Transport Equations and Multi-D Hyperbolic Conservation Laws

This book collects the lecture notes of two courses and one mini--course held in a winter school in Bologna in January 2005. The aim of this school was to popularize techniques of geometric measure theory among researchers and PhD students in hyperbolic differential equations. Though initially developed in the context of the calculus of variations, many of these techniques have proved to be quite powerful for the treatment of some hyperbolic problems. Obviously, this point of view can be reversed: the hope of the editors is that the topics of these notes will also capture the interest of some members of the elliptic community, willing to explore the links to the hyperbolic world. The three contributions of the volume gravitate all around the theory of functions of bounded variation which play a fundamental role in the subject of hyperbolic conservation laws, and focus in particular on the structure and fine properties of BV functions in more than one space dimension.