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

Abstract

We study uniqueness property for the Cauchy problemx′∈ϖV(x), x(0)=ξ, whereV∶R ⁿ→R is a locally Lipschitz continuous, quasiconvex function (i.e. the sublevel sets {V≤c} are convex) and ϖV(x) is the generalized gradient ofV atx. We prove that if 0∉ϖV(x) forV(x)≥b, then the set of initial data ξ∈{V=b} yielding non uniqueness of solution in a geometric sense has (n−1)-dimensional Hausdorff measure zero in {V=b}.