The ∂-neumann Problem in the Sobolev Topology

Here, and throughout the paper, we useD to denote the α-order derivative, where α is a multi-index and we are using standard multi-index notation. Moreover, γα := |α|!/α! denotes the polynomial coefficient. [The naturality of this choice of the Sobolev inner product will be pointed out and discussed below.] We define the Sobolev space W (Ω) to be the closure of C(Ω̄) with respect to the above inner product. We denote by W s (0,q)(Ω) the space of (0, q) forms whose coefficients are in W (Ω). If φ = ∑ |J |=q φJdz̄ J and ψ = ∑ |J |=q ψJdz̄ J , then the inner product in W s (0,q)(Ω) is defined by