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

TL;DR

This paper establishes criteria for the functional dissipativity of the Lamé operator in PDE systems, providing necessary and sufficient conditions and applying them to regularity results in elasticity theory.

Contribution

It introduces the concept of functional dissipativity for the Lamé system and derives precise conditions for its characterization.

Findings

Necessary and sufficient conditions for functional dissipativity of the Lamé system.

Application of the theory to regularity of displacement vectors in elasticity.

Extension of dissipativity concepts to systems of PDEs with complex coefficients.

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.