Graduate Lecture Series "Applied Analysis"

Welcome to our series of concise graduate lecture courses exploring a range of topics in Applied Analysis. These courses address both classical results and ongoing research in the field. Our first set of courses focuses on the theory of homogenization. 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

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