Dirichlet-to-Neumann graphs on partial boundaries
G. Gordon · ResearchSpace (University of Auckland) · 2019
This thesis considers how one can construct a Dirichlet-to-Neumann operator on a sufficiently regular subset C of the boundary of a bounded Lipschitz domain . We consider two methods of construction. First by placing a Dirichlet condition on the complement of C. Second by placing a Neumann condition on the complement of C. After constructing these operators we prove basic properties as well as Krein-type resolvent formulae. We also define their L2(C) realisations using form methods. Next we prove several convergence theorems for these Dirichlet-to-Neumann operators. We focus on two situations. We either fix the domain Ω and consider a sequence of subsets (Cn)n∈ℕ of ∂Ω or we fix the partial boundary C and consider a sequence of bounded Lipschitz domains (Ωn)n∈ℕ such that C ⊆ ∂Ωn for all n ∈ ℕ. Types of convergence proved include operator convergence, resolvent convergence and semigroup convergence. Finally we consider the situation where Ω is a Lipschitz hypograph. We define a Dirichlet-to-Neumann operator on the Lipschitz graph ∂Ω. Then we approximate Ω by a sequence of truncations of Ω and prove convergence theorems such as resolvent convergence of Dirichlet-to-Neumann operators on the truncated Ω to the Dirichlet-to-Neumann operator on Ω.