Vorlesung Semi linear Elliptic PDEs 2

This course is a continuation of my lecture Semi-linear Elliptic PDEs in the past semester. This course studies existence of weak solutions of semi-linear elliptic Partial Differential Equations (PDEs). Examples of semi-linear elliptic PDEs are abundant, in particular from Physics, Geometry, and Biology. They in particular describe solitary (or, stationary) waves for nonlinear time-dependent equations from Physics, such as the Klein-Gordon equation and the nonlinear Schrödinger equation (sometimes called 'nonlinear scalar field equations' in these cases). They also appear as stationary states for nonlinear heat equations, or in nonlinear diffusion in population genetics. On the other hand, such equations often appear in problems in Differential Geometry, such as the Yamabe Problem. There are also connections with constant mean curvature and minimal surfaces, as well as to stationary solutions for various geometric flows. In this course we will continue the study of various techniques to prove existence of weak solutions to such equations in bounded and unbounded domains. Keywords: Variational methods (Minimization Techniques: constrained minimization (on spheres and Nehari manifolds), lack of compactness; Minimax Methods: Saddle Point Theorem). Non-variational methods (Fixpoint theory, Method of lower and upper solutions). Audience (Hörerkreis): Master students of Mathematics (WP 17.2, 18.1, 18.2, 44.3, 45.2, 45.3), TMP-Master. Credits: 3 ECTS. Exam (Prüfung): There will be an oral exam of 30min (Es wird eine mündliche Prüfung von 30min geben). Prerequisites (Vorkenntnisse): Knowledge of Sobolev spaces (also on domains) and the theory of weak solutions of linear elliptic PDEs, as normally presented in (some version of) PDE2 will be an advantage. (This is basically the content of Chapters 1.2, 1.4, and 1.7 in the book by Badiale and Serra mentioned below.) The course will start with a (quick!) review of the material covered last semester. Students who wish to follow this course, but did not follow the course last semester, should (in due time!) contact the Lecturer (Prof. Sørensen) via email to discuss the prerequisites needed. (These are basically the content of Chapters 1, 2.1, 4.1-4.3, and 4.4.1 in the book by Badiale and Serra mentioned below.) [BS] M. Badiale, E. Serra (2011), Semilinear Elliptic Equations for Beginners, Springer (Universitext), 2011. Further information on literatur to come. For more information, see http://www.math.lmu.de/~sorensen/teaching.html Voraussetzungen Knowledge of Sobolev spaces (also on domains) and the theory of weak solutions of linear elliptic PDEs, as normally presented in (some version of) PDE2 will be an advantage. (This is basically the content of Chapters 1.2, 1.4, and 1.7 in the book by Badiale and Serra mentioned above.) The course will start with a (quick!) review of the material covered last semester. Students who wish to follow this course, but did not follow the course last semester, should (in due time!) contact the Lecturer via email to discuss the prerequisites needed. (These are basically the content of Chapters 1, 2.1, 4.1-4.3, and 4.4.1 in the book by Badiale and Serra mentioned above.) Leistungsnachweis Gilt für Masterprüfung Mathematik (WP17), Masterprüfung (WP35) im Studiengang Theor. und Math. Physik. Zielgruppe Master students of Mathematics (WP 17.2, 18.1, 18.2, 44.3, 45.2, 45.3), TMP-Master. Department Mathematisches Institut Knowledge of Sobolev spaces (also on domains) and the theory of weak solutions of linear elliptic PDEs, as normally presented in (some version of) PDE2 will be an advantage. (This is basically the content of Chapters 1.2, 1.4, and 1.7 in the book by Badiale and Serra mentioned above.) The course will start with a (quick!) review of the material covered last semester. Students who wish to follow this course, but did not follow the course last semester, should (in due time!) contact the Lecturer via email to discuss the prerequisites needed. (These are basically the content of Chapters 1, 2.1, 4.1-4.3, and 4.4.1 in the book by Badiale and Serra mentioned above.) Gilt für Masterprüfung Mathematik (WP17), Masterprüfung (WP35) im Studiengang Theor. und Math. Physik. LMU München SoSe 2016 Univ.Prof.Dr. Sørensen Thomas