K-Quasiconvexity Reduces to Quasiconvexity

F. Cagnetti Department of Mathematical Sciences Carnegie Mellon University cagnetti@andrew.cmu.edu

Abstract: The relation between quasiconvexity and k-quasiconvexity, k ≥ 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.