Phase Field Approximation of Plateau's Problem

• Phase field approximation models minimal surfaces for Plateau's problem using diffuse interfaces and variational energy functionals with topological penalties.
• It employs Ambrosio-Tortorelli energy for area and a geodesic distance penalty to enforce complex topological spanning constraints.
• Mathematical analysis confirms convergence to minimal surfaces, and numerical methods effectively handle various complex topologies and boundaries.

A phase field approximation of Plateau’s problem refers to a class of variational models that encode minimal surfaces (i.e., solutions to Plateau’s problem) as diffuse interfaces of smooth functions, where area minimization and boundary spanning conditions are imposed via energy functionals and topological penalization. The central aim is to construct, analyze, and numerically realize phase field energies whose sharp-interface ($\varepsilon \to 0$) limits yield surfaces with the required minimal area and prescribed boundary, extending the successful phase field methods from codimension one (e.g., Modica-Mortola, Ambrosio-Tortorelli) to fully topologically prescribed settings such as those arising in Plateau’s classical and modern generalizations.

1. Variational Model: Ambrosio-Tortorelli Energy with Geodesic Distance Penalty

Plateau’s problem seeks a surface of minimal area spanning a given set of boundary curves. Classical phase field models approximate area using Ambrosio-Tortorelli energies, representing the surface as the transition layer of a smooth function $u:\Omega \to [0,1]$. The functional is

$$\mathrm{AT}_\varepsilon(u) = \int_{\Omega} \varepsilon |\nabla u|^2 + \frac{1}{4\varepsilon}(1-u)^2\,dx,$$

where $u \approx 1$ far from the interface and $u \approx 0$ near the desired surface. As $\varepsilon \to 0$, the energy concentrates on the region where $u$ transitions, mimicking the Hausdorff area of the limiting sharp surface.

To encode topological spanning constraints for Plateau’s problem, the model introduces a geodesic distance penalty. Given $d$ prescribed Lipschitz boundary curves $\gamma_0, \gamma_1, ..., \gamma_d$, the model incorporates, for each boundary pair, the $p$-geodesic “height”: $$d_{u, \varepsilon}^{p}(\gamma_i, \gamma_j) := \inf \left\{ \int_{S_l} (|u|^p + \delta_\varepsilon) d\mathcal{H}^2 : l \in \mathrm{Hom}(\gamma_i, \gamma_j) \right\}$$ where $l$ ranges over Lipschitz homotopies between $\gamma_i$ and $\gamma_j$, $S_l$ is the swept-out surface, and $\delta_\varepsilon \to 0$ is a vanishing regularizing constant.

The full energy functional is

$$F_\varepsilon(u) = \mathrm{AT}_\varepsilon(u) + \frac{1}{c_\varepsilon} \sum_{i=1}^d d_{u,\varepsilon}^p(\gamma_0, \gamma_i)$$

with $c_\varepsilon \to 0^+$ chosen suitably related to $\delta_\varepsilon$. The first term drives concentration along a minimal surface; the second ensures that the support of the interface, as $\varepsilon\to 0$, separates and spans the prescribed curves in the sense of Reifenberg’s formulation or its generalizations.

2. Mathematical Analysis: $\Gamma$-Convergence and the Simple-Curve Cylinder Case

The paper presents rigorous $\Gamma$-convergence analysis for the model in the setting of a single closed curve $\Gamma$ on the boundary of a cylinder. Let $C_0$ be a cylindrical domain and $\Gamma$ a closed Lipschitz curve on its side. The classical Plateau problem is to find a minimal-area surface $S$ in $C_0$ with boundary $\Gamma$.

The main result establishes that for a sequence $\{u_\varepsilon\}$ minimizing $F_\varepsilon$, the measure-theoretic support of the set $\{u_\varepsilon > t_\varepsilon\}$ converges (in measure) to a surface $S$ that solves the classical Plateau problem in this domain:

• Liminf inequality: For any sequence $u_\varepsilon \to u$ (in $L^1$) with bounded energy,

$$\liminf_{\varepsilon\to 0} F_\varepsilon(u_\varepsilon) \geq \mathcal{H}^2(S)$$

where $S$ is the limit interface. The geodesic penalty enforces the required spanning property.

• Limsup inequality (recovery sequence): For any admissible $S$, there exists a sequence $u_\varepsilon$ approximating $S$ with

$$\limsup_{\varepsilon\to 0} F_\varepsilon(u_\varepsilon) \leq \mathcal{H}^2(S)$$

realized by phase fields with sharp transition across $S$.

