On the lower semicontinuity and approximation of \({L^{\infty}}\)-functionals

  • Francesca PrinariEmail author


In this paper we show that if the supremal functional
$$F(V,B)= \mathop{\rm ess sup} \limits_{x \in B} f(x,DV (x))$$
is sequentially weak* lower semicontinuous on \({W^{1,\infty}(B, \mathbb{R}^d)}\) for every open set \({B \subseteq \Omega}\) (where \({\Omega}\) is a fixed open set of \({\mathbb{R}^N}\)), then \({f(x,\cdot)}\) is rank-one level convex for a.e \({x \in \Omega}\). Next, we provide an example of a weak Morrey quasiconvex function which is not strong Morrey quasiconvex. Finally we discuss the L p -approximation of a supremal functional F via \({\Gamma}\)-convergence when f is a non-negative and coercive Carathéodory function.


Supremal functionals Lower semicontinuity Strong Morrey quasiconvexity \({\Gamma}\)-convergence 

Mathematics Subject Classification

49J45 49A50 


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Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Dip. di Matematica e InformaticaUniversità di FerraraFerraraItaly

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