Tristan Rivière: Katalogdaten im Frühjahrssemester 2012 |
| Name | Herr Prof. Dr. Tristan Rivière |
| Lehrgebiet | Mathematik |
| Adresse | Professur für Mathematik ETH Zürich, HG G 48.1 Rämistrasse 101 8092 Zürich SWITZERLAND |
| Telefon | +41 44 632 06 71 |
| tristan.riviere@math.ethz.ch | |
| URL | http://www.math.ethz.ch/~triviere |
| Departement | Mathematik |
| Beziehung | Ordentlicher Professor |
| Nummer | Titel | ECTS | Umfang | Dozierende | |
|---|---|---|---|---|---|
| 401-4354-12L | Introduction to Elliptic Partial Differential Equations | 6 KP | 3V | T. Rivière | |
| Kurzbeschreibung | We will describe the basic regularity theory for elliptic equations and the basic partial regularity result for elliptic systems, with a special focus on epsilon-regularity results. | ||||
| Lernziel | |||||
| Inhalt | In certain situations, a condition on the derivatives of a function imposes automatically that such function is smooth. Illustra- tions of this phenomenon are given by holomorphic functions (where the constraint is given by the Cauchy-Riemann equa- tions), or harmonic functions (where the constraint is that the Laplacian is equal to zero). More generally, ellipticity in Partial Differential Equations is the property that characterizes situa- tions when this improvement occurs. The goal of this course is to give basic and robust methods for deducing the regularity of solutions to elliptic PDE. We will first consider the case of one equation and give stra- tegies for passing from the lowest regularity assumption to the highest possible regularity. We will prove the basic regularity result for weak solutions to elliptic equations. We will describe two different approaches to the regularity, due respectively to E. De Giorgi and to J. Moser. The case of more equations, i.e. Elliptic Partial Differential Systems, is much more involved and does not always satisfy the best scenario in terms of regularity: sometimes one can only expect partial regularity. We will look at elliptic systems in the second part of the course: after a discussion on different ellipticity conditions we will give a counterexample to regularity and prove the basic partial regularity result, focusing on the idea of 'epsilon-regularity'. The third part of the course will consist of a presentation of ex- amples where the methods of the second part are successfully used. Depending on the time constraint, we plan to describe the case of energy-minimizing harmonic maps and that of area-minimizing surfaces or more. | ||||
| Literatur | 1. Luigi Ambrosio, Alessandro Carlotto, Annalisa Massaccesi: Lecture notes on Partial Differential Equations Available online at http://cvgmt.sns.it/media/ doc/paper/1280/Corso2009.pdf 2. Qing Han, Fanghua Lin: Elliptic Partial Diffe- rential Equations 2nd ed., Courant Institute of Mathematical Sciences, Courant lecture notes in mathematics; 1, Ed. 2; 2011. 3. Mariano Giaquinta, Luca Martinazzi: An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs, Edizioni della Normale; 2005. 4. Qing Han: A basic course in Partial Differen- tial Equations, Graduate studies in mathema- tics; vol. 120; 2011. (more notes and literature will be provided, if needed, during the course) | ||||
| Voraussetzungen / Besonderes | PREREQUISITES: Knowledge achieved from the course Functional Analysis II. See the notes of Prof. Dr. M. Struwe http://www.math.ethz.ch/~struwe/Skripten/FA- I-II-26-8-08.pdf for the precise program. We will use the following without proof: - Definition and existence of weak solutions - Poincaré and Sobolev inequalities, Sobolev embedding theorem We will recall the following at the beginning of the course, but having looked at these topics beforehand will prove particularly helpful: - Elementary properties of harmonic functions (i.e. value property, smoothness, maximum principle, fundamental solutions) - Definition and relations between Morrey, Campanato, Sobolev and Hölder spaces of functions. | ||||
| 401-5350-00L | Analysis Seminar  | 0 KP | 1K | M. Struwe, F. Da Lio, M. Eichmair, N. Hungerbühler, T. Kappeler, T. Rivière, D. A. Salamon | |
| Kurzbeschreibung | Forschungskolloquium | ||||
| Lernziel | |||||
| Inhalt | Research seminar in Analysis | ||||