In the present paper we consider the Dirichlet problem for the second order differential operator E = del(A del), where A is a matrix with complex valued L-infinity entries. We introduce the concept of dissipativity of E with respect to a given function phi : R+ -> R+. Under the assumption that the ImA is symmetric, we prove that the condition vertical bar s phi (s)vertical bar vertical bar < ImA (x)xi, xi >vertical bar <= 2 root phi(s)[s phi(s)] < ReA (x)xi, xi > (for almost every x is an element of Omega subset of R-N and for any s > 0, xi is an element of R-N) is necessary and sufficient for the functional dissipativity of E. (c) 2020 Elsevier Ltd. All rights reserved.