The Two-Phase Membrane Problem -- an Intersection-Comparison Approach to the Regularity at Branch Points

TL;DR

The paper analyzes the two-phase membrane problem and proves in 2D that near branch points the free boundary splits into at most two $C^1$-graphs, using an intersection-comparison method based on Aleksandrov reflection. It also proves a stability result under boundary data perturbations and, in higher dimensions, establishes finiteness of the $(n-1)$-dimensional Hausdorff measure of the free boundary. Key tools include Weiss’s monotonicity formula, the Alt-Caffarelli-Friedman monotonicity, and a global blow-up classification that leads to uniform graphical regularity near branch points. The work advances the understanding of branch-point regularity for free boundary problems with two phases and provides a framework for stability and measure-theoretic control of the free boundary.

Abstract

For the two-phase membrane problem $ \Delta u = {\lambda_+\over 2} \chi_{\{u>0\}} - {\lambda_-\over 2} \chi_{\{u<0\}} ,$ where $\lambda_+> 0$ and $\lambda_->0 ,$ we prove in two dimensions that the free boundary is in a neighborhood of each ``branch point'' the union of two $C^1$-graphs. We also obtain a stability result with respect to perturbations of the boundary data. Our analysis uses an intersection-comparison approach based on the Aleksandrov reflection. In higher dimensions we show that the free boundary has finite $(n-1)$-dimensional Hausdorff measure.

Figures (6)

• Figure 1: A solution in 1d
• Figure 2: Examples of branch points
• Figure 3: Turning free boundary
• Figure 4: Example of $v$
• Figure 5: Example of $\phi_0$

Theorems & Definitions (8)

• Theorem 2.1: Weiss's Monotonicity Formula
• Theorem 2.2: Alt-Caffarelli-Friedman Monotonicity Formula
• Lemma 2.3
• Theorem 3.1
• Theorem 4.1
• Proposition 4.2
• Theorem 5.1
• Theorem 6.1