Vorlesung

Spaces of Radon Measures: Functional Analysis and Applications to PDEs

Structured population models are transport--type equations used to describe dynamics of populations with respect to a specific structural variable such as age, size or cell maturity. In this approach, the solution can be interpreted as a distribution of the population over possible values of the structural variable. This generality results in their wide applicability in demography, cell biology, immunology or ecology. The goal of this lecture is the introduction of such equations in the setting of Radon measures over an abstract metric space (S,d) which is less restrictive for possible applications. To this end, profound knowledge on the functional analytic properties of the underlying measure space M(S) is needed and will be provided in the first part of the lecture. Prerequisites: A basic course on functional analysis is required; Knowledge on measure theory and weak partial differential equations will be helpful but not necessary. The lecture will cover the following topics: Basics on Lipschitz functions and Functional Analysis: Norms for Lipschitz functions, approximations, proper metric spaces Basics on Measure Theory: signed measures, narrow convergence, tightness, total variation norm Properties of M(S) and the cone M+(S): introduction of the flat norm, separability, completeness, (local) compactness results Structured Population Models on R+: existence, uniqueness, dependence on model functions, numerical algorithms Structured Population Model on a general proper space

Update: Due to the current situation, the semester will be organised online. Lectures will be streamed live on Microsoft Teams during the regular lecture time. \\ Every one or two weeks, there is discussion session on Wednesday 9.15 o'clock. It is also organised via Microsoft Teams, in the same channel as the lecture. The aim of the discussion session is to answer your questions arising during the lecture and to present further examples. On Monday, 20.4.2020, an email containing more organisatorial information concerning the lecture will be send to all students enlisted in our Müsli course. So if you haven't registered yet, please do so until Monday morning.

Lecture Lecturer: Professor Dr. Anna Marciniak-Czochra Head Assistant: Christian Düll Time: Tuesday 11:15-12:45 (online) und Thursday 11:15-12:45 (online) Tutorials: Wednesday, 9.15-10.45

Please register via Müsli to the lecture.

Literature:

• A. Klenke: Probability Theory
• P. Billingsley: Probability and Measure
• H.W. Alt: Linear Functional Analysis
• V.I. Bogachev: Measure Theory Vol I and II
• R.M. Dudley: Real Analysis and Probability

Organising the lecture is very beneficial for our book on which the lecture is based. We would like to thank all students who helped us improving the text by handing in questions, pointing out mistakes and putting in suggestions.