BV Functions and Sets of Finite Perimeter, Variational Calculus
(Lecture course in Summer Term 2022)

Instructors: Thomas Schmidt (lecture), Jule Schütt (exercise class).

Lecture (weekly, first on April 6th):

• Wed, 8-10, H5

Exercise class (biweekly (!), first on April 13th):

• Wed, 12-14, Room 435

Relevance: The course is eligible in the Master Program in Mathematics, Industrial Mathematics, Mathematical Physics, or Economathematics. Further, it is mandatory in the Intermaths Program and generally recommended to everyone with interests in analysis.

ECTS points: The module (lecture plus exercise class) counts 6 ECTS points.

Prerequisites: The lecture builds on basic knowledge in analysis (including the theory of Lebesgue integration) and linear algebra.

Contents: The lecture gives an introduction to the theory of functions of bounded variation and sets of finite perimeter and to the calculus of variations.

BV functions are (single-variable or multiple-variable) functions whose weak derivatives are representable as measures. The concept generalizes Sobolev functions (whose weak derivatives are representable as L^p functions) and allows to express derivative information even for functions with essential jump discontinuities. Sets of finite perimeter have finite surface area in a suitably generalized sense and turn out to be intimately related to BV functions.

The calculus of variations is a classical mathematical discipline, which has its roots in the 19th century but has remained an active research area ever since. It is concerned with minimization problems which depend on a function as the unknown (or, in the geometric branch, rather on a surface or a set) and thus cope with infinitely many degrees of freedom. Existence theorems for (generalized) solutions to such problems can typically be obtained among Sobolev or BV functions (or sets of finite perimeter), and the first-order conditions for solutions open up a close and distinctive connection to ODEs and PDEs.

Literature (partially introductory, partially more advanced):

• L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, 2000,
• G. Buttazzo, M. Giaquinta, S. Hildebrandt, One-Dimensional Variational Problems, Oxford University Press, 1998,
• B. Dacorogna, Introduction to the Calculus of Variations, Second Edition, Springer, 2008,
• L.C. Evans, R.F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, 1992,
• G.B. Folland, Real Analysis, Wiley, 1999,
• E. Giusti, Functions of Bounded Variation and Minimal Surfaces, Birkhäuser, 1984,
• F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems, Cambridge University Press, 2012,
• F. Rindler, Calculus of Variations, Springer, 2018.