Criterion for the functional dissipativity of the Lam\'e operator

Abstract

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.