Gaussian Upper Bounds on Heat Kernels of Uniformly Elliptic Operators on Bounded Domains

We obtain Gaussian upper bounds for heat kernels of higher order differential operators with Dirichlet boundary conditions on bounded domains in ${\mathbb{R}}^N$. The bounds exhibit explicitly the nature of the spatial decay of the heat kernel close to the boundary as well as the long-time exponential decay implied by the spectral gap. We make no smoothness assumptions on our operator coefficients which we assume only to be bounded and measurable Keywords : Heat Kernel, Parabolic, Uniformly Elliptic, Gaussian.