Rank-one convex non-quasiconvex integrands are explicitly constructed in R^{2x4}, R^{4x4}_sym, and R^{3x3}_sym, via restrictions and transpositions of Grabovsky's example and a modification of Sverak's example.
Quasiconvexity for the Dacorogna--Marcellini Energy
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abstract
We prove that the planar Dacorogna--Marcellini energy $f_\gamma(A)=|A|^4-2\gamma|A|^2\det A$ is quasiconvex exactly when it is rank-one convex, i.e. if and only if $|\gamma|\leq\frac{2}{\sqrt3}$. The proof uses a monotonicity property of the energy functional along the componentwise heat flow. As a corollary of our method, we show that for homogeneous quartic polynomials on $2 \times 2$ matrices invariant by left and right rotation, quasiconvexity is equivalent to rank one convexity.
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Morrey's problem in $\mathbb{R}^{2 \times 4}$ and $\mathbb{R}^{3 \times 3}_\mathrm{sym}$
Rank-one convex non-quasiconvex integrands are explicitly constructed in R^{2x4}, R^{4x4}_sym, and R^{3x3}_sym, via restrictions and transpositions of Grabovsky's example and a modification of Sverak's example.