PORTUGALIAE MATHEMATICA Vol. 53, No. 1, pp. 7387 (1996) 

Evolution Problems Associated with Nonconvex Closed Moving Sets with Bounded VariationC. Castaing and Manuel D.P. Monteiro MarquesDépartement de Mathématiques, Case 051,Université de Montpellier II, F34095 Montpellier Cédex 5  FRANCE C.M.A.F. and Faculdade de Ciências da Universidade de Lisboa, Av. Prof. Gama Pinto, 2, P1699 Lisboa  PORTUGAL 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 rightcontinuous 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 rightcontinuous 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; ScorzaDragoni's theorem; Dugundji's extension theorem. Classification (MSC2000): 35K22, 34A60, 34G20 Full text of the article:
Electronic version published on: 29 Mar 2001. This page was last modified: 27 Nov 2007.
© 1996 Sociedade Portuguesa de Matemática