The proof leverages the co-area formula, compactness results in $BV$, and analysis of the spanning property imposed by geodesic penalization.

3. Penalization of Spanning via Geodesic Functionals

The geodesic term transposes topological constraints from the geometric setting to the phase field. For each pair $(\gamma_0, \gamma_i)$, the penalty function

$$d_{u,\varepsilon}^p(\gamma_0, \gamma_i) = \inf_{l \in \mathrm{Hom}(\gamma_0, \gamma_i)} \int_{S_l} (|u|^p+\delta_\varepsilon) d\mathcal{H}^2$$

penalizes configurations where surfaces of low $|u|$-“weight” do not exist between prescribed boundary components. As $\varepsilon\to 0$, only sets with interfaces spanning the curves have finite energy. This generalizes the mechanism of phase field approximations for the Steiner problem and adapts it to the context of minimal surfaces spanning higher-codimension boundaries.

This approach allows direct enforcement of Reifenberg-type and other homotopical spanning criteria, avoids explicit parametrizations, and is compatible with both orientable and non-orientable (and possibly disconnected or singular) limiting surfaces.

4. Numerical Algorithms and Implementation

A robust alternating minimization scheme is developed to numerically minimize $F_\varepsilon$:

• Phase field evolution: $u$ is updated (e.g., via Allen–Cahn/gradient flow, or higher-order WiLLMore-Cahn-Hilliard flow variants for regularity), keeping geodesic terms fixed.
• Geodesic update: For each $(\gamma_0, \gamma_i)$, minimal $u$-weighted geodesic homotopies are (approximately) computed using Fast Marching or Eikonal solvers, treating $(1-4u)^2 + \delta_\varepsilon$ as local cost in the ambient space or reduced parameterizations (especially when the homotopy space is low-dimensional).
• Regularization: WiLLMore-Cahn-Hilliard variants sharpen the interface and improve numerical stability, particularly when small $\varepsilon$ is required.

Experimental results include reproductions of the catenoid between two circles, minimal surfaces spanning multiple boundaries, non-orientable films, and singular minimizers (e.g., cube configurations). The method accommodates complex topologies, including disconnected boundaries and multiple components, and is effective even beyond the settings rigorously analyzed by $\Gamma$-convergence.

5. Connections, Generalizations, and Theoretical Implications

The model extends the codimension-one approximation schemes (e.g., Allen–Cahn, Ambrosio-Tortorelli) to settings with nontrivial topological constraints, and is closely related in spirit to phase field approximations of the Steiner problem by Bonnivard, Lemenant, and Santambrogio. The geodesic penalty, inspired by Reifenberg’s approach, may be adapted to encode additional homological or cohomological constraints (e.g., as in Harrison-Pugh or Reifenberg homology/Čech conditions).

The $\Gamma$-convergence result confirms that the method selects minimal-area surfaces satisfying the prescribed spanning property in the sharp-interface limit. For more general or higher-genus Plateau problems, the approach potentially extends by adapting the notion of admissible homotopies/homologies and suitable phase field functionals.

The framework thus provides a rigorous and computationally effective method for approximating and realizing minimal surfaces in Plateau’s problem, incorporating topological constraints robustly within a diffused interface (phase field) setting, with support from both theoretical convergence results and numerical experiments.

6. Summary Table: Model Ingredients and Properties

Term/Component	Role	Implementation/Comment
$\mathrm{AT}_\varepsilon(u)$	Approximates area of interface	Standard Modica–Mortola/Ambrosio–Tortorelli
Geodesic penalty $d_{u,\varepsilon}^p$	Enforces topological spanning constraint	Fast Marching, reduced-space Eikonal solvers
$p$ parameter, $\delta_\varepsilon$	Regularization, flexibility in topology	Must vanish suitably as $\varepsilon \to 0$
WiLLMore-Cahn-Hilliard variant	Improved regularity for computations	Higher-order regularization of $u$
Alternating optimization scheme	Efficient numerical minimization	Time-splitting: update $u$ and geodesics

7. References to Related Work

The approach generalizes earlier phase field approximations for Steiner’s problem and draws from convergence/stability theory for Ambrosio–Tortorelli energies. It is designed to be compatible with main geometric measure theory advances in Plateau’s problem (Reifenberg, Harrison–Pugh, De Lellis et al.), and provides an effective bridge between variational PDEs and geometric minimization with topological constraints.

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In sum, this phase field geodesic-distance penalty approach furnishes a mathematically rigorous, numerically robust framework for approximating Plateau-minimizing surfaces with complex topology, validated by $\Gamma$-convergence analysis and practical realization in challenging geometric settings.