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Periodic solutions of evolution problem associated with moving convex sets. (English) Zbl 0854.35134

This paper requires for its understanding a good familiarity with the theory of multivalued operators. In section 2, the authors introduce on two pages fifteen notions and concepts which are constantly used in later parts. We reproduce just few of them. Let \(H\) be a separable Hilbert space. Then \(cwk(H)\) is the family of subsets \(H\) which are nonempty, weakly closed and convex. If \(C\subseteq H\) is \(\neq \phi\) and convex then \(N_C(u)\) is the normal cone of \(C\) at \(u\in H\).
The problems to be solved are of the following type. Let there be given a multivalued function \(C: [0, T]\to H\) such that \(C(t)\in cwk(H)\) for \(t\in [0, T]\) and \(C(0)= C(T)\) (periodicity). Moreover, let there be given a Caratheodory function \(f: [0, T]\times H\to H\). One seeks a continuous function \(u: [0, T]\to H\) of bounded variation and a Radon measure \(\nu\) on \([0, T]\) such that the following holds: (1) \(u(0)= u(T)\), (2) \(u(t)\in C(t)\), \(t\in [0, T]\), (3) \(Du\), \(dt\) are absolutely continuous with respect to \(\nu\), where \(Du\) is the differential measure of \(u\) and \(dt\) Lebesgue measure on \([0, T]\), \[ - {d\over dt} Du(t)- f(t, u(t)) {dt\over d\nu}\in N_{C(t)}(u(t)),\quad d\nu\text{-a.e.}\tag{4} \] After a preparatory section which contains standard results from the field, the authors prove a series of results concerning the above problem. We cite just one. Let \(C: [0, T]\to cwk(H)\) be \(k\)-Lipschitzian and \(T\)-periodic, i.e. \(C(0)= C(T)\). Let \(\omega: [0, T]\to \mathbb{R}\) be integrable and \(\int^T_0 \omega ds> 0\). Then there exists a unique, absolutely continuous, \(T\)-periodic function \(u: [0, T]\to H\) such that \[ - {du\over dt}\in N_{C(t)}(u(t))+ \omega(t) u(t),\quad dt\text{-a.e.} \] Further results along these lines are proved and discussed.

MSC:

35R70 PDEs with multivalued right-hand sides
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
34A60 Ordinary differential inclusions
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