TY - JOUR

T1 - Multidimensional graph completions and cellina approximable multifunctions

AU - Bressan, Alberto

AU - Deforest, Russell

PY - 2011

Y1 - 2011

N2 - 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 [12] to functions of several space variables. The second part introduces the notion of Cellina W1,p-approximable multifunction. To show its relevance, we consider the Cauchy problem on the plane x ∈ F(x), x(0) = 0 ∈ R2. 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 W1,1-approximable.

AB - 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 [12] to functions of several space variables. The second part introduces the notion of Cellina W1,p-approximable multifunction. To show its relevance, we consider the Cauchy problem on the plane x ∈ F(x), x(0) = 0 ∈ R2. 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 W1,1-approximable.

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U2 - 10.1216/RMJ-2011-41-2-411

DO - 10.1216/RMJ-2011-41-2-411

M3 - Article

AN - SCOPUS:79958727898

VL - 41

SP - 411

EP - 444

JO - Rocky Mountain Journal of Mathematics

JF - Rocky Mountain Journal of Mathematics

SN - 0035-7596

IS - 2

ER -