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ma8105:2019v:lectures [2019-04-05]
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ma8105:2019v:lectures [2019-04-08] (nåværende versjon)
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 ^ 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 \(W^{1,​p}\)]] ​ \\ \\ | ^ 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  | 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]] ​ \\ \\  | ^ 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  | 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** ​ \\ \\ | +^ 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]] ​ \\ \\ | 
2019-04-08, Espen Robstad Jakobsen