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Soneji, Parth (2018): On growth conditions for quasiconvex integrands. In: Quarterly Journal of Mathematics, Vol. 69, No. 2: pp. 611-630
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We prove that, for 1 <= p < 2, if a W-1,W-p-quasiconvex integrand f : R-Nx n -> R has linear growth from above on the rank-one cone, then it must satisfy this growth for all matrices in R-Nx 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. This result was first anticipated in (P. Soneji, Relaxation in BV of integrals with superlinear growth, ESAIM Control Optim. Calc. Var. 20 (2014), 1078-1122), with some partial results given. 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 generalize this result for W-1,W-p-quasiconvex integrands, where 1 <= p < k <= min {n, N}.