Green’s Identity in Curvilinear Polygons

For PDE examples illuminating the feedback (Redheffer product) theory of conservative boundary control systems, we need to verify conservativity of various hyperbolic PDE’s in domains Ω that have Lipschitz boundaries. A properly rigged version of Green’s Identity is then required on such domains where technical assumptions on functions at the corners of ∂Ω cannot be tolerated; such as present in [1, Theorem 1.5.3.11]. The sufficient generalization of this theorem is given for curvilinear polygons of R2. The approach seems to give the same generalization for any polytope of Rn including tubular domains studied in [2, 3]. 1. Statement of the problem The point of this note is to give a sufficient generalization of [1, Theorem 1.5.3.11] so that the wave equation system on Ω ⊂ R can be shown to be an internally wellposed boundary node where ∂Ω is a piecewise smooth curvilinear polygon. We shall need the following sets defined for any open Ω ⊂ R with a Lipschitz boundary: D(Ω) = {f = g|Ω : g ∈ D(R)}; D(∆, L(Ω)) := {u ∈ L(Ω) : ∆u ∈ L(Ω)}; and E(∆, L(Ω)) := {u ∈ H(Ω) : ∆u ∈ L(Ω)}. The sets D(∆, L(Ω)) and E(∆, L(Ω)) are equipped with their natural inner products, and clearly E(∆, L(Ω)) ⊂ D(∆, L(Ω)) with a continuous inclusion. We call D(∆, L(Ω)) the maximal domain of Laplacian. Whenever ∂Ω = ∪j=1Γj where each Γj is an open, finite length, and smooth curve, we have ∂ ∂ν ∈ L(D(∆;L (ω));H(Γj)) for all j = 1, . . . , k; see [1, Theorem 1.5.3.4] where H(Γj) is the dual of H 3/2 0 (Γj). We mean by the requirement ∂φ ∂ν ∈ L (∂Ω) simply that ∂φ ∂ν ∈ L (Γj) for all j = 1, . . . , k. We assume that the arcs Γj for j = 1, 2, . . . , k are ordered so that Γj∩Γj+1 = {rj} for j = 1, 2, . . . , k − 1 and Γ1 ∩ Γk = {rk}. Thus CΩ := {r1, r2, . . . , rk, } ⊂ ∂Ω is the set of corner points of the curvilinear polygon ∂Ω, and we define the minimum separation between the points of CΩ by δΩ := min 1≤j1<j2≤k |rj1 − rj2 |. 1991 Mathematics Subject Classification. foo.