Quasi-Lp-contractive analytic semigroups generated by elliptic operators with complex unbounded coefficients on arbitrary domains

Let Ω be a domain in ℝ N and consider a second order linear partial differential operator A in divergence form on Ω which is not required to be uniformly elliptic and whose coefficients are allowed to be complex, unbounded and measurable. Under rather general conditions on the growth of the coefficients we construct a quasi-contractive analytic semigroup on L 2 ( Ω , dx ), whose generator A V gives an operator realization of A under general boundary conditions. Under suitable additional conditions on the imaginary parts of the diffusion coefficients, we prove that for a wide class of boundary conditions, the semigroup is quasi- L p -contractive for 1< p