| Contents |
| According to Jannelli (Comm. P.D.E. 1989) and D'Ancona Spagnolo (Boll. U.M.I. 1998), we say that an $N\times N$ matrix $A$ admits a quasisymmetrizer if there exists a family $\{Q_\varepsilon\}_{0\lt\varepsilon\le1}$ of positive definite matrices such that
\begin{eqnarray*}
1)
& &
(Q_\varepsilon u,u)
\ge {}^\exists C \, \varepsilon^{N-1} \, \|u\|^2 \\
2)
& &
\bigl|(R_\varepsilon u , v)\bigr|
\le {}^\exists C \, \varepsilon \,
(Q_\varepsilon u,u)^{1/2} \, (Q_\varepsilon v,v)^{1/2}
\end{eqnarray*}
where $R_\varepsilon := Q_\varepsilon A - (Q_\varepsilon A)^*$.
In this talk, we recall how to construct a quasisymmetrizer for some classes of matrices and we illustrate how to use it in order to derive apriori estimates for hyperbolic systems. |
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