Linearized equation and generic regularity in the Alt-Caffarelli problem

TL;DR

This paper investigates the regularity of free boundaries in the Alt-Caffarelli problem, demonstrating analyticity in six dimensions for generic data and improving bounds on singular set dimensions through analysis of a linearized equation.

Contribution

It introduces a new analysis of the linearized equation around homogeneous minimizers, leading to improved regularity results and bounds on singular sets in the Alt-Caffarelli problem.

Findings

Free boundaries are analytic in six dimensions for generic boundary data.

Established a Harnack inequality for solutions to the linearized equation.

Provided a lower bound for the principal eigenvalue of the linearized operator.

Abstract

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.