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

In the present paper we consider the Dirichlet problem for the second order differential operator E = ∇(A ∇), where A is a matrix with complex valued L^∞ entries. We introduce the concept of dissipativity of E with respect to a given function φ : R+ → R+. Under the assumption that the ImA is symmetric, we prove that the condition |sφ′(s)| |⟨ImA (x)ξ, ξ⟩| ⩽ 2√φ(s)[sφ(s)]′ ⟨ReA (x)ξ, ξ⟩ (for almost every x∈Ω⊂R^N) and for any s>0,ξ∈^R^N) is necessary and sufficient for the functional dissipativity of E