Abstract: We consider the problem of minimizing the first nonzero eigenvalue of an elliptic operator with Neumann boundary conditions with respect to the distribution of two conducting materials with a prescribed area ratio in a given domain. In one dimension, we show monotone properties of the first nonzero eigenvalue with respect to various parameters and find the optimal distribution of two conducting materials on an interval under the assumption that the region that has lower conductivity is simply connected. On a rectangular domain in two dimensions, we show that the strip configuration of two conducting materials can be a local minimizer. For general domains, we propose a rearrangement algorithm to find the optimal distribution numerically. Many results on various domains are shown to demonstrate the efficiency and robustness of the algorithms. Topological changes of the optimal configurations are discussed on circles, ellipses, annuli, and L-shaped domains.