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Evolution problems associated with nonconvex closed moving sets with bounded variation. (English) Zbl 0848.35052
Summary: We consider the following new differential inclusion $- du\in N_{C(t)} (u(t))+ F(t, u(t)),$ where $$u: [0, T]\to \mathbb{R}^d$$ is a right-continuous function with bounded variation and $$du$$ is its Stieltjes measure; $$C(t)= \mathbb{R}^d\backslash \text{Int } K(t)$$, where $$K(t)$$ is a compact convex subset of $$\mathbb{R}^d$$ with nonempty interior; $$N_{C(t)}$$ denotes Clarke’s normal cone and $$F(t, u)$$ is a nonempty compact convex subset of $$\mathbb{R}^d$$. 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 or $$F$$ is bounded or has sublinear growth.
In particular, these results extend the Theorem 4.1 in C. Castaing, T. X. Dúc Hā and M. Valadier [Evolution equations governed by the sweeping process, Set-Valued Anal. 1, No. 2, 109-139 (1993; Zbl 0813.34018)].

##### MSC:
 35G10 Initial value problems for linear higher-order PDEs 35K25 Higher-order parabolic equations 34G20 Nonlinear differential equations in abstract spaces 34A60 Ordinary differential inclusions
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