Variational source conditions for inverse Robin and flux problems by partial measurements

De-Han Chen; Daijun Jiang; Irwin Yousept; Jun Zou

doi:10.3934/ipi.2021050

Article Contents

2022, Volume 16, Issue 2: 283-304. Doi: 10.3934/ipi.2021050

This issue Previous Article Addendum to: "Variational source conditions for inverse Robin and flux problems by partial measurements" Next Article Inverse problems for the fractional Laplace equation with lower order nonlinear perturbations

Variational source conditions for inverse Robin and flux problems by partial measurements

1.
School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China
2.
Universität Duisburg-Essen, Fakultät für Mathematik, Thea-Leymann-Str. 9, D-45127 Essen, Germany
3.
Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, China

^* Corresponding author: Jun Zou
^* Corresponding author: Jun Zou

Received: January 31, 2021

Revised: April 30, 2021

Early access: July 2021

Published: April 2022

The first author is supported by National Natural Science Foundation of China (Nos. 11701205 and 11871240). The second author is supported by National Natural Science Foundation of China (No. 11871240), NSFC-RGC (China-Hong Kong, No. 11661161017) and self-determined research funds of CCNU from the colleges' basic research and operation of MOE (No. CCNU20TS003). The third author is supported by the German Research Foundation (DFG grants YO159/2-2 and YO159/4-1). The fourth author is supported by Hong Kong RGC grant (project 14306719) and the NSFC/Hong Kong RGC Joint Research Scheme 2016/17 (N_CUHK437/16)

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Abstract

This work is devoted to the convergence analysis of the Tikhonov regularization for the inverse Robin and flux problems.Both inverse problems aim at recovering a respective physical quantity on an inaccessible part of the boundary through some measurement on a partial accessible boundary.The convergence and convergence rate in the desirable L²-norm are derived based on two new logarithmic type stabilities (only in some weak norms,e.g.,the negative Sobolev norms), which enable us to construct and rigorously verify the required variational source conditions.

Addendum: The abstract is added since it was missing when the article was published online. To reflect this addition an Addendum is published in this same IPI 16-2 April 2022 regular issue. We apologize for any inconvenience this may cause.

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Mathematics Subject Classification: Primary: 35R35, 41A25; Secondary: 65M32.

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Figure 1. An example of $ \Omega $ in $ {{\mathbb R}}^2 $ and $ \Gamma_a $ (resp. $ \Gamma_b $) is an open part of $ \Gamma_1 $ (resp. $ \Gamma_2 $)

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