Morrey's conjecture for the planar volumetric-isochoric split. Part I: least convex energy functions

We consider Morrey’s open question whether rank-one convexity already implies quasiconvexity in the planar case. For some specific families of energies, there are precise conditions known under which rank-one convexity even implies polyconvexity. We will extend some of these findings to the more general family of energies $W:{\mathrm{GL}}^{+}(n)\to \mathbb{R}$ with an additive volumetric-isochoric split, i.e. $$W(F)=W_{\mathrm{i}\mathrm{s}\mathrm{o}}(F)+W_{\mathrm{v}\mathrm{o}\mathrm{l}}(\det F)={\tilde{W}}_{\mathrm{i}\mathrm{s}\mathrm{o}}\left(\frac{F}{\sqrt{\det F}}\right)+W_{\mathrm{v}\mathrm{o}\mathrm{l}}(\det F),$$ which is the natural finite extension of isotropic linear elasticity. Our approach is based on a condition for rank-one convexity which was recently derived from the classical two-dimensional criterion by Knowles and Sternberg and consists of a family of one-dimensional coupled differential inequalities. We identify a number of \enquote{least} rank-one convex energies and, in particular, show that for planar volumetric-isochorically split energies with a concave volumetric part, the question of whether rank-one convexity implies quasiconvexity can be reduced to the open question of whether the rank-one convex energy function $$W_{\mathrm{m}\mathrm{a}\mathrm{g}\mathrm{i}\mathrm{c}}^{+}(F)=\frac{{\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}}{{\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}}-\log \frac{{\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}}{{\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}}+\log \det F=\frac{{\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}}{{\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}}-2\log {\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}$$ is quasiconvex. In addition, we demonstrate that under affine boundary conditions, $W_{\mathrm{m}\mathrm{a}\mathrm{g}\mathrm{i}\mathrm{c}}^{+}(F)$ allows for non-trivial inhomogeneous deformations with the same energy level as the homogeneous solution, and show a surprising connection to the work of Burkholder and Iwaniec in the field of complex analysis.