Improved $C^{1,1}$ regularity for multiple membranes problem

TL;DR

This paper establishes $C^{1,1}$ regularity for stationary solutions to the multiple membrane problem, which is crucial for progress on Yau's four minimal spheres conjecture.

Contribution

It provides the first proof of $C^{1,1}$ regularity for solutions to the multiple membrane problem, advancing understanding of free boundary regularity.

Findings

Proves $C^{1,1}$ regularity for stationary solutions

Supports progress on Yau's four minimal spheres conjecture

Enhances regularity theory for free boundary problems

Abstract

We prove the $C^{1,1}$-regularity for stationary $C^{1,\alpha}$ ($\alpha\in(0,1)$) solutions to the multiple membrane problem. This regularity estimate was essentially used in our recent work on Yau's four minimal spheres conjecture.