# John-von-Neumann lecture - Sommersemester 2017

### An introduction to Functions of Bounded Variations and Sets of Finite Perimeter

Prof. Nicola Fusco ## Tentative program

The course will provide a comprehensive introduction to the modern theory of functions of bounded variations (BV functions) and of sets of finite perimeter.
• BV functions: approximation by smooth functions, extension, composition with Lipschitz functions, compact imbedding. Pointwise variation and BV functions of one variable.
• Sets of finite perimeter: basic properties, reduced boundaries, convergence in measure. Coarea formula for BV functions, approximation of sets of finite perimeter by smooth sets. Embedding theorems and isoperimetric inequalities. Structure of sets of finite perimeter. Approximate continuity, jump sets and approximate differentiability of BV functions. Traces and decomposition of BV functions. Slicing.

## Time and Location

• Tuesday, 14-16, MI 03.08.011
• Wednesday, 14-16, MI 03.08.011
• starting May, 2nd, 2017

## Evaluation of the students

During the course exercises will be given to the students and there will be a final exam consisting of an oral presentation on some selected topics.

Prerequisites: Students should be familiar with the basic concepts of Measure Theory (Borel and Radon measures, Lebesgue integral, derivatives of measures, Lebesgue points) and with the basic concepts of Functional Analysis. A previous knowledge of Haudorff measures and Sobolev spaces is not mandatory, but can be helpful.

## References

• L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, 2000.
• L.C. Evans, R.F. Gariepy, Measure theory and fine properties of functions. CRC Press, Inc., Boca Raton, Florida, 1992.
• F. Maggi, Sets of finite perimeter and geometric variational problems. An introduction to geometric measure theory. Cambridge Studies in Advanced Mathematics 135. Cambridge University Press, Cambridge, 2012.