Extending representation formulas for boundary voltage perturbations of low volume fraction to very contrasted conductivity inhomogeneities
Yves Capdeboscq, Shaun Chen Yang Ong · Comptes Rendus Mathématique · 2022
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 γ 0 background conductivity matrix on a measurable set well within the domain, and we assume ( γ n - γ 0 ) γ n - 1 ( γ n - γ 0 ) → 0 in L 1 ( Ω ) . Adapting the limit measure, we show that the general representation formula introduced for bounded contrasts in a previous work from 2003 can be extended to unbounded sequences of matrix valued conductivities.