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 Lp 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.
