On the Dirichlet-Neumann operator for nearly spherical domains

Abstract

We consider the Dirichlet-Neumann operator for a nearly spherical domain in R^n, and prove sharp analytic and tame estimates in Sobolev class. The novelty of this paper concerns technical improvements, the most important of which are the independence of the analyticity radius on the high norms and the regularity loss of one in the elevation function. These properties are expectable but nontrivial to prove. The result is obtained by introducing local charts and a convenient class of non-isotropic Sobolev spaces of high, possibly fractional tangential regularity and integer, limited regularity in the normal direction.

Comments: This article contains the results of Sections 5 and 7 of arXiv:2408.02333v1 (the first version of arXiv:2408.02333), which are here improved and presented as a separate article. The improvements concern the loss of regularity for the elevation function and the generalization to any dimension