This paper presents a numerical result for generalized eigenvalue problems in the optimum shape obtained by conducting the maximizing and minimizing problems of the energy dissipation. The shape optimization problem addressed in this paper is defined as a two-dimensional lid-driven cavity flow. As an objective cost function, the energy dissipation is used, and the domain volume is used as a constraint cost function, where the shape of the boundaries aside the top boundary is optimized. By solving a generalized eigenvalue problem, linear neutral curves between the initial domain Omega(0) and the optimum shape domains Omega(1) and Omega(2) for maximizing and minimizing problems respectively are compared. Numerical results reveal that the shape is optimized by satisfying the volume constraint. Based on generalized eigenvalue problems, the critical Reynolds numbers of the optimum shapes Omega(1) and Omega(2) are larger and smaller than that of the initial shape Omega(0).