Quasiconvexity for the Dacorogna--Marcellini Energy

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 $|\gamma |\le \frac{2}{\sqrt{3}}$. 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.