  After introducing the concept of functional dissipativity of the Dirichlet
problem in a domain $\Omega\subset {\mathbb R}^N$ for systems of partial
differential operators of the form $\partial_{h}({\mathscr
A}^{hk}(x)\partial_{k})$ (${\mathscr A}^{hk}(x)$ being $m \times m$ matrices
with complex valued $L^\infty$ entries), we find necessary and sufficient
conditions for the functional dissipativity of the two-dimensional Lam\'e
system. As an application of our theory we provide two regularity results for
the displacement vector in the $N$-dimensional equilibrium problem, when the
body is fixed along its boundary.
