Optimality conditions for a related layout optimization problem

In this Note, we consider a layout optimization problem, where the dependen ce of the material coefficient with respect to the density is nonlinear. Th e particular case when the density only depends on one space variable is st udied. We show that the original design problem can be relaxed into an opti mization that only involves three shape variables. The structure of the opt imal designs is characterized by expliciting optimality conditions of a rel axed functional. (C) Academie des Sciences/Elsevier, Paris.