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 $\de_{h}(A^{hk}(x)\de_{k})$, were $A^{hk}(x)$ 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 $$\left({1\over 2}-{1\over p}\right)^{2} \leq {2(\nu-1)(2\nu-1)\over (3-4\nu)^{2}},$$ $\nu$ being the Poisson ratio. Finally we find a necessary and sufficient algebraic condition for the $L^{p}$-dissipativity of the operator $\de_{h} (\A^{h}(x)\de_{h})$, where $\A^{h}(x)$ are $m\times m$ matrices with complex $L^{1}_{\rm loc}$ entries, and we describe the maximum angle of $L^{p}$-dissipativity for this operator.