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

Abstract

We give complete algebraic characterizations of the $L^p$-dissipativity of the Dirichlet problem for some systems of partial differential operators of the form ${\partial }_h({\mathcal{A}}^{hk}(x){\partial }_k)$, were ${\mathcal{A}}^{hk}(x)$ 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 ${\partial }_h({\mathcal{A}}^h(x){\partial }_h)$, where ${\mathcal{A}}^h(x)$ are $m\times m$ matrices with complex $L_{\mathrm{loc}}^1$ entries, and we describe the maximum angle of $L^p$-dissipativity for this operator.

42 pages, LaTeX, no figures