Rigidity and lack of rigidity for solenoidal matrix valued fields

The problem of finding a curl–free matrix–valued field E with values in an assigned set of matrices К has received considerable attention. Given a bounded connected open set Ω, and a compact set К of m × n matrices, in this paper we establish existence or non–existence results for the following problem: find B ∈ L∞(Ω, М^{m × n}) such that Div B = 0 in Ω in the sense of distributions under the constraint that B(x) ∈ К almost everywhere in Ω. We consider the case of К={A,B}, rank(A−B) = n, and we establish non–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 {A1,A2,A3} of matrices with rank{Ai −Aj} =n, i≠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.