Gaussian estimates for second order elliptic operators with boundary conditions

We prove Gaussian estimates for the kernel of the semigroup generated by a second order operator A in divergence form with real, not necessarily symmetric, second order coefficients on an open subset $\Omega$ of $\mathbb R^d$ satisfying various boundary conditions. Moreover, we show that $A + \omegaI$ has a bounded $H_\infty$-functional calculus and has bounded imaginary powers if $\omega$ is large enough