The three divergence free matrix fields problem

created on 20 Oct 2003
modified by ponsiglio on 05 Jan 2005

[BibTeX]

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.