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ma8105:2019v:lectures [2019-03-15]
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ma8105:2019v:lectures [2019-03-18] (nåværende versjon)
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 ^ 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]]. ​ \\ \\ | ^ 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  | 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]]. ​ \\ \\  | ^ 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  | 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.  | +^ 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  | General Sobolev inequalities, ​ \\ embedding, compactness in \(W^{1,​p}\) ​ \\ \\  | | 116-118, 18-19, 112-114 ​ \\ \\ | **Obs:** 1 only one lecture this week.  \\ \\  | +^ 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/​2017v/​notes/​LecNote03.04.2017_GenSobolevIneq_and_RellichKondrachovCompThm.pdf|General Sobolev inequalities and Strong comactness in \(W^{1,​p}\)]] ​ \\ \\ | 
-^ 14  | Compactness (cont.), ​ \\ Sobolev chain rule, finite differences. ​ \\ \\ | |  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 1:** linear 2nd order elliptic PDEs, existence and uniqueness of weak solutions in \(W^{1,​2}_0(\Omega)\),​ \(W^{2,​2}(\Omega)\) interior regularity. ​ \\ \\ | |  114-116, 126, 128  \\ \\ \\ Evans: PDEs, parts of chp. 6.1-6.3 ​ \\ \\ | **My lecture notes** from this week:  \\ \\ [[http://​www.math.ntnu.no/​emner/​MA8105/​2017v/​notes/​LecNote06.04.2017_Rellich_ChainRule_and_DifferenceQuotients.pdf|Strong comactness, chain rule, and difference quotients]] ​ \\ \\  | 
-^ 15  | Application:​ To be decided | | | | +^ 15  | **Application ​2:** To be decided | | | | 
2019-03-15, Espen Robstad Jakobsen