
Journal of Convex Analysis 16 (2009), No. 2, 441472 Copyright Heldermann Verlag 2009 On the Lower Semicontinuous Quasiconvex Envelope for Unbounded Integrands (II): Representation by Generalized Controls Marcus Wagner Brandenburg University of Technology, Dept. of Mathematics, P. O. Box 101344, 03013 Cottbus, Germany wagner@math.tucottbus.de [Abstractpdf] \def\K{\mathord{{\rm K}}} \def\R{{\mathbb R}^{nm}} \def\S{\mathord{{\rm S}}} [For the first part of this paper see ESAIM, Control, Optimisation and Calculus of Variations.]\par Motivated by the study of multidimensional control problems of Dieudonn\'eRashevsky type, e.g. nonconvex correspondence problems from image processing, we raise the question how to understand to notion of quasiconvexity for a continuous function $f$ with a convex body $\K \subset \R$ instead of the whole space $\R$ as the range of definition. Extending $f$ by $(+\infty)$ to the complement $\R\setminus\,\K$, the appropriate quasiconvex envelope turns out to be\par \medskip\hskip10mm $f^{(qc)}(w) = \sup\, \bigl\{g(w)\, \big\vert\, g \colon \R \to \mathbb{R} \cup \{(+\infty)\}$ quasiconvex\par \vskip2mm\hskip10mm and lower semicontinuous, $g(v) \le f(v)\ \forall v \in \R\bigr\}.$\par In the present paper, we prove that $f^{(qc)}$ admits a representation as\par \medskip\hskip10mm $f^{(qc)}(w) =$ Min $\bigl\{\int_{\K} f(v)\,d\nu(v)\, \big\vert\, \nu \in \S^{(qc)}(w)\bigr\} \quad \forall w \in \K$\par where the sets $\S^{(qc)} (w)$ are nonempty, convex, weak$^*$sequentially compact subsets of probability measures. This theorem, forming a natural counterpart to the author's previous results about the representation of $f^{(qc)}$ in terms of Jacobi matrices, has been proven indispensable for the derivation of Jensens' integral inequality as well as of differentiability theorems for the envelope $f^{(qc)}$. The paper is mainly concerned with a detailed analysis of the setvalued map $\S^{(qc)}$, which will be explicitely described in terms of averages of generalized controls. Keywords: Unbounded function, quasiconvex envelope, probability measure, generalized control, mean value theorem, setvalued map, representation theorem. MSC: 26B25, 26B40, 26E25, 49J45, 49J53 [ Fulltextpdf (274 KB)] for subscribers only. 