The three divergence free matrix fields problem

Abstract

We prove that for any connected open set $Ω\subset \R^n$ and for any set of matrices $K=\{A_1,A_2,A_3\}\subset M^{m\times n}$, with $m\ge n$ and rank$(A_i-A_j)=n$ for $i\neq j$, there is no non-constant solution $B\in L^{\infty}(Ω,M^{m\times n})$, called exact solution, to the problem Div B=0 \quad \text{in} D'(Ω,\R^m) \quad \text{and} \quad B(x)\in K \text{a.e. in} Ω. In contrast, A. Garroni and V. Nesi \cite{GN} exhibited an example of set $K$ for which the above problem admits the so-called approximate solutions. We give further examples of this type. We also prove non-existence of exact solutions when $K$ is an arbitrary set of matrices satisfying a certain algebraic condition which is weaker than simultaneous diagonalizability.