Brandenburg University of Technology, Cottbus; Department of Mathematics, P.O.B. 10 13 44, 03013 Cottbus, Germany. e-mail: email@example.com
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.
(Received August 30 2006)
(Revised June 19 2007)
(Revised October 23 2007)
(Online publication January 23 2009)
Mathematics Subject Classification: