Generalized Gaussian Estimates for Elliptic Operators with Unbounded Coefficients on Domains

We consider second-order elliptic operators A in divergence form with coefficients belonging to L^∞loc(Ω), when Ω⊆Rd is a sufficiently smooth (unbounded) domain. We prove that the realization of A in L^2( Ω), with Neumann-type boundary conditions, generates a contractive, strongly continuous and analytic semigroup (T(t)) which has a kernel k satisfying generalized Gaussian estimates, written in terms of a distance function induced by the diffusion matrix and the potential term. Examples of operators where such a distance function is equivalent to the Euclidean one are also provided.