Advanced Topics in Partial Differential Equations (WiSe 2023/24)
Welcome to the "Advanced Topics in Partial Differential Equations" course page. In this course, we will delve into the intricacies of partial differential equations, exploring advanced concepts and techniques that are crucial for understanding complex mathematical and scientific phenomena. The course will be conducted in person, and we encourage all interested students to register for the corresponding Moodle course. This is where you will find exercise sheets, course materials, and important updates related to the course.
This course comes with a particular emphasis on elliptic regularity, specifically Hölder regularity for solutions of elliptic PDEs. The foundation of this course is the book "Direct Methods in the Calculus of Variations" by E. Giusti. Throughout this course, you will develop a deep understanding of the Calculus of Variations, sharpen your analytical skills, and gain valuable insights into the Hölder regularity of solutions to elliptic PDEs.
In the first part of our journey, we will build the necessary tools and concepts essential for the main results on Hölder regularity. We'll begin with a concise introduction to Calculus of Variations, honing our focus on direct methods. From there, we'll delve into the study of convex functionals and semicontinuity, followed by an exploration of quasiconvexity and the concept of quasi-minima.
The second, and central, part of our course is where our main focus lies. Here, we will study various results on Hölder continuity of minimizers.
- E. Giusti: Direct methods in the calculus of variations. World Scientific Publishing Co., Inc., River Edge, NJ, 2003.
If you are interested in this course, you should be familiar with the following concepts:
- Lebesgue integration (as taught in the course "Maßtheorie")
- Sobolev spaces (usually introduced in "Funktionalanalsis" or "Partial Differential Equations")
The mode of the exam will be announced later during the semester.