Morrey’s Conjecture for the Planar Volumetric-Isochoric Split: Least Rank-One 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 : GL + ( n ) → R with an additive volumetric-isochoric split, i.e. W ( F ) = W iso ( F ) + W vol ( det F ) = W ~ iso ( F det F ) + W vol ( 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 “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 magic + ( F ) = λ max λ min - log λ max λ min + log det F = λ max λ min + 2 log λ min is quasiconvex. In addition, we demonstrate that under affine boundary conditions, W magic + ( 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.