori estimates for the approximation. \\ \\ | | 114-116, 126, 128 | **My lecture notes** from this week: \\ \\ [[http://www.math.ntnu.no/emner/MA8105/2019v/notes/LecNote06.04.2017_Rellich_ChainRule_and_DifferenceQuotients.pdf|Strong comactness, chain rule, and difference quotients]] \\ \\ [[http://www.math.ntnu.no/emner/MA8105/2019v/notes/LecNote05.04.2019_Application_PorousMediumEquation_I.pdf|Porous Medium Equations I: Intro, finite difference approximation, a priori estimates]] \\ \\ | |
| ^ 15 | Application: To be decided | | | | | ^ 15 | D. Interpolation in time and compactness. \\ \\ E. Convergence of the method. \\ \\ Plan the oral exam. \\ Dates and organization. \\ \\ | | | **Only one lecture this week - Monday** \\ \\ **Reference group meeting** \\ \\ **Decide on exam date** \\ \\ | |