Stability of p-area minimizing surfaces in the Heisenberg group

Technical Summary: Stability of p-Area Minimizing Surfaces in the Heisenberg Group

Problem Statement
This paper investigates the quantitative stability of minimizers for weighted $p$-area functionals associated with surfaces of prescribed $p$-mean curvature in the Heisenberg group $\mathbb{H}^n$. While the existence and uniqueness of such minimizers are well-established in the literature (notably for the minimal surface case $H=0$ and the prescribed curvature case $H \neq 0$), the behavior of these minimizers under perturbations of the mean curvature function $H$ remains largely unexplored. Specifically, the authors address the lack of quantitative stability estimates for the non-zero $H$ regime, where the coupling between the geometry and the potential term $Hu$ introduces significant analytical challenges, including degeneracy along characteristic sets and the breakdown of homogeneity.

Methodology
The authors employ a Rockafellar–Fenchel duality framework to analyze the variational problem. The primary functional under consideration is:
$$F(u) = \int_{\Omega} (a|\nabla u - X^*| + Hu) \, dx$$
where $X^*$ encodes the intrinsic rotation of horizontal directions, $a$ is a positive weight function, and $\Omega \subset \mathbb{R}^{2n}$.

The core of the methodology involves:

1. Dual Formulation: Utilizing duality to identify a unique underlying vector field $J$ associated with any minimizer $u$. This field satisfies $|J| \leq a$ and aligns with the direction of the horizontal gradient $\nabla u - X^*$ almost everywhere.
2. Stability of the Vector Field: Proving that the vector field $J$ is stable under perturbations of $H$. By establishing bounds on the difference between the vector fields $J$ and $\tilde{J}$ corresponding to curvatures $H$ and $\tilde{H}$, the authors derive control over the direction of the horizontal gradient.
3. Geometric Assumptions on Level Sets: To extend stability from the vector field to the minimizers themselves, the authors impose structural hypotheses on the level sets of the minimizers.
   • In dimension $n=2$, they assume the length of level sets is uniformly bounded and that almost every level set reaches the boundary.
   • In dimension $n=3$, they assume level sets can be foliated into one-dimensional curves with specific regularity and boundedness properties.
4. Regularity and Coarea Formulas: The proofs utilize the coarea formula, integration by parts for BV functions, and elliptic regularity estimates (assuming sufficient smoothness of data) to derive $L^1$ and $W^{1,1}$ stability estimates.

Key Contributions and Results
The paper establishes the first quantitative stability theory for $p$-area minimizing graphs with prescribed nonzero $p$-mean curvature, even in the unweighted case ($a \equiv 1$). The specific results include:

• Vector Field Stability: The authors prove that the underlying vector field $J$ is stable in the $L^1$ norm with respect to perturbations in $H$. Specifically, $\|J - \tilde{J}\|_{L^1(\Omega)} \leq C \|H - \tilde{H}\|_{L^\infty(\Omega)}^{1/2}$. This implies that the direction of the horizontal gradient is quantitatively controlled by the perturbation of the mean curvature.
• $L^1$ Stability of Minimizers: Under natural geometric assumptions on the level sets (bounded length in 2D, foliation in 3D), the authors establish $L^1$ stability for admissible minimizers:
  $$\|u - \tilde{u}\|_{L^1(\Omega)} \leq C \|H - \tilde{H}\|_{L^\infty(\Omega)}^{1/2}$$
• $W^{1,1}$ Stability: Under additional regularity hypotheses (requiring $C^2$ boundary, $C^2$ vector fields, and "well-structured" level sets), the paper derives $W^{1,1}$ stability estimates for the gradients:
  $$\|\nabla u - \nabla \tilde{u}\|_{L^1(\Omega)} \leq C \|H - \tilde{H}\|_{L^\infty(\Omega)}^{1/4}$$
  Consequently, stability for the weight function $\sigma$ (related to the magnitude of the gradient) is also derived with the same rate.
• Numerical Validation: The theoretical results are illustrated through numerical simulations using an alternating split Bregman algorithm. The authors construct a test case with a known exact solution and introduce stochastic noise to the mean curvature $H$. The simulations demonstrate that the numerical approximations remain robust, with relative $L^2$ errors increasing predictably with the noise level, thereby supporting the theoretical stability claims.

Significance
The authors claim that these results provide a more refined understanding of the structure of pseudohermitian $p$-mean curvature graphs. By extending the theory beyond mere existence and uniqueness to include quantitative stability, the work addresses a critical gap in the analysis of sub-Riemannian minimal surfaces. The identification of the unique underlying vector field $J$ serves as a foundational tool for this analysis. The results are presented as new even in the special case where the weight function is constant, offering explicit rates of convergence in terms of the perturbation of the mean curvature. The inclusion of numerical simulations serves to illustrate the robustness of these theoretical findings in the presence of data perturbations.