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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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