A family of boundary value problems is considered in domains Ω(E)=Ω\ωE⊂ Rn, n≥3, with cavities ωE depending on a small parameter E∈(0,E0]. An approximation U(E,x), x∈Ω(E), of the solution u(E,x), x∈Ω(E), to the boundary value problem is obtained by an application of the methods of matched and compound asymptotic expansions. The asymptotic expansion is constructed with precise a priori estimates for solutions and remainders in Hölder spaces, i.e., pointwise estimates are established as well. The asymptotic solution U(E,x) is used in order to derive the first term of the asymptotic expansion with respect to E for the shape functional J(Ξ(E))=JE (u)≅JE(U). In particular, we obtain the topological derivative T(x) of the shape functional J(Ξ) at a point x∈Ω. Volume and surface functionals are considered in the paper. © 2003 Éditions scientifiques et médicales Elsevier SAS. All rights reserved.
Nazarov, S. A., & Sokołowski, J. (2003). Asymptotic analysis of shape functionals. Journal Des Mathematiques Pures et Appliquees, 82(2), 125–196. https://doi.org/10.1016/S0021-7824(03)00004-7