For every p>1 there are p-homogeneous rank-one convex integrands on R^{2xm}, and on large square matrix spaces, that are nowhere quasiconvex; for m large enough and in R^{4x2} when p≠2.
Sets of gradients with no rank-one connections,
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A solution to Morrey's problem in $\mathbb{R}^{2\times m}$
For every p>1 there are p-homogeneous rank-one convex integrands on R^{2xm}, and on large square matrix spaces, that are nowhere quasiconvex; for m large enough and in R^{4x2} when p≠2.