Analyticity of Dirichlet--Neumann Operators on Hölder and Lipschitz Domains

Abstract

In this paper we take up the question of analyticity properties of Dirichlet--Neumann operators with respect to boundary deformations. In two separate results, we show that if the deformation is sufficiently small and lies either in the class of $C^{1+\alpha}$ (any $\alpha>0$) or Lipschitz functions, then the Dirichlet--Neumann operator is analytic with respect to this deformation. The proofs of both results utilize the "domain flattening" change of variables recently advocated by Nicholls and Reitich for the stable, high-order numerical simulation of Dirichlet--Neumann operators. We extend their analyticity results through the use of more specialized function spaces, and our new theorems are optimal in terms of boundary regularity. In the case of $C^{1+\alpha}$ boundary perturbations the underlying field also lies in the Hölder class $C^{1+\alpha}$ and the theorem follows by appealing to familiar Schauder theory arguments. In contrast, for Lipschitz deformations the field must lie in an $L^p$-based Sobolev space ($W^{1,p}$), so the relevant elliptic estimates come from Sobolev theory. Additionally, in the case of Lipschitz domains, the Dirichlet--Neumann operator must be reformulated weakly in order to accommodate the lack of regularity at the boundary which these Sobolev-class fields possess.