**A. Cialdea and V. Maz'ya**

##
L^{p}-Dissipativity of the Lamé
Operator

**abstract:**

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