Evolution Problems Associated with Nonconvex Closed Moving Sets with Bounded Variation

C. Castaing and Manuel D.P. Monteiro Marques

Abstract: We consider the following new differential inclusion $$ -du\in N_{C(t)}(u(t))+F(t,u(t)), $$ where $u: [0,T]\to\R^{d}$ is a right-continuous function with bounded variation and $du$ is its Stieltjes measure; $C(t)=\R^{d}\backslash\Int K(t)$, where $K(t)$ is a compact convex subset of $\R^{d}$ with nonempty interior; $N_{C(t)}$ denotes Clarke's normal cone and $F(t,u)$ is a nonempty compact convex subset of $\R^{d}$. We give a precise formulation of the inclusion and prove the existence of a solution, under the following assumptions: $t\mapsto K(t)$ has right-continuous bounded variation in the sense of Hausdorff distance; $u\mapsto F(t,u)$ is upper semicontinuous and $t\mapsto F(t,u)$ admits a Lebesgue measurable selection (Theorem 3.4); $F$ is bounded (Theorem 3.2) or has sublinear growth (Remark 3.3). In particular, these results extend the Theorem 4.1 in [6].

Keywords: Evolution problems; sweeping processes; nonconvex sets; Clarke's normal cone; bounded variation; Scorza-Dragoni's theorem; Dugundji's extension theorem.

Classification (MSC2000): 35K22, 34A60, 34G20

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