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+α (any α > 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

more »

... table, 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+α boundary perturbations the underlying field also lies in the Hölder class C 1+α 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.