Measure properties of the set of initial data yielding non uniqueness for a class of differential inclusions (Q1817352)

Uniqueness of solutions of the differential inclusion \(x'\in-\partial V(x)\), \(x(0)= x_0\) is studied. It is a well known fact that for \(V\) convex and locally Lipschitz the subdifferential \(\partial V\) is a maximal monotone operator and the above differential inclusion has uniquely defined solutions on \([0,\infty)\). This result is generalized in the paper. Instead of assuming convexity of \(V\) it is assumed only that the level sets of \(V\) are convex. Then it is proved that under some additional assumptions the set of those initial conditions from which two integral curves of the differential inclusion start has zero measure.