# teaching

### WS 2022/23 - Multidimensional Calculus of Variations

**Lectures**:

- Mon 11:00am, JvN, RUD25, Room 2.006,
- Wed 1:00pm, JvN, RUD25, Room 2.006.

**Exercise**:

- Mon 1:00pm, JvN, RUD25, Room 2.006.

**Lectures:**

- Dr. Thomas Eiter
- Dr. Matthias Liero

**Office hours:** Mon 4:00pm, Room: tba

**Tutors:**

- Melanie Koser
- Anastasija PeĹˇiÄ‡

**Contents.** The calculus of variations is a classical branch of calculus that is concerned with the study of minimization problems for (nonlinear) functionals defined on suitable function spaces. This means, one is looking for a function (of multiple variables) that minimizes a given functional. The functionals are often expressed as definite integrals involving the functions and their derivatives. The origin of the calculus of variations was the solution of the brachystochrone problem in 1697 by Jakob Bernoulli.

This lecture covers:

- first and second variations of multidimensional integral functionals
- necessary and sufficient conditions for weak and strong minimizers
- abstract theory of minimization, direct methods of calculus of variations, weak lower semi-continuity
- various notions of convexity: rank-1-, poly- and quasi-convexity
- existence theorems for global minimizers in Sobolev spaces
- extrema under constraints, eigenvalue problems
- applications from physics: quantum mechanics, linear and nonlinear elasticity

**References.**
The lecture will not follow a specific book. However, the details for the topics treated in the lecture can be found in the following books:

- B. Dacorogna. Introduction to the calculus of variations. Imperial College Press, London, 2004.
- B. Dacorogna. Direct Methods in the Calculus of Variations. Springer 1989.
- F. Rindler. Calculus of Variations. Springer, 2018

### SS 2021 - Partial Differential equations

The lectures and exercise will be held online via Zoom.

**Lectures**:

- Wed 9am â€“ 11am (starting at 9h15)
- Thu 11am â€“ 13pm (starting at 11h30)

**Exercise**:

- Thu 13-15 (starts at 13h30)

**Lecturers**:

- Prof. Dr. Alexander Mielke
- Dr. Matthias Liero

**Tutors**:

- Lukas Abel
- Anastasija PeĹˇiÄ‡

**Contents.**
This course provides a thorough introduction to linear partial differential equations.

- Scalar first order equation
- Elementary PDEs: heat equation, wave equation, Laplace equation
- Solutions of linear problems via orthogonal series (separation)
- Elliptic problems via Laxâ€“Milgram theory, maximum principles
- Existence and smoothing properties of parabolic equations
- Hyperbolic equations in several space dimensions
- Semilinear equations

**Literature**

- Ben Schweizer. Partielle Differentialgleichungen, Springer Spektrum, 2013
- Lawrence C. Evans. Partial Differential Equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1998.
- Fritz John. Partial Differential Equations, volume 1 of Applied Mathematical Sciences. Springer-Verlag, New York, fourth edition, 1982.
- JĂĽrgen Jost. Partielle Differentialgleichungen. Springer, Berlin, 1998. (analysis!).
- Michael Renardy and Robert C. Rogers. An Introduction to Partial Differential Equations, volume 13 of Texts in Applied Mathematics. Springer-Verlag, New York, second edition, 2004.
- Walter A. Strauss. Partielle Differentialgleichungen (Eine EinfĂĽhrung). Vieweg, 1995. (many classical examples, less deep).

### SS 2018 - Multidimensional Calculus of Variations

**Lectures**:

- Tue 1pm (ct), RUD 25, 3.007
- Thu, 9am (ct), RUD 25, 3.007

**Exercise**:

- Tue 3pm (ct), RUD 25, 3.007

**Lecturers**:

- Dr. Matthias Liero

**Tutors**:

- Dr. Matthias Liero

**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. Introduction to the calculus of variations. Imperial College Press, London, 2004.
- B. Dacorogna. Direct Methods in the Calculus of Variations. Springer 1989.

### WS 2017/18 - Optimal transport and Wasserstein Gradient Flows

**Lecture**:

- Tue 9 am (ct), RUD 25, 4.007, weekly

**Tutorial**:

- Tue 11 am (ct), RUD 25, 4.007, every 2nd week

**Lecturers**:

- Dr. Matthias Liero

**Tutors**:

- Dr. Matthias Liero

**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 systemâ€™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, SavarĂ©, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lecture in Mathematics ETH ZĂĽrich, 2005
- Santambrogio, Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling, BirkhĂ¤user, 2015