On the Dirichlet-Neumann operator for nearly spherical domains

TL;DR

This paper establishes sharp Sobolev estimates for the Dirichlet-Neumann operator on nearly spherical domains, with technical improvements ensuring independence of analyticity radius and controlled regularity loss.

Contribution

It introduces novel analytic and tame estimates for the operator, notably achieving independence of the analyticity radius and handling regularity loss through local charts and specialized Sobolev spaces.

Findings

Proves sharp Sobolev estimates for the Dirichlet-Neumann operator.

Achieves independence of the analyticity radius from high norms.

Handles regularity loss using local charts and non-isotropic Sobolev spaces.

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.