TOPOLOGICAL ASYMPTOTIC ANALYSIS FOR A CLASS OF QUASILINEAR ELLIPTIC EQUATIONS

Samuel Amstutz (1) , Alain R. Bonnafé (2, 3)

1 LMA - EA2151 Laboratoire de Mathématiques d'Avignon
2 IMT - Institut de Mathématiques de Toulouse UMR5219
3 INSA Toulouse - Institut National des Sciences Appliquées - Toulouse

Topological asymptotic expansions for quasilinear elliptic equations have not been studied yet. Such questions arise from the need to apply topological asymptotic methods in shape optimization to nonlinear elasticity equations as in imaging to detect sets with codimensions ≥ 2 (e.g. points in 2D or segments in 3D). Our main contribution is to provide topological asymptotic expansions for a class of quasilinear elliptic equations, perturbed in non-empty subdomains. The obtained topological gradient can be split into a classical linear term and a new term which accounts for the non linearity of the equation. With respect to topological asymptotic analysis, moving from linear equations onto quasilinear ones requires to heavily revise the implemented methods and tools. By comparison with the steps carried out to obtain such expansions with the Laplace equation, the core issue for a quasilinear equation lies in the ability to defi ne the variation of the direct state at scale 1 in R^N . Accordingly we build dedicated weighted quotient Sobolev spaces, which semi-norms encompass both the L^p norm and the L^2 norm of the gradient in R^N. Then we consider an appropriate class of quasilinear elliptic equations, to ensure that the problem de ning the direct state at scale 1 enjoys a combined p and 2 ellipticity property. The needed asymptotic behavior of the solution of the nonlinear interface problem in R^N is then proven. An appropriate duality scheme is set up between the direct and adjoint states at each stage of approximation.

Mots clés: quasilinear elliptic equations; quasilinear interface problems; asymptotic behavior of solutions; topological asymptotic analysis; topological derivatives; two-norm discrepancy

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Samuel Amstutz, Alain R. Bonnafé. TOPOLOGICAL ASYMPTOTIC ANALYSIS FOR A CLASS OF QUASILINEAR ELLIPTIC EQUATIONS. Journal de Mathématiques Pures et Appliquées, 2017, 107 (4), pp.367-408. ⟨hal-00952109⟩