[Paper Review] Some remarks about the existence of an Alt-Caffarelli-Friedman monotonicity formula in the Heisenberg group

Fausto Ferrari, Nicolò Forcillo|arXiv (Cornell University)|Jan 13, 2020

Geometric Analysis and Curvature Flows26 references4 citations

TL;DR

This paper investigates the existence of an Alt-Caffarelli-Friedman-type monotonicity formula in the Heisenberg group $\mathbb{H}^1$, deriving necessary and sufficient conditions for its validity. Using spherical coordinates adapted to the Heisenberg geometry and analyzing subharmonic functions with radial and angular dependence, the authors establish a monotonicity formula for solutions of two-phase free boundary problems involving the Kohn sublaplacian, proving that the product of weighted $L^2$ norms of gradients remains bounded and monotonic under specific symmetry and homogeneity conditions.

ABSTRACT

The aim of this paper is to study the existence of an Alt-Caffarelli-Friedman monotonicity type formula in the Heisenberg group.

Motivation & Objective

• To determine necessary and sufficient conditions for the existence of an Alt-Caffarelli-Friedman monotonicity formula in the Heisenberg group $\mathbb{H}^1$.
• To analyze the structure of solutions to two-phase free boundary problems involving the Kohn sublaplacian $\Delta_{\mathbb{H}^1}$.
• To extend the classical Euclidean monotonicity formula to the sub-Riemannian setting of $\mathbb{H}^1$ using geometric and analytic techniques.
• To provide a detailed computation of the monotonicity formula in $\mathbb{H}^1$ for functions depending on $\theta$ and $\varphi$, particularly in symmetric caps.

Proposed method

• Parametrize the Heisenberg group $\mathbb{H}^1$ using spherical coordinates $\rho, \theta, \varphi$, with $\xi = (x,y,t) = (\rho\sqrt{\sin\varphi}\cos\theta, \rho\sqrt{\sin\varphi}\sin\theta, \rho^2\cos\varphi)$.
• Compute the Jacobian determinant of the transformation to obtain the volume element $|\det J_T| = \rho^3$, enabling integration in spherical coordinates.
• Derive the expression for the Kohn sublaplacian in $\mathbb{H}^1$ and analyze homogeneous solutions $u = \rho^\alpha f(\theta, \varphi)$ with $f$ independent of $\rho$, focusing on radial and angular dependence.
• Use the change of variables and geometric measure theory to compute weighted $L^2$ norms of gradients over Korányi balls and their boundaries.
• Establish a monotonicity formula by analyzing the product $\int_{B_R} \frac{|\nabla_{\mathbb{H}^1} u^+|^2}{|\xi|_{\mathbb{H}^1}^2} d\xi \cdot \int_{B_R} \frac{|\nabla_{\mathbb{H}^1} u^-|^2}{|\xi|_{\mathbb{H}^1}^2} d\xi$, showing it is bounded and increasing.
• Prove that for symmetric caps and specific functions like $u = x$, the ratio of weighted gradient and function norms on the boundary equals 2, confirming monotonicity.

Experimental results

Research questions

• RQ1 Under what conditions does an Alt-Caffarelli-Friedman monotonicity formula exist in the Heisenberg group $\mathbb{H}^1$?
• RQ2 How does the structure of solutions to two-phase free boundary problems in $\mathbb{H}^1$ affect the monotonicity of energy products?
• RQ3 What role does angular and radial symmetry play in the validity of the monotonicity formula in the sub-Riemannian setting?
• RQ4 Can the classical Euclidean monotonicity formula be generalized to the Kohn sublaplacian in $\mathbb{H}^1$?
• RQ5 What is the precise value of the ratio of weighted $L^2$ norms of gradients and functions on the boundary for symmetric caps in $\mathbb{H}^1$?

Key findings

• The product $\int_{B_R^{\mathbb{H}^1}(0)} \frac{|\nabla_{\mathbb{H}^1} u^+|^2}{|\xi|_{\mathbb{H}^1}^2} d\xi \cdot \int_{B_R^{\mathbb{H}^1}(0)} \frac{|\nabla_{\mathbb{H}^1} u^-|^2}{|\xi|_{\mathbb{H}^1}^2} d\xi$ equals $4\pi^2 \alpha^2 \beta^2 R^8$ for homogeneous solutions $u = \rho^\alpha f(\theta, \varphi)$, establishing a quantitative monotonicity behavior.
• For the function $u = x$, the ratio $\frac{\int_{\partial B_1^{\mathbb{H}^1} \cap \{u>0\}} \frac{|\nabla_{\mathbb{H}^1}^\varphi u|^2}{\sqrt{x^2+y^2}} d\sigma_{\mathbb{H}^1}}{\int_{\partial B_1^{\mathbb{H}^1} \cap \{u>0\}} u^2 \sqrt{x^2+y^2} d\sigma_{\mathbb{H}^1}}$ evaluates exactly to 2, confirming the monotonicity formula in a concrete case.
• The volume element in $\mathbb{H}^1$ under spherical coordinates is $\rho^3 d\rho d\theta d\varphi$, derived from the Jacobian determinant $|\det J_T| = \rho^3$, which is essential for integration over Korányi balls.
• The integral $\int_{B_R^{\mathbb{H}^1}(0) \cap \{t>0\}} \frac{x^2 + y^2}{|\xi|_{\mathbb{H}^1}^2} d\xi = \frac{\pi R^4}{2}$ is computed explicitly, providing a key component in the energy product estimate.
• The monotonicity formula is shown to hold under specific symmetry and homogeneity assumptions, with the ratio of weighted norms being bounded and increasing in $R$.
• The paper establishes that the existence of the monotonicity formula in $\mathbb{H}^1$ is equivalent to certain geometric and analytic conditions on the boundary data and the angular dependence of the solution.