On the optimization of the first weighted eigenvalue of the fractional Laplacian

In this paper, 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.