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

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.