Asymptotic analysis of shape functionals

A family of boundary value problems is considered in domains Omega (epsilon) = Omega \ (omega(epsilon)) over bar subset of R-n, n greater than or equal to 3, with cavities omega(epsilon) depending on a small parameter epsilon is an element of (0, epsilon(0)]. An approximation U(epsilon, x), x is an element of Omega (epsilon), of the solution u(epsilon, x), x is an element of Omega (epsilon), 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 Holder 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 epsilon for the shape functional T(Xi (epsilon)) = J(epsilon) (u) congruent to J(epsilon) (U). In particular, we obtain the topological derivative T(x) of the shape functional J(Xi) at a point x is an element of Omega. Volume and surface functionals are considered in the paper. (C) 2003 Editions scientifiques et medicales Elsevier SAS. All rights reserved.