Graduate Lecture Series "Applied Analysis"

Welcome to our series of concise graduate lecture courses exploring a range of topics in Applied Analysis and related fields. These courses address both classical results and ongoing research in the field.

Each course consists of one or more 90-minute sessions led by different speakers. While the courses are interconnected, they are designed to be attended individually as well.

The courses are open and tailored for graduate students and researchers with a solid background in the analysis of Partial Differential Equations (PDEs).

Schedule - winter 2024/2025

The graduate lectures in the winter term 2024/2025 focus on mathematical methods in continuum mechanics.

Please check the schedule regularly, as it may be subject to short-notice changes.
Unless stated otherwise, the lectures take place on the specified dates on
Thursday, 3. DS (11:10 - 12:40), room: Z21-381.

The lectures can be streamed on request.

Abstracts

Discrete dislocation models

Prof. Dr. Florian Theil

Dislocation models offer insights into the microscopic processes underpinning elasto-plastic materials.
The plan is to

1. Give a heuristic introduction to dislocation in atomistic systems.
2. Recall basic concepts of continuum mechanics: Geometrically linear and non-linearity models
3. Introduce some basic concepts of algebraic topology and exterior calculus
4. Introduce the  Ariza-Ortiz model and fundamental energy scalings
5. Discuss dislocation dipoles and walls for scalar and vectorial versions of the Ariza-Ortriz model
6. Introduce equilibrium Statistical Mechanics and the concepts of positional and orientational order
7. Give a proof that for low temperature the AO model exhibits positional order
8. Present first results on rate-independent evolution of dislocations

Rate-independent systems and Finsler geometry

Prof. Dr. Oliver Sander

Rate-independent systems are a particular type of differential equations
that are useful to describe certain dissipative systems in mechanics. From the perspective of numerical analysis, one particularly attractive feature is that time discretizations naturally lead to sequences of minimization problems.  However, the objective functionals
of these minimization problems are nonsmooth, and can be very nonconvex.

The natural geometric setting of rate-independent systems is Finsler geometry, which investigates differentiable manifolds that have a Minkowski functional (a sort of generalized norm) attached to each tangent space.  Despite being much more general, Finsler manifolds retain a surprising number of features of Riemannian manifolds;
in particular, there are the notions of geodesics and exponential maps.

In this series of talks we will explain rate-independent systems and Finsler geometry, and show how Finsler exponential maps can be used to make the minimization problems of time-discrete rate-independent systems easier to solve.
 

Schedule - summer 2024

The graduate lectures in the summer term 2024 focus on the theory of homogenization.

Please check the schedule regularly, as it may be subject to short-notice changes.
Unless stated otherwise, the lectures take place in Z21-380.

Abstracts

Linearization after Homogenization of nonlinear elasticity with prestrain

Kai Richter, M.Sc.
In this graduate lecture we study an energy functional related to nonlinear elasticity with a prestrain that is a perturbation of a periodic stress-free joint. The homogenization of such non-convex functions evokes a mulit-cell formula that is not practical for analysis and numerics. The goal of this this lecture is to study a quadratic expansion of the homogenized energy using Gamma-convergence that results in an explicit linearized limit. From this we can infer explicit first order information about the model. The analysis of this expansion invokes studying serveral interesting properties of maps with periodic derivative which are related to periodic stress-free joints.

Transport equation: renormalisation and quantitative regularity estimates
Prof. Tomasz Dębiec (U Warsaw)
In these lectures, we will focus on the transport equation with non-smooth velocity fields. The idea of renormalised solutions, introduced in the seminal paper of DiPerna and Lions, plays an important role not only in the study of the transport equation and the related ordinary differential equations, but also for a variety of larger systems of PDEs that contain the transport equation (usually in the conservative form). We will review the classical notion of renormalised solutions and use it to obtain well-posedness for transport equations with Sobolev-regular velocity fields. Subsequently, we will discuss quantitative regularity estimates for the trajectories of the flow.

Boundary corrector in stochastic homogenization

Dr. Claudia Raithel
In the homogenization of linear elliptic PDEs on domains there is a well-known boundary layer phenomenon -in particular, in a layer around the boundary the heterogeneous coefficient solution behaves qualitatively differently than in the bulk of the domain. This is most easily seen in the fact that the standard ansatz that is used to describe the heterogeneous coefficient solution in the interior does not have the correct boundary data. To remedy this, one can introduce homogenization correctors satisfying homogeneous Dirichlet boundary data -this is necessary to prove higher-order homogenization rates. In a series of three lectures we construct a Dirichlet corrector for the half-space in the stochastic setting -we do so by correcting an existing whole-space corrector. We will then discuss how to obtain optimal decay estimates for the boundary correction.

Contact person:

Prof. Dr. Stefan Neukamm

Prof. Dr. Stefan Neukamm