Abstract:The classical one-phase free boundary problem (FBP) is one of the prototypical free boundary problems. Starting from the pioneering work of Alt and Caffarelli (1981), its energy-minimizing solutions have been fairly well studied and understood. The focus of the work that I will present in this talk is on solutions of the one-phase FBP that are not necessarily energy minimizing. In joint work with J. Basulto (PUC-Chile), we investigated entire solutions of bounded Morse index and obtained a complete classification theorem in the plane as well as a partial rigidity result for stable solutions in Euclidean 3-space. Our results are free boundary counterparts to classical theorems in the minimal surface literature.