  We give complete algebraic characterizations of the $L^{p}$-dissipativity of
the Dirichlet problem for some systems of partial differential operators of the
form $\partial_{h}({\mathscr A}^{hk}(x)\partial_{k})$, were ${\mathscr
A}^{hk}(x)$ are $m\times m$ matrices. First, we determine the sharp angle of
dissipativity for a general scalar operator with complex coefficients. Next we
prove that the two-dimensional elasticity operator is $L^{p}$-dissipative if
and only if $$
  ({1\over 2}-{1\over p})^{2} \leq {2(\nu-1)(2\nu-1)\over (3-4\nu)^{2}},
  $$
  $\nu$ being the Poisson ratio. Finally we find a necessary and sufficient
algebraic condition for the $L^{p}$-dissipativity of the operator $\partial_{h}
({\mathscr A}^{h}(x)\partial_{h})$, where ${\mathscr A}^{h}(x)$ are $m\times m$
matrices with complex $L^{1}_{\rm loc}$ entries, and we describe the maximum
angle of $L^{p}$-dissipativity for this operator.
