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

One-phase free boundary solutions of finite Morse index