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In the present paper we consider the Dirichlet problem for the second orderdifferential operator $E=\nabla(A \nabla)$,where $A$ is a matrix with complexvalued $L^\infty$ entries. We introduce the concept of dissipativity of $E$with respect to a given function $\varphi:R^+ \to R^+$. Under the assumptionthat the $Im\, A$ is symmetric, we prove that the condition $|s\, \varphi'(s)|\, | \langle Im\, A (x)\, \xi,\xi\rangle |\leq 2\, \sqrt{\varphi(s)\, [s\,\varphi(s)]'}\, \langle Re\, A(x) \, \xi,\xi\rangle $ (for almost every$x\in\Omega\subset R^N$ and for any $s>0$, $\xi\in R^N$) is necessary andsufficient for the functional dissipativity of $E$.

Criterion for the functional dissipativity of second order differential operators with complex coefficients