Gaussian upper bounds for heat kernels of second order complex elliptic operators with unbounded diffusion coefficients on arbitrary domains

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.