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=∇(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〉|⩽2φ(s)[sφ(s)]′〈ReA(x)ξ,ξ〉 (for almost every x∈Ω⊂RN and for any s>0, ξ∈RN) is necessary and sufficient for the functional dissipativity of E.