Calculus of Variations and Geometric Measure Theory

G. Dal Maso - I. Fonseca - G. Leoni - M. Morini

Higher Order Quasiconvexity Reduces to Quasiconvexity

created on 09 May 2003
modified on 16 Jan 2004


Published Paper

Inserted: 9 may 2003
Last Updated: 16 jan 2004

Journal: Archive Rational Mech. Anal.
Volume: 171
Pages: 55-81
Year: 2004


In this paper it is shown that higher order quasiconvex functions suitable in the variational treatment of problems involving second derivatives may be extended to the space of all matrices as classical quasiconvex functions. Precisely, it is proved that a smooth strictly $2$-quasiconvex function with $p$-growth at infinity, $p>1$, is the restriction to symmetric matrices of a $1$-quasiconvex function with the same growth. As a consequence, lower semicontinuity results for second-order variational problems are deduced as corollaries of well-known first order theorems.

Keywords: quasiconvexity