Consider a function on a Euclidean space with a gradient of
locally bounded variation and such that the gradient takes values in a
fixed finite set almost everywhere. Then the preimages of the elements
of that set form a finite Caccioppoli partition of the Euclidean space.
Such functions can arise in certain models from materials science, for
example in the context of an Allen-Cahn (Modica-Mortola) type energy
involving the gradient.
The theory of Caccioppoli partitions gives some information about the
structure of such a function. The fact that we are dealing with a
gradient, however, gives rise to much more rigidity, especially when we
assume that the set of possible values for the gradient is either convex
independent or affine independent. Then the given function is piecewise
affine away from a small subset of the domain.
