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On the structure of the quasiconvex hull in planar elasticity
Let K and L be compact sets of real 2x2 matrices with positive determinant. Suppose that both sets are frame invariant, meaning invariant under the left action of the special orthogonal group. Then we give an algebraic characterization for K and L to be incompatible for homogeneous gradient Young measures. This result permits a simplified characterization of the quasiconvex hull and the rank-one convex hull in planar elasticity.
Computations of quasiconvex hulls of isotropic sets
We design an algorithm for computations of quasiconvex hulls of isotropic compact sets in in the space of 2x2 real matrices. Our approach uses a recent result by the first author [Adv. Calc. Var. (2014), DOI: 10.1515acv-2012-0008] on quasiconvex hulls of isotropic compact sets in the space of 2x2 real matrices. We show that our algorithm has the time complexity of O(N log N ) where N is the number of orbits of the set. We show some applications of our results to relaxation of L∞ variational problems.
Quasiconvexity equals rank-one convexity for isotropic sets of 2 x 2 matrices
Let K be a given compact set of real 2x2 matrices that is isotropic, meaning invariant under the left and right action of the special orthogonal group. Then we show that the quasiconvex hull of K coincides with the rank-one convex hull (and even with the lamination convex hull of order 2). In particular, there is no difference between quasiconvexity and rank-one convexity for K. This is a generalization of a known result for connected sets.