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ma8105:2019v:lectures [2019-02-13]
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ma8105:2019v:lectures [2019-04-08] (nåværende versjon)
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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** ​ \\ \\  **My lecture notes** from this week:  \\ \\ [[http://​www.math.ntnu.no/​emner/​MA8105/​2019v/​notes/​LecNote08.04.2019_Application_PorousMediumEquation_II.pdf|Porous Medium Equations II: compactness,​ convergence]] ​ \\ \\ 
2019-02-13, Espen Robstad Jakobsen