EIGENVALUE PROBLEMS

Keywords: shape optimization, topology optimization, asymptotic analysis, shape and topological derivative, free boundary problems

## Minimization of the ground state for two phase conductors

We consider the problem of the optimal distribution of
two conducting materials with given volume inside a fixed domain, in
order to minimize the first eigenvalue (the ground state) of
a Dirichlet operator. It is known, when the domain is a ball, that the
solution is radial, and it was conjectured that the optimal
distribution of the materials consists in putting the material with
the highest conductivity in a ball around the center. We show that
this conjecture is not true in general. For this, we consider the
particular case where the two conductivities are close to each other
(low contrast regime) and we perform an asymptotic expansion
with respect to the difference of conductivities. We find that the
optimal solution is the union of a ball and an outer ring when the
amount of the material with the higher density is large enough.

> Minimization of the ground state for two phase conductors in low contrast regime

> Global minimizer of the ground state for two phase conductors in low contrast regime

> Minimization of the ground state for two phase conductors in low contrast regime

> Global minimizer of the ground state for two phase conductors in low contrast regime

## Eigenvalue problems with indefinite weight

We focus on the study of a linear eigenvalue problem with indefinite weight and Robin type boundary conditions. We investigate the minimization of the positive principal eigenvalue under the constraint that the absolute value of the weight is bounded and the total
weight is a fixed negative constant. Biologically, this minimization
problem is motivated by the question of determining the optimal spatial
arrangement of favorable and unfavorable regions for a species to
survive. For rectangular domains with Neumann boundary condition, it is known that there exists a threshold value such that if the total weight is below this threshold value then the optimal favorable region is like a section of a disk at one of the four corners; otherwise, the optimal favorable region is a strip attached to the shorter side of the rectangle. Here, we investigate the same problem with mixed Robin-Neumann type boundary conditions and study how this boundary condition affects the optimal spatial arrangement.

> Principal Eigenvalue Minimization for an Elliptic Problem

> Properties of optimizers of the principal eigenvalue with indefinite weight and Robin conditions

> Principal Eigenvalue Minimization for an Elliptic Problem

> Properties of optimizers of the principal eigenvalue with indefinite weight and Robin conditions