Topological derivatives for a class of quasilinear elliptic equations

Abstract

Topological derivatives for quasilinear elliptic equations have not been studied yet. Such results are needed to apply topological asymptotic methods in shape optimization to nonlinear elasticity equations and in imaging to detect sets with codimensions ≥2 (e.g. points in 2D or curves in 3D). Our contribution is to provide for the first time topological derivatives for a class of quasilinear elliptic equations. The topological derivative can be split into a classical linear term and a new term which accounts for the nonlinearity of the principal part. Shifting from linear equations to nonlinear ones requires to significantly change the asymptotic analysis method. One core issue lies in the ability to approximate the variation of the direct state at scale 1 in RN. Accordingly we construct dedicated function spaces and rely on comparison techniques. Moreover one has to set up a duality scheme between the direct and adjoint states that applies in the nonlinear framework at each stage of the analysis.