A classical problem in analysis is to relate information about the spectrum of the Laplacian of a domain to that domain's shape. One approach to this program is by studying sets which minimize some function of their Laplacian's eigenvalues, with, say, a volume constraint. I will explain how to study the local structure of the boundaries of such minimizers by interpreting the configuration as a vector-valued free boundary problem. Then I will present some new regularity results for these free boundaries. This is based on joint work with Fanghua Lin.