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Stability analysis in the inverse Robin transmission problem
Author(s) -
Meftahi Houcine
Publication year - 2016
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4173
Subject(s) - mathematics , parameterized complexity , piecewise , monotone polygon , stability (learning theory) , monotonic function , perturbation (astronomy) , boundary (topology) , degenerate energy levels , boundary value problem , mathematical analysis , combinatorics , geometry , physics , quantum mechanics , machine learning , computer science
In this paper, we consider the conductivity problem with piecewise‐constant conductivity and Robin‐type boundary condition on the interface of discontinuity. When the quantity of interest is the jump of the conductivity, we perform a local stability estimate for a parameterized non‐monotone family of domains. We give also a quantitative stability result of local optimal solution with respect to a perturbation of the Robin parameter. In order to find an optimal solution, we propose a Kohn–Vogelius‐type cost functional over a class of admissible domains subject to two boundary values problems. The analysis of the stability involves the computation of first‐order and second‐order shape derivative of the proposed cost functional, which is performed rigorously by means of shape‐Lagrangian formulation without using the shape sensitivity of the states variables. © 2016 The Author. Mathematical Methods in the Applied Sciences Published by John Wiley & Sons Ltd.