Linearized equation and generic regularity in the Alt-Caffarelli problem

For the Alt-Caffarelli problem, we study free boundary regularity of energy minimizers. In six dimensions, we show that free boundaries are analytic for generic boundary data. In general, we improve previous generic Hausdorff dimensions of the singular sets. To achieve this, we analyze positive solutions to the linearized equation around homogeneous minimizers (possibly with singular sections on the sphere). For this equation, we prove a Harnack inequality and establish a dimensional lower bound for its principal eigenvalue.