Multidimensional Graph Completions and Cellina Approximable Multifunctions

Relying on the continuous approximate selection method of Cellina, ideas and techniques from Sobolev spaces can be applied to the theory of multifunctions and differential inclusions. The first part of this paper introduces a concept of graph completion, which extends the earlier construction in [10] to functions of several space variables. The second part introduces the notion of Cellina W 1,p approximable multifunction. To show its relevance, we consider the Cauchy problem on the plane ẋ ∈ F (x), x(0) = 0 ∈ IR. If F is an upper semicontinuous multifunction with compact but possibly non-convex values, this problem may not have any solution, even if F is Cellina-approximable in the usual sense. However, we prove that a solution exists under the assumption that F is Cellina W 1,1 approximable.