Estimates in the Hardy-Sobolev space of the annulus and stability result

TL;DR

This paper establishes optimal logarithmic estimates in Hardy-Sobolev spaces of an annulus and applies these to prove a stability result for an inverse boundary value problem involving Robin parameters.

Contribution

It extends previous results from the unit disk to annular domains, providing new logarithmic estimates and stability results for inverse boundary problems.

Findings

Logarithmic estimates of optimal type in Hardy-Sobolev spaces of an annulus.

A stability result for identifying Robin boundary parameters.

Extension of prior disk-based results to annular domains.

Abstract

The main purpose of this work is to establish some logarithmic estimates of optimal type in the Hardy-Sobolev space $H^{k, \infty}; k \in {\mathbb{N}}^*$ of an annular domain. These results are considered as a continuation of a previous study in the setting of the unit disk by L. Baratchart and M. Zerner: On the recovery of functions from pointwise boundary values in a Hardy-sobolev class of the disk. J.Comput.Apll.Math 46(1993), 255-69 and by S. Chaabane and I. Feki: Logarithmic stability estimates in Hardy-Sobolev spaces $H^{k,\infty}$. C.R. Acad. Sci. Paris, Ser. I 347(2009), 1001-1006. As an application, we prove a logarithmic stability result for the inverse problem of identifying a Robin parameter on a part of the boundary of an annular domain starting from its behavior on the complementary boundary part.