Rigidity and lack of rigidity for solenoidal matrix fields

The problem of finding a curl-free matrix-valued field E with values in an assigned set of matrices K has received considerable attention. Given a bounded connected open set Omega, and a compact set K of m x n matrices, in this paper we establish existence or non-existence results for the following problem: find B is an element of L-infinity(Omega, M-m x n) such that Div B = 0 in Omega in the sense of distributions under the constraint that B(x) is an element of K almost everywhere in Q. We consider the case of K = {A,B}, with rank(A-B) = n, and we establish lion-existence both for the case of exact solutions described above and for the case of approximate solutions described in 1. We also prove existence of approximate solutions for a suitably chosen triple {A(1), A(2) A(3)} of matrices with rank(A(i) - A(j)) = n, i not equal j and i,j = 1, 2 3. We give examples when the differential constraints are of a different type and present some applications to composites.