*Published Paper*

**Inserted:** 20 oct 2003

**Last Updated:** 5 jan 2005

**Journal:** Asymptot. Anal.

**Volume:** 40

**Number:** 1

**Pages:** 37-49

**Year:** 2004

**Abstract:**

We prove that for any connected open set $\Omega$ in ${R}^n$
and for any set of matrices
$K=\{A_1,A_2,A_3\}$ in ${M}^{n\times n}$, with
rank$(A_i-A_j)=n$, there is no non-constant solution
$B$ from $\Omega$ to ${M}^{m\times n}$, called exact solution,
to the problem
\begin{equation**}
Div B=0 \quad in D` (\Omega,{R} ^{n)} \quad
and \quad B(x) \,\,in \,\,K a.e. in \Omega\,.
\end{equation**}
In contrast, A. Garroni and V. Nesi $[10]$
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.

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