Decomposition and integral representation of Cauchy interactions associated with measures

It is well-known that the modelization of interactions in Continuum Physics deals with set functions associated with physical quantities rather than with functions evaluated at single points (see e.g. [7]). Very important examples of these are the stress and the heat flux. This, in turn, implies that the concept of subbody of a material body B has to be taken into account. However, subbodies are not completely physical (although they may be used to describe the situation arising in the body in a very satisfactory way), since the class of subsets which have to represent them is a matter of choice. For such set functions should satisfy some reasonable additivity condition, it is natural to put the approach into the framework of Measure Theory. An example of how this way of thinking has been developed is given by the Cauchy Stress Theorem, leading in [3] to the notion of Cauchy flux. For further developments, we refer the reader to [3, 8, 5, 1] and the references quoted therein. In [4], Gurtin, Williams and Ziemer proposed to choose the normalized sets of finite perimeter as subbodies and introduced the concept of Cauchy interaction in order to represent an interaction between two disjoint subbodies, possibly having a part of their boundary in common. This is, roughly speaking, a set function I of two variables, the subbodies, which is additive on each variable and which is Lipschitz continuous with respect to the area measure of the common part of the boundaries and the volume measure. In that paper it is proved that: