The three divergence free matrix fields problem
| dc.creator | Palombaro, Mariapia | |
| dc.creator | Ponsiglione, Marcello | |
| dc.date | 2003-10-23 | |
| dc.date.accessioned | 2026-07-07T05:02:12Z | |
| dc.date.available | 2026-07-07T05:02:12Z | |
| dc.description | 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. | |
| dc.description | 15 pages, 1 figure | |
| dc.identifier | https://arxiv.org/abs/math/0310374 | |
| dc.identifier | http://arxiv.org/abs/math/0310374 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/68964 | |
| dc.subject | Analysis of PDEs | |
| dc.title | The three divergence free matrix fields problem | |
| dc.type | text |