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

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.