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

Abstract

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 \((e^{-t A_{V}})_{t\geqslant0}\) 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 \((e^{-t A_{V}})_{t\geqslant0}\) is quasi-L p -contractive for 1<p<∞. Similar results hold for second order nondivergence form operators whose coefficients satisfy conditions similar to those on the coefficients of the operator A, except for some further requirements on the diffusion coefficients. Some examples where our results can be applied are provided.