On the lower semicontinuous quasiconvex envelope for unbounded integrands (I)
Brandenburg University of Technology, Cottbus; Department of Mathematics, P.O.B. 10 13 44, 03013 Cottbus, Germany. e-mail: firstname.lastname@example.org
Revised: 19 June 2007
Revised: 23 October 2007
Motivated by the study of multidimensional control problems of Dieudonné-Rashevsky type, we raise the question how to understand to notion of quasiconvexity for a continuous function f with a convex body K instead of the whole space as the range of definition. In the present paper, we trace the consequences of an infinite extension of f outside K, and thus study quasiconvex functions which are allowed to take the value +∞. As an appropriate envelope, we introduce and investigate the lower semicontinuous quasiconvex envelope quasiconvex and lower semicontinuous, Our main result is a representation theorem for which generalizes Dacorogna's well-known theorem on the representation of the quasiconvex envelope of a finite function. The paper will be completed by the calculation of in two examples.
Mathematics Subject Classification: 26B25 / 26B40 / 49J45 / 52A20
Key words: Unbounded function / quasiconvex function / quasiconvex envelope / Morrey's integral inequality / representation theorem
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