This paper is concerned with minimization and maximization problems of eigenvalues. The principal eigenvalue of a differential operator is minimized or maximized over a set which is formed by intersecting a rearrangement class with an affine subspace of finite co-dimension. A solution represents an optimal design of a 2-dimensional composite membrane Omega, fixed at the boundary, built out of two different materials, where certain prescribed regions (patches) in Omega are occupied by both materials. We prove existence results, and present some features of optimal solutions. The special case of one patch is treated in detail.