Numerical approaches for investigating quasiconvexity in the context of Morrey's conjecture

Deciding whether a given function is quasiconvex is generally a difficult task. Here, we discuss a number of numerical approaches that can be used in the search for a counterexample to the quasiconvexity of a given function $W$. We will demonstrate these methods using the planar isotropic rank-one convex function $$W_{\mathrm{m}\mathrm{a}\mathrm{g}\mathrm{i}\mathrm{c}}^{+}(F)=\frac{{\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}}{{\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}}-\log \frac{{\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}}{{\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}}+\log \det F=\frac{{\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}}{{\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}}+2\log {\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}},$$ where ${\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}\ge {\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}$ are the singular values of $F$, as our main example. In a previous contribution, we have shown that quasiconvexity of this function would imply quasiconvexity for all rank-one convex isotropic planar energies $W:{\mathrm{GL}}^{+}(2)\to \mathbb{R}$ with an additive volumetric-isochoric split of the form $$W(F)=W_{\mathrm{i}\mathrm{s}\mathrm{o}}(F)+W_{\mathrm{v}\mathrm{o}\mathrm{l}}(\det F)={\tilde{W}}_{\mathrm{i}\mathrm{s}\mathrm{o}}\left(\frac{F}{\sqrt{\det F}}\right)+W_{\mathrm{v}\mathrm{o}\mathrm{l}}(\det F)$$ with a concave volumetric part. This example is therefore of particular interest with regard to Morrey’s open question whether or not rank-one convexity implies quasiconvexity in the planar case.