The five gradient problem: rigidity and large T_5 sets

It is known that the four gradient problem is rigid; however the situation changes dramatically when five gradients are allowed.

 

We briefly discuss how nontrivial solutions can be constructed using rank-one convex integration method, based on Székelyhidi Jr. and Förster work. To this end, we introduce the notions of T_5 configurations and large T_5 sets.

 

The relationship between the size of the rank-one convex hull and rigidity in the five gradient problem remains rather mysterious in general. Motivated by an idea of Kirchheim and Chlebik, we study a particular class of the five gradient problems in which each of the five gradients satisfies two linear relations, not necessarily implying symmetry of the problem.

 

First, we show that the problem is always rigid when the determinants coincide. Second, in the case of unequal determinants, we discuss existence results for T_4 and T_5 configurations.

 

On a final note, we comment on the case in which each gradient satisfies only one linear relation and all the determinants are equal, with particular emphasis on the degenerate case of zero determinant.