Criteria for the $L^p$-dissipativity of systems of second order differential equations

We give complete algebraic characterizations of the $L^p$-dissipativity of the Dirichlet problem for some systems of partial differential operators of the form ${\mathrm{\partial }}_h({\mathcal{A}}^{hk}(x){\mathrm{\partial }}_k)$, were ${\mathscr A}^{hk}(x)$are$m\times m$ matrices. First, we determine the sharp angle of dissipativity for a general scalar operator with complex coefficients. Next we prove that the two-dimensional elasticity operator is $L^p$-dissipative if and only if $$(\frac{1}{2}-\frac{1}{p}{)}^2\le \frac{2(\nu -1)(2\nu -1)}{(3-4\nu {)}^2},$$ $\nu$ being the Poisson ratio. Finally we find a necessary and sufficient algebraic condition for the $L^p$-dissipativity of the operator $\partial_{h} ({\mathscr A}^{h}(x)\partial_{h})$,where${\mathscr A}^{h}(x)$are$m\times m$matriceswithcomplex$L^{1}_{\rm loc}$ entries, and we describe the maximum angle of $L^p$-dissipativity for this operator.