After introducing the concept of functional dissipativity of the Dirichlet problem in a domain Ω⊂RN for systems of partial differential operators of the form ∂h(Ahk(x)∂k) (Ahk(x) being m×m matrices with complex valued L∞ entries), we find necessary and sufficient conditions for the functional dissipativity of the two-dimensional Lamé 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.