  In this article, we consider the minimization problem for the first
eigenvalue of the fractional Laplacian with respect to the weight functions
lying in the rearrangement classes of fixed weight functions. We prove the
existence of minimizing weights in the rearrangement classes of weight
functions satisfying some assumptions. Also, we provide characterizations of
these minimizing weights in terms of the eigenfunctions. Furthermore, we
establish various qualitative properties, such as Steiner symmetry, radial
symmetry, foliated Schwarz symmetry, etc., of the minimizing weights and
corresponding eigenfunctions.
