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

Abstract

In the present paper we consider the Dirichlet problem for the second order differential operator $E=\nabla (A\nabla )$,where $A$ is a matrix with complex valued $L^{\infty }$ entries. We introduce the concept of dissipativity of $E$ with respect to a given function $\phi :R^{+}\to R^{+}$. Under the assumption that the $ImA$ is symmetric, we prove that the condition $|s{\phi }^{\prime }(s)||⟨ImA(x)\xi ,\xi ⟩|\le 2\sqrt{\phi (s)[s\phi (s){]}^{\prime }}⟨ReA(x)\xi ,\xi ⟩$ (for almost every $x\in \Omega \subset R^N$ and for any $s>0$, $\xi \in R^N$) is necessary and sufficient for the functional dissipativity of $E$.