Teaching

Winter Semester 2025/26: 

V5B5 Advanced Topics in Calculus of Variations: Free boundary problems

  • Tue 10.15-11.45: room 2.040
  • Thu 08.30-10.00: room 1.007

 

Link to Vorlesungsverzeichnis

 

Summary: Free boundary problems are partial differential equations that exhibit a priori unknown interfaces, the so-called free boundaries. Such problems arise for instance in physics, biology, or finance, with a classical example being the melting of ice in water. A central mathematical challenge is to understand the structure and regularity of the free boundary, using tools from PDEs, calculus of variations, and geometric measure theory.

This course gives an introduction to the theory of free boundary problems, focusing on two fundamental models: the one-phase Bernoulli problem and the obstacle problem. For these problems we will discuss basic properties of solutions, the classification of blow-ups, and the smoothness of the free boundary near regular points. If time permits, we will also study the thin obstacle problem and its relation to nonlocal PDEs.

 

Literature:

 

Lecture notes: (last update: 22.01.2026)

 

Summer Semester 2026: 

S4B2 Graduate Seminar on PDEs: Kinetic theory and the Boltzmann equation

  • Wed 10.15-11.45: room 0.011
     

Preliminary meeting: 03 February 2026, 16-18, room 1.007
(If you are interested in the seminar but cannot attend the meeting, 
please let me know before 03 Feb.)


Link to Vorlesungsverzeichnis

 

Summary: Kinetic theory emerged in the 19th century through the fundamental works of Maxwell and Boltzmann to statistically describe large systems of particles. The Boltzmann equation is one of the central equations in this field and it models the evolution of a dilute gas at a mesoscopic scale. The unknown in the Boltzmann equation is a probability density f(t,x,v) which keeps track of the number of particles that have velocity v at time t at the point x.

The goal of this seminar is to give an introduction to the Boltzmann equation and to other relevant kinetic models, exploring this topic from various angles. We will focus on selected topics in the field, some of which are motivated by recent developments, including for instance:

  • linear kinetic models (Fokker-Planck Kolmogorov equations)
  • Boltzmann collision kernels
  • hydrodynamic and diffusion limits
  • convergence to equilibrium
  • short time well-posedness
  • conditional regularity theory
  • well-posedness for space homogeneous equations
  • Landau equation

 

Literature:

  • C. Cercignani. The Boltzmann equation and its applications. 1988
  • C. Villani. A review of mathematical topics in collisional kinetic theory. 2001
  • F. Anceschi, S. Polidoro. A survey on the classical theory for Kolmogorov equation. 2019
  • C. Imbert, L. Silvestre. Regularity for the Boltzmann equation conditional to macroscopic bounds. 2020
  • C. Villani. Fisher information in kinetic theory. 2025

Links