Calculus of Variations and Geometric Measure Theory
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F. Cagnetti

k-Quasiconvexity Reduces to Quasiconvexity

created by cagnetti on 24 May 2010
modified on 13 Nov 2013


Published Paper

Inserted: 24 may 2010
Last Updated: 13 nov 2013

Journal: Proc. Roy. Soc. Edinburgh Sect. A
Volume: 141
Pages: 673-708
Year: 2011


The relation between quasiconvexity and $k$-quasiconvexity, $k \geq 2$, is investigated. It is shown that every smooth strictly $k$-quasiconvex integrand with $p$-growth at infinity, $p>1$, is the restriction to $k$-th order symmetric tensors of a quasiconvex function with the same growth. When the smoothness condition is dropped, it is possible to prove an approximation result. As a consequence, lower semicontinuity results for $k$-th order variational problems are deduced as corollaries of well-known first order theorems. This generalizes a previous work by Dal Maso, Fonseca, Leoni and Morini, in which the case $k=2$ was treated.

Keywords: quasiconvexity, higher order variational problems


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