Stability for an inverse flux and an inverse boundary coefficient problems

TL;DR

This work analyzes stability for inverse flux and inverse boundary coefficient problems on a domain endowed with a Riemannian metric $g$, establishing both logarithmic and Lipschitz stability estimates for recovering the inaccessible boundary data from boundary measurements. The authors derive a logarithmic-type interpolation bound $\mathbf{c}\|\mathfrak{a}\|_{L^2(\mathcal{S})} \le e^{cs}\mathcal{C}(u(0,\mathfrak{a})) + s^{-\eta}\|\mathfrak{a}\|_{H^{1/2}(\mathcal{S})}$ for $0<\eta<1/4$, and a Lipschitz bound on a finite-bandwidth class $\mathcal{A}$; extensions to interior problems and to inverse boundary coefficient problems are discussed. The results are then applied to an inverse boundary coefficient problem, proving Lipschitz stability for the coefficient on the inaccessible boundary under spectral constraints or boundary-equality conditions, complemented by a logarithmic bound without these constraints via a function $\Phi_{\eta,c}$. These findings provide a framework for reconstructing boundary corrosion coefficients from measurements on the accessible boundary, with implications for non-destructive testing.

Abstract

We establish both Lipschitz and logarithmic stability estimates for an inverse flux problem and subsequently apply these results to an inverse boundary coefficient problem. Furthermore, we demonstrate how the stability inequalities derived for the inverse boundary coefficient problem can be utilized in solving an inverse corrosion problem. This involves determining the unknown corrosion coefficient on an inaccessible part of the boundary based on measurements taken on the accessible part of the boundary.

Figures (2)

• Figure 1: Illustration of the domain for Equation \ref{['BVP']}. $\mathcal{S}:=\partial B$, $B \Subset \Omega$, $\Gamma:=\partial \Omega$ and $U:=\mathbb{R}^n\setminus\overline{B}$. $\Gamma$ is an accessible and $\mathcal{S}$ is an inaccessible part of the boundary.
• Figure 2: Illustration of the domain for Equation \ref{['BVP_2']}. $B\subset\mathbb{R}^n$, $\Omega\Supset B$ and $D:=\Omega\setminus\overline{B}$. The boundaries are $\mathcal{S}:=\partial B$ and $\Gamma:=\partial \Omega$ so that $\partial D=\mathcal{S}\cup \Gamma$.

Theorems & Definitions (19)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Remark 1.4
• Lemma 2.1
• proof
• proof : Proof of Theorem \ref{['thm_single_log_ifp']}
• proof : Proof of Theorem \ref{['thm_Lipschitz_ifp']}
• Theorem 3.1
• Corollary 3.2