Optimal logarithmic estimates in the Hardy-Sobolev space of the disk and stability results

TL;DR

This paper establishes a new logarithmic estimate in the Hardy-Sobolev space of the disk, extending previous results, and applies it to stability analysis in inverse boundary value problems and interpolation schemes.

Contribution

It introduces a generalized logarithmic estimate in $H^{k, 2}$ and demonstrates its application to inverse problems and boundary interpolation stability.

Findings

Derived a logarithmic estimate in $H^{k, 2}$ space.

Established stability results for Robin coefficient identification.

Applied estimates to boundary interpolation problems.

Abstract

We prove a logarithmic estimate in the Hardy-Sobolev space $H^{k, 2}$, $k$ a positive integer, of the unit disk ${\mathbb D}$. This estimate extends those previously established by L. Baratchart and M. Zerner in $H^{1,2}$ and by S. Chaabane and I. Feki in $H^{k,\infty}$. We use it to derive logarithmic stability results for the inverse problem of identifying Robin's coefficients in corrosion detection by electrostatic boundary measurements and for a recovery interpolation scheme in the Hardy-Sobolev space $H^{k, 2}$ with interpolation points located on the boundary ${\mathbb T}$ of the unit disk.