An optimization algorithm applied to the Morrey conjecture in nonlinear elasticity

For a long time it has been studied whether rank-one convexity and quasiconvexity give rise to different families of constitutive relations in planar nonlinear elasticity. Stated in 1952 the Morrey conjecture says that these families are different, but no example has come forward to prove it. Now we attack this problem by deriving a specialized optimization algorithm based on two ingredients: first, a recently found necessary condition for the quasiconvexity of fourth-degree polynomials that distinguishes between both classes in the three dimensional case, and secondly, upon a characterization of rank-one convex fourth-degree polynomials in terms of infinitely many constraints. After extensive computational experiments with the algorithm, we believe that in the planar case, the necessary condition mentioned above is also necessary for the rank-one convexity of fourth-degree polynomials. Hence the question remains open.