2021/06/22 by Voss, Jendrik, Martin, Robert J., Ghiba, Ionel-Dumitrel +1
#26B25 #74A10 #74B20 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2106.11887
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\rm iso(F)+W\rm vol(det F)=\widetilde W\rm iso((F)/(√(det F)))+W\rm 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 \enquoteleast 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\rm magic+(F)=\fracλ\rm maxλ\rm min-log\fracλ\rm maxλ\rm min+logdet F=\fracλ\rm maxλ\rm min-2logλ\rm min is quasiconvex. In addition, we demonstrate that under affine boundary conditions, W\rm 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.