One-phase free boundary solutions of finite Morse index

We study global solutions to the classical one-phase free boundary problem that have finite Morse index relative to the Alt-Caffarelli functional. We show that such solutions are stable outside a compact set and characterize the index as the maximal number of linearly independent $L^2$ integrable eigenfunctions of the corresponding Robin eigenvalue problem, associated to negative eigenvalues. As an application, we obtain a complete classification of global solutions of finite Morse index in the plane. Our results are counterparts to the minimal surface theorems of Fischer-Colbrie and Gulliver.