A. Cialdea and V. Maz'ya

Lp-Dissipativity of the Lamé Operator

We study conditions for the $L^p$-dissipativity of the classical linear elasticity operator. In the two-dimensional case we show that $L^p$-dissipativity is equivalent to the inequality
$$ \Big(\frac{1}{2}-\frac{1}{p}\Big)^{2} \leq \frac{2(\nu-1)(2\nu-1)}{(3-4\n)^{2}}. $$
Previously \cite{cialmaz2} this result has been obtained as a consequence of general criteria for elliptic systems, but here we give a direct and simpler proof. We show that this inequality is necessary for the $L^p$-dissipativity of the three-dimensional elasticity operator with variable Poisson ratio. We give also a more strict sufficient condition for the $L^p$-dissipativity of this operator. Finally we find a criterion for the $n$-dimensional Lam\'e operator to be $L^p$-negative with respect to the weight $|x|^{-\alpha}$ in the class of rotationally invariant vector functions.

Mathematics Subject Classification: 74B05, 47B44

Key words and phrases: Elasticity system, $L^p$-dissipativity