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ma8105:2019v:lectures [2019-03-18]
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ma8105:2019v:lectures [2019-03-18]
erj |
| ^ 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. \\ \\ **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 \(W^{1,p}\)]] \\ \\ | | ^ 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. \\ \\ Application: 2nd order linear 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]] \\ \\ | | ^ 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 | | | | |