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{\bf $K$-Quasiconvexity Reduces to Quasiconvexity}\\

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F. Cagnetti \\
Department of Mathematical Sciences\\
Carnegie Mellon University\\
cagnetti@andrew.cmu.edu
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\noindent {\bf Abstract}: 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.

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