Teaching
Summer term 2021: Partial differential equations
All information and exercise sheets are on the Moodle page.
Summer term 2018: Multidimensional calculus of variations (M4)
Lectures: Tuesday, 13:00 (ct), RUD 25, 3.007 and Thursday, 09:00 (ct), RUD 25, 3.007 Tutorial: Tuesday, 15.00 (ct), RUD 25, 3.007
Content
In the theory of calculus of variations, we consider functionals defined on sets of functions with the goal to find critical points of these functionals. Typical applications are: The circle maximizes the area for a given perimeter. The soap film minimizes the area for given volume. A stable elastic deformation minimizes the elastic energy. The lecture starts by recalling the classical theory starting from Bernoulli to Weierstrass for the one-dimensional setting. Using simple functional analytic methods we extend the theory to the multidimensional case including nonlinear elasticity theory.
Literature
B. Dacorogna, Direct Methods in the Calculus of Variations
B. Dacorogna, Introduction to the Calculus of Variations
Exercises
(Exercise 1) for April 24, 2018
(Exercise 2) for May 8, 2018
(Exercise 3) for May 15, 2018
(Exercise 4) for May 22, 2018
(Exercise 5) for May 29, 2018
(Exercise 6) for June 5, 2018
(Exercise 7) for June 12, 2018
(Exercise 8) for June 19, 2018
(Exercise 9) for June 26, 2018
(Exercise 10) for July 3, 2018
(Exercise 11) for July 10, 2018
(Exercise 12) for July 17, 2018
Winter term 2017/2018: Optimal transport and Wasserstein gradient flows
Lecture: Tuesday, 9.00 am (ct), RUD 25, 4.007, weekly Tutorial: Tuesday, 11.00 am (ct), RUD 25, 4.007, every 2nd week
Content
The optimal transport problem was already formulated by Gaspard Monge in the 18 century. It deals with the relocation of an initial distribution of mass to a final distribution, such that the cost of transport is minimal. The formulation of this problem was generalized by Kantorovich in 1942. Besides the original applications in economy, new connections to Problems in geometry, probability theory, and analysis emerged. In particular, in the recent decades a strong connection between partial differential equations, that describe diffusion processes, could be made. These diffusion problems can be formulated as gradient flows of the system's entropy and the so-called Wasserstein distance.
In this module, we introduce the problem of optimal transport, discuss basic results and applications: Monge- and Kantorovich formulation, existence of optimal transport plans, dual formulation, dynamical formulation, diffusion equations as Wasserstein gradient flows.
Literature
Ambrosio, Gigli, Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lecture in Mathematics ETH Zürich, 2005
Santambrogio, Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling, Birkhäuser, 2015