Mathematical Institute of the University of Bonn

V5B5 - Advanced Topics in Analysis and Calculus of Variations: Regularity for Elliptic Variational Problems

Instructor: Dr. Franz Gmeineder

Timetable:

Mondays, 16 (c.t.) - 18, Endenicher Allee 60, SR 1.007
Thursdays, 16(c.t.) - 18, Endenicher Allee 60, SR 1.008
Note: There will be no lectures in the week of May 20-24.

Exam (oral):

The exams to this course will take place in calendar weeks 29 and 30 (or by personal arrangement). Please sign up via e-mail (fgmeined(at)math.uni-bonn.de)!

Course Synopsis:

The central aim of regularity theory is to establish that solutions of certain partial differential equations or variational problems are better behaved than generic competitor maps. This course intends to give an introduction to the regularity theory for elliptic variational problems, and to present recent developments on an accesible level. After a quick recap of the direct method and the required notions of convexity, we will discuss which additional regularity properties minimisers feature. Besides higher Sobolev regularity, we will recap the by now classical De Giorgi-Nash-Moser theory for (scalar) equations. As the corresponding Hölder continuity and boundedness results do not necessarily hold for systems of equations – and hereafter vectorial variational problems – anymore, we focus on which regularity properties still survive. This comprises, amongst others, partial Hölder continuity and aspects of nonlinear potential theory for vectorial problems. The focus is on modern techniques and will – in some instances – lead to state-of-the-art results.

Prerequisites:

• Analysis 1–3 and Functional Analysis. An introductory course on partial differential equations is helpful, but not mandatory. Students that took the course 'Nonlinear PDE 1' from winter term 2018/19 will discover some overlaps, but this course is essentially self-contained.

Specific course material:

• Lecture Notes (Updated Thu, May 9)

Background Reading:
• L. Ambrosio, N. Fusco, D. Pallara: Functions of bounded variation and free discontinuity problems. Oxford University Press, 2000.
• L. Beck: Elliptic Regularity Theory - A first course. Lecture Notes of the Italian Mathematical Union, Springer, 2016.
• E. Giusti: Direct Methods in the Calculus of Variations. World Scientific, 2003.
• G. Mingione: Regularity of minima: an invitation to the dark side of the calculus of variations. Appl. Math. 51 (2006), no. 4, 355–426.
• A more extensive list of relevant background material is available here.
• Lecture notes will be available and shall be updated weekly.