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 cover both classical results and current research in the field.

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

The courses are open to and tailored for graduate students and researchers with a solid background in the analysis of partial differential equations (PDEs).

Schedule - Winter 2026/2027

Under planning

Schedule - Winter 2025/2026

The graduate lectures in the summer term 2024/2025 focus on the analysis of applied evolution equations using variational methods.

Please check the schedule regularly, as it may be subject to changes on short notice.
Unless otherwise noted, the lectures will be held on the specified dates on
Wednesday, 2nd lecture period, room: Z21-242.

Schedule - Summer 2025

The graduate lectures in the summer term 2024/2025 focus on the analysis of PDEs and the calculus of variations.

Please check the schedule regularly, as it may be subject to changes on short notice.
Unless otherwise noted, the lectures will be held on the specified dates on
Wednesdays at 9:30 a.m. in room Z21-243.

The lectures can be streamed upon request.

!!!! The lecture on May 14, 2025, is from 10:30 a.m. to 11:20 a.m. !!!!

Schedule - Winter 2024/2025

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

Please check the schedule regularly, as it may be subject to last-minute changes.
Unless otherwise stated, the lectures take place on the specified dates on
Thursday, 3rd lecture period (11:10 a.m. – 12:40 p.m.), room: Z21-381.

The lectures can be streamed upon request.

Abstracts

Discrete dislocation models

Prof. Dr. Florian Theil

Dislocation models provide insights into the microscopic processes underlying elastoplastic materials.
The plan is to

1. Provide a heuristic introduction to dislocations in atomistic systems.
2. Review basic concepts of continuum mechanics: geometrically linear and nonlinear 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 vector versions of the Ariza-Ortiz model
6. Introduce equilibrium statistical mechanics and the concepts of positional and orientational order
7. Provide a proof that, at low temperatures, the AO model exhibits positional order
8. Present initial results on the rate-independent evolution of dislocations

Rate-independent systems and Finsler geometry

Prof. Dr. Oliver Sander

Rate-independent systems are a specific type of differential equation
that are useful for describing 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 highly nonconvex.

The natural geometric setting for rate-independent systems is Finsler geometry, which studies differentiable manifolds that have a Minkowski functional (a kind of generalized norm) associated with each tangent space. Although they are much more general, Finsler manifolds retain a surprising number of features of Riemannian manifolds;
in particular, the concepts 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 of 2024 will focus on the theory of homogenization.

Please check the schedule regularly, as it may be subject to changes at short notice.
Unless otherwise noted, the lectures will be held in Z21-380.

Abstracts

Linearization after homogenization of nonlinear elasticity with prestain

Kai Richter, M.Sc.
In this graduate lecture, we study an energy functional related to nonlinear elasticity with prestress, which is a perturbation of a periodic stress-free joint. The homogenization of such non-convex functions yields a multi-cell formula that is impractical for both analysis and numerical computation. The goal of this lecture is to study a quadratic expansion of the homogenized energy using Gamma-convergence, which yields an explicit linearized limit. From this, we can derive explicit first-order information about the model. The analysis of this expansion involves studying several interesting properties of maps with periodic derivatives that are related to periodic stress-free joints.

Transport Equation: Renormalization and Quantitative Regularity Estimates
Prof. Tomasz Dębiec (University of Warsaw)
In these lectures, we will focus on the transport equation with nonsmooth velocity fields. The concept of renormalized solutions, introduced in the seminal paper by DiPerna and Lions, plays an important role not only in the study of the transport equation and related ordinary differential equations, but also for a variety of larger systems of PDEs that include the transport equation (usually in its conservative form). We will review the classical notion of renormalized solutions and use it to establish 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 solution with heterogeneous coefficients behaves qualitatively differently than in the bulk of the domain. This is most easily seen in the fact that the standard ansatz used to describe the solution with heterogeneous coefficients in the interior does not have the correct boundary conditions. To remedy this, one can introduce homogenization correctors that satisfy homogeneous Dirichlet boundary conditions—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

Mailing list:

We maintain a mailing list for the Graduate Lecture Series. All announcements—including upcoming talks, schedule updates, and any last-minute changes—are distributed through this list. If you would like to subscribe, simply send an email to Prof. Dr. Stefan Neukamm.