Stability for an inverse flux and an inverse boundary coefficient problems

Mourad Choulli; Shuai Lu; Hiroshi Takase

doi:10.3934/ipi.2025058

Article Contents

2026, Volume 23: 152-162. Doi: 10.3934/ipi.2025058

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Stability for an inverse flux and an inverse boundary coefficient problems

1.
Université de Lorraine, 34 cours Léopold, 54052 Nancy cedex, France
2.
School of Mathematical Sciences, Fudan University, 220 Handan Road, Shanghai 200433, China
3.
Institute of Mathematics for Industry, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan

^*Corresponding author: Hiroshi Takase
^*Corresponding author: Hiroshi Takase

Received: February 2025

Revised: November 2025

Early access: November 12, 2025

Published online: November 12, 2025

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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.

Keywords:

Inverse flux problem,
inverse boundary coefficient problem,
quantitative uniqueness of continuation,
elliptic BVPs.

Mathematics Subject Classification: Primary: 35R30, 35R25; Secondary: 35J15, 58J05.

Citation:

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Figure 1. Illustration of the domain for equation (2). $ \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

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Figure 2. Illustration of the domain for equation (6). $ 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 $

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