Soneji, Parth
(2018):
On growth conditions for quasiconvex integrands.
In: Quarterly Journal of Mathematics, Vol. 69, No. 2: pp. 611630

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Abstract
We prove that, for 1 <= p < 2, if a W1,Wpquasiconvex integrand f : RNx n > R has linear growth from above on the rankone cone, then it must satisfy this growth for all matrices in RNx 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 rankone directions. This result was first anticipated in (P. Soneji, Relaxation in BV of integrals with superlinear growth, ESAIM Control Optim. Calc. Var. 20 (2014), 10781122), with some partial results given. The key element of this proof involves constructing a Sobolev function which maps points in a cube to some onedimensional 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 generalize this result for W1,Wpquasiconvex integrands, where 1 <= p < k <= min {n, N}.