Abstract
In this paper, we study a shape optimization problem in two dimensions where the objective function is the convex combination of two sequential Steklov eigenvalues of a domain with a fixed area constraint. We show the existence of the optimal domain and the nondecreasing, Lipschitz continuity, and convexity of the optimal objective function with respect to the convex combination constant. On one-parameter family of rectangular domains, asymptotic behaviors of lower eigenvalues are found. For general shapes, numerical approaches are used to find optimal shapes. The range of the first two Steklov eigenvalues are discussed for several one-parameter families of shapes including Cassini oval shapes and Hippopede shapes.