Extending Representation Formulae for Boundary Voltage Perturbations of Low Volume Fraction to Very Contrasted Conductivity Inhomogeneities

2021/03/17 by Yves Capdeboscq, Capdeboscq, Yves, Shaun Chen Yang Ong +1
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Composite Material Mechanics #FOS: Mathematics #Numerical methods in engineering #Numerical methods in inverse problems #math.AP

paper · pdf · doi:10.48550/arxiv.2103.09644

arxiv created 2021/03/17 · openalex publication_date 2021/03/17 · arxiv updated 2021/03/18 · openalex created_date 2022/09/30 · openalex updated_date 2026/07/28

Abstract

Imposing either Dirichlet or Neumann boundary conditions on the boundary of a smooth bounded domain Ω, we study the perturbation incurred by the voltage potential when the conductivity is modified in a set of small measure. We consider (γ_n)_n∈ℕ, a sequence of perturbed conductivity matrices differing from a smooth γ₀ background conductivity matrix on a measurable set well within the domain, and we assume (γ_n-γ₀)γ_n^-1(γ_n-γ₀)→0 in L¹(Ω). Adapting the limit measure, we show that the general representation formula introduced for bounded contrasts in \citepcapdeboscq-vogelius-03a can be extended to unbounded sequencesof matrix valued conductivities.

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• High-Order Terms in the Asymptotic Expansions of the Steady-State Voltage Potentials in the Presence of Conductivity Inhomogeneities of Small Diameter
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• Recovering a quasilinear conductivity from boundary measurements
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