Criterion for the functional dissipativity of the Lamé 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 ${\partial }_h({\mathcal{A}}^{hk}(x){\partial }_k)$ (${\mathcal{A}}^{hk}(x)$ 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 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.

References in corpus (3)

• On p-elliptic divergence form operators and holomorphic semigroups
• Extrapolation of the Dirichlet problem for elliptic equations with complex coefficients
• The $p$-ellipticity condition for second order elliptic systems and applications to the Lamé and homogenisation problems