Abstract:
Via form methods we investigate the Dirichlet-to-Neumann operator N associated with a uniformly elliptic pure second-order operator on an exterior domain Ω with Lipschitz boundary Γ. We consider two versions of the Dirichlet-to-Neumann operator and a variational problem on Ω associated with each case. We prove that for bounded data, solutions of the variational problem are continuous on Ω and decay at infinity. We then characterise the Dirichlet-to-Neumann operator N in terms of a j-elliptic sesquilinear form and establish that −N generates an asymptotically stable submarkovian holomorphic C0-semigroup on L2(Γ) that leaves C(Γ) invariant. Finally we prove that the associated heat kernel is jointly continuous on Σθ × Γ × Γ, satisfies uniform bounds in complex time and converges uniformly on Γ × Γ to an equilibrium, where Σθ ⊂ C is an open sector of angle θ ∈ (0, π 2 ).