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On the projections of multivalued maps. (English) Zbl 0885.49001
The problem considered in the paper can be described very roughly as follows. There are given an open set \(\Omega\subset{\mathbb{R}}^n,\) an integer \(m,\) a multivalued map \(\Pi:\Omega\rightarrow 2^{{\mathbb{R}}^{m\times n}}\) and the set \(\Pi'\) of square integrable selections of \(\Pi.\) The author gives sufficient conditions for inclusions of the type \(0\in \text{cl } P\Pi'\) to be valid in the case of a piecewise constant map \(\Pi\) and the map \(\Pi\) is given by a Nemitskii operator, respectively, where \(P\) is a certain projection operator. Problems of such a type occur for example in the control theory for partial differential equations, where the main part of the differential operator depends on the control parameter. An application to such a control problem for elliptic systems is given.
Reviewer: M.Goebel (Halle)

49J20 Existence theories for optimal control problems involving partial differential equations
35J50 Variational methods for elliptic systems
Full Text: DOI
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