Homogenization is concerned with the rigorous derivation of effective macroscopic models for materials with microscturctures, for example porous media or elastic composite material. A prototype problem is the limiting behavior of an elliptic PDE with rapidly oscillating coefficients.In this seminar, we will study methods such as two-scale convergence and Gamma-convergence and apply them to different homogenization problems.
Irrfahrten sind diskrete stochastische Prozesse, die zufällige Bewegungen auf dem Gitter $\Z^d$ modellieren. Sie sind ein wichtiges Thema der Wahrscheinlichkeitstheorie mit Anwendungen in der Finanzmathematik und Physik. In diesem Seminar werden wir mit elementaren stochastischen Methoden Eigenschaften von Irrfahrten untersuchen. Gegebenenfalls werden wir uns zudem mit theoretischem Hintergund und Verallgemeinerungen wie Markovketten und Martingalen befassen.
The lecture series gives an introduction to the theory of partial differential equations (PDEs). In the first part we will study classical solution theories for PDEs. In particular we will discuss some fundamental equations and examples and show limitations of classical solution concepts. In the second part of the lecture series an introduction to the modern theory of PDEs is given, which is based on a weaker notion of solutions and functional analytic concepts. In particular we will study elliptic PDEs.
We continue the study of elliptic partial differential equations from the lecture series "Partial Differential Equation I". The content is disjoint from the content of the lecture series "Partial Differential Equation I" of previous years. The following content is planned: - Schauder and L^p estimates for linear equations by the Campanato approach - Harnack inequality and DeGiorgi-Nash-Moser Theorem - Meyers' estimate - Existence and regularity for some nonlinear elliptic equations e.g. by compactness and variational methods