ESAIM: Control, Optimisation and Calculus of Variations

Table of Contents - January 2009 - Volume 15, Issue 01  

Research Article

On the lower semicontinuous quasiconvex envelope for unbounded integrands (I)

Wagner, Marcus

Brandenburg University of Technology, Cottbus; Department of Mathematics, P.O.B. 10 13 44, 03013 Cottbus, Germany. e-mail: wagner@math.tu-cottbus.de

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 $\subset \mathbb{R}^{nm}$ instead of the whole space $\mathbb{R}^{nm}$ 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 $f^{(qc)} (v) = {\rm sup} \{ \,g(v)\, \vert \,g : \mathbb{R}^{nm} \rightarrow \mathbb{R} \cup \{ + \infty \}$ quasiconvex and lower semicontinuous, $g(v) \leq f(v) \,\,\,\,\forall v \in \mathbb{R}^{nm}\,\}.$ Our main result is a representation theorem for $f^{({\it qc})}$ 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 $f^{({\it qc})}$ in two examples.


(Received August 30 2006)

(Revised June 19 2007)

(Revised October 23 2007)

(Online publication January 23 2009)

Key Words:

  • Unbounded function;
  • quasiconvex function;
  • quasiconvex envelope;
  • Morrey's integral inequality;
  • representation theorem

Mathematics Subject Classification:

  • 26B25;
  • 26B40;
  • 49J45;
  • 52A20