Abstract
In this paper we obtain Gaussian upper bounds for the integral kernel of the semigroup associated with second order elliptic differential operators with complex unbounded measurable coefficients defined in a domain Omega of a"e (N) and subject to various boundary conditions. In contrast to the previous literature the diffusions coefficients are not required to be bounded or regular. A new approach based on Davies-Gaffney estimates is used. It is applied to a number of examples, including degenerate elliptic operators arising in Financial Mathematics and generalized Ornstein-Uhlenbeck operators with potentials.