Pseudo-concave optimization of the first eigenvalue of elliptic operators with application to topology optimization by homogenization

Abstract: We consider optimization problems of the first eigenvalue of elliptic operators with applications to two-phase optimal design problems (also known as topology optimization problems) of conductivity and elasticity relaxed by homogenization. Under certain assumptions, we show that the first eigenvalue is a pseudo-concave function. Due to pseudo-concavity, every stationary point is a global maximizer, and there exists a global minimizer that is an extreme point (corresponding to a 0-1 solution in optimal design problems). We perform simple numerical experiments on optimal design problems to demonstrate that global optimal solutions or 0-1 solutions can be obtained by a simple gradient method.