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

We study optimization problems for the first eigenvalue of a linear elliptic operator.

As applications, we consider homogenized two-phase optimal design problems, also known as topology optimization problems, for conductivity and simplified elasticity settings.

Under suitable assumptions, we prove that the first eigenvalue is pseudo-concave with respect to the density-like parameter.

This pseudo-concavity implies that every stationary point of the corresponding maximization problem is a global maximizer.

Also, for a certain pseudo-concave minimization problem in the conductivity setting, a classical $0$-$1$ minimizer exists.

Finally, we present simple numerical experiments illustrating the theoretical results.