Measure properties of the set of initial data yielding non uniqueness for a class of differential inclusions

We study uniqueness property for the Cauchy problemxV(x), x(0)=, whereVR ⁿR is a locally Lipschitz continuous, quasiconvex function (i.e. the sublevel sets {Vc} are convex) and V(x) is the generalized gradient ofV atx. We prove that if 0V(x) forV(x)b, then the set of initial data {V=b} yielding non uniqueness of solution in a geometric sense has (n–1)-dimensional Hausdorff measure zero in {V=b}.