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

Marcus Wagner

ESAIM: Control, Optimisation and Calculus of Variations (2009)

  • Volume: 15, Issue: 1, page 68-101
  • ISSN: 1292-8119

Abstract

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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 n m instead of the whole space n m 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 ( q c ) ( v ) = sup { g ( v ) | g : n m { + } quasiconvex and lower semicontinuous, g ( v ) f ( v ) v n m } . Our main result is a representation theorem for f ( 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 ( 𝑞𝑐 ) in two examples.


How to cite

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Wagner, Marcus. "On the lower semicontinuous quasiconvex envelope for unbounded integrands (I)." ESAIM: Control, Optimisation and Calculus of Variations 15.1 (2009): 68-101. <http://eudml.org/doc/250566>.

@article{Wagner2009,
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.
},
author = {Wagner, Marcus},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Unbounded function; quasiconvex function; quasiconvex envelope; Morrey's integral inequality; representation theorem; unbounded function},
language = {eng},
month = {1},
number = {1},
pages = {68-101},
publisher = {EDP Sciences},
title = {On the lower semicontinuous quasiconvex envelope for unbounded integrands (I)},
url = {http://eudml.org/doc/250566},
volume = {15},
year = {2009},
}

TY - JOUR
AU - Wagner, Marcus
TI - On the lower semicontinuous quasiconvex envelope for unbounded integrands (I)
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2009/1//
PB - EDP Sciences
VL - 15
IS - 1
SP - 68
EP - 101
AB - 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.

LA - eng
KW - Unbounded function; quasiconvex function; quasiconvex envelope; Morrey's integral inequality; representation theorem; unbounded function
UR - http://eudml.org/doc/250566
ER -

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