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 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 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.
Mots clés : unbounded function, quasiconvex function, quasiconvex envelope, Morrey's integral inequality, representation theorem
@article{COCV_2009__15_1_68_0, author = {Wagner, Marcus}, title = {On the lower semicontinuous quasiconvex envelope for unbounded integrands {(I)}}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {68--101}, publisher = {EDP-Sciences}, volume = {15}, number = {1}, year = {2009}, doi = {10.1051/cocv:2008067}, mrnumber = {2488569}, zbl = {1173.26009}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv:2008067/} }
TY - JOUR AU - Wagner, Marcus TI - On the lower semicontinuous quasiconvex envelope for unbounded integrands (I) JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2009 SP - 68 EP - 101 VL - 15 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2008067/ DO - 10.1051/cocv:2008067 LA - en ID - COCV_2009__15_1_68_0 ER -
%0 Journal Article %A Wagner, Marcus %T On the lower semicontinuous quasiconvex envelope for unbounded integrands (I) %J ESAIM: Control, Optimisation and Calculus of Variations %D 2009 %P 68-101 %V 15 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2008067/ %R 10.1051/cocv:2008067 %G en %F COCV_2009__15_1_68_0
Wagner, Marcus. On the lower semicontinuous quasiconvex envelope for unbounded integrands (I). ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 1, pp. 68-101. doi : 10.1051/cocv:2008067. http://www.numdam.org/articles/10.1051/cocv:2008067/
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