  In the present paper we consider the functional dissipativity of the
Dirichlet problem for systems of partial differential operators of the form
$\partial_{h} ({\mathop{\mathscr A}\nolimits}^{hk}(x)\partial_{k})$
(${\mathop{\mathscr A}\nolimits}^{hk}$ being $m\times m$ matrices with complex
valued $L^{1}_{\text{loc}}$ entries). In the particular case of the operator
$\partial_{h} ({\mathop{\mathscr A}\nolimits}^{h}(x)\partial_{h})$ (where
${\mathop{\mathscr A}\nolimits}^{h}$ are $m\times m$ matrices) we obtain
algebraic necessary and sufficient conditions. We give also three different
notions of functional ellipticity and investigate the relations between them
and the functional dissipativity for the operators in question.
