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: wagner@math.tu-cottbus.de

Received: 30 August 2006
Revised: 19 June 2007
Revised: 23 October 2007

Abstract

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.