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.

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 nm instead of the whole space 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)= sup {g(v)|g: nm {+} quasiconvex and lower semicontinuous, g(v)f(v)v nm }. 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.

DOI : 10.1051/cocv:2008067
Classification : 26B25, 26B40, 49J45, 52A20
Mots clés : unbounded function, quasiconvex function, quasiconvex envelope, Morrey's integral inequality, representation theorem
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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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