The activity in mathematical analysis is mainly focussed on ordinary and partial differential equations, on dynamical systems, on the calculus of variations, and on control theory. Connections of these topics with differential geometry are also developed.The activity in mathematical modelling is oriented to subjects for which the main technical tools come from mathematical analysis. The present themes are multiscale analysis, mechanics of materials, micromagnetics, modelling of biological systems, and problems related to control theory.The applications of mathematics developed in this course are related to the numerical analysis of partial differential equations and of control problems. This activity is organized in collaboration with MathLab for the study of problems coming from the real world, from industrial applications, and from complex systems.

## On the Cauchy problem for the wave equation on time-dependent domains

## BV functions

- Vector valued Radon measures
- Notions of weak convegrence of measures
- Definition of BV and of perimeter
- Semicontinuity of the perimeter and of the total variation
- Approximation of BV functions by smooth functions
- Approximation of sets with finite perimeter by smooth sets
- Embedding theorems and isoperimetric inequalities
- Coarea formula
- Traces of BV functions
- Carathéodory costruction
- Hausdorff measures and comparison with Lebesgue mesasure

## Hyperbolic systems of conservation laws

The course is an introduction to systems of conervation/balace laws. The list of subjects are

- Balance and conservation laws

- Cauchy problem

- Entropy solutions

- Scalar equations

- Admissible solutions and Riemann problem

- Glimm scheme and wavefront tracking

- Uniqueness

- Vanishing viscosity

- Compensated compactness