On growth conditions for quasiconvex integrands

TL;DR

This paper proves that for certain integrands with linear growth in rank-one directions, the growth must be linear everywhere, revealing restrictions on the form of quasiconvex integrands with subquadratic growth.

Contribution

It establishes that linear growth in rank-one directions implies linear growth everywhere for $W^{1,p}$-quasiconvex integrands with $1 extless p extless 2$, using a novel Sobolev mapping construction.

Findings

No quasiconvex integrand with superlinear growth can have only linear rank-one growth for 1<p<2.

The proof involves constructing Sobolev functions mapping cubes to one-dimensional frames.

The technique generalizes to $W^{1,p}$-quasiconvex integrands with $1 extless p extless k extless n,N$.

Abstract

We prove that, for $1\leq p< 2$, if a $W^{1,p}$-quasiconvex integrand $\,f\colon\mathbb{R}^{N\times n}\rightarrow\mathbb{R}$ has linear growth from above on the rank-one cone, then it must satisfy this growth for all matrices in $\mathbb{R}^{N\times n}$. An immediate corollary of this is, for example, that there can be no quasiconvex integrand that has genuinely superlinear $p$ growth from above for $1<p<2$, but only linear growth in rank-one directions. The key element of this proof involves constructing a Sobolev function which maps points in a cube to some one-dimensional frame, and moreover preserves boundary values. This construction is an inductive process on the dimension $n$, and involves using a Whitney decomposition. This technique also allows us to generalise this result for $W^{1,p}$-quasiconvex integrands where $1\leq p < k\leq \min\{n,N\}$.