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

Wagner, Marcus

ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 1, pp. 68-101.

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 $\subset {ℝ}^{nm}$ instead of the whole space ${ℝ}^{nm}$ 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 $+\infty$. As an appropriate envelope, we introduce and investigate the lower semicontinuous quasiconvex envelope $f^{\left(qc\right)}\left(v\right)=\sup \left\{g\left(v\right)\right|g:{ℝ}^{nm}\to ℝ\cup \{+\infty \}$ quasiconvex and lower semicontinuous, $g\left(v\right)\le f\left(v\right)\forall v\in {ℝ}^{nm}\}.$ Our main result is a representation theorem for $f^{\left(\mathrm{𝑞𝑐}\right)}$ 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 $f^{\left(\mathrm{𝑞𝑐}\right)}$ in two examples.

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