This paper is concerned with optimal design problems with aspecial assumption on the coefficients of the state equation.Namely we assume that the variations of these coefficientshave a small amplitude. Then, making an asymptotic expansionup to second order with respect to the aspect ratio of thecoefficients allows us to greatly simplify the optimal designproblem. By using the notion of H-measures we are able toprove general existence theorems for small amplitudeoptimal design and to provide simple and efficient numericalalgorithms for their computation. A key feature of thistype of problems is that the optimal microstructures arealways simple laminates.
