On the symmetric Morrey problem in three dimensions

We study the relations between seven generalized notions of convexity for continuous integrands in the space of symmetric $3\times 3$ matrices. Here, symmetric quasiconvexity is defined through symmetrized gradients, and symmetric rank-one convexity through symmetrized rank-one lines. We construct a symmetric quasiconvex integrand that is not polyconvex at symmetric matrices, and an integrand that is both symmetric rank-one convex and Hessian quasiconvex but not symmetric quasiconvex. We also establish a transfer principle that converts smooth rank-one convex functions that lack Hessian quasiconvexity into symmetric rank-one convex integrands while preserving this lack of of Hessian quasiconvexity. Applying this scheme to already existing counterexamples yields the existence of symmetric rank-one convex integrands that are not Hessian quasiconvex. Together with known examples, these results determine all pairwise implications among the seven notions considered here.