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Phase Field Approximation of Plateau's Problem

Updated 1 July 2025
  • 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 (ε0\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:Ω[0,1]u:\Omega \to [0,1]. The functional is

ATε(u)=Ωεu2+14ε(1u)2dx,\mathrm{AT}_\varepsilon(u) = \int_{\Omega} \varepsilon |\nabla u|^2 + \frac{1}{4\varepsilon}(1-u)^2\,dx,

where u1u \approx 1 far from the interface and u0u \approx 0 near the desired surface. As ε0\varepsilon \to 0, the energy concentrates on the region where uu 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 dd prescribed Lipschitz boundary curves γ0,γ1,...,γd\gamma_0, \gamma_1, ..., \gamma_d, the model incorporates, for each boundary pair, the pp-geodesic “height”: du,εp(γi,γj):=inf{Sl(up+δε)dH2:lHom(γi,γj)}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 ll ranges over Lipschitz homotopies between γi\gamma_i and γj\gamma_j, SlS_l is the swept-out surface, and δε0\delta_\varepsilon \to 0 is a vanishing regularizing constant.

The full energy functional is

Fε(u)=ATε(u)+1cεi=1ddu,εp(γ0,γi)F_\varepsilon(u) = \mathrm{AT}_\varepsilon(u) + \frac{1}{c_\varepsilon} \sum_{i=1}^d d_{u,\varepsilon}^p(\gamma_0, \gamma_i)

with cε0+c_\varepsilon \to 0^+ chosen suitably related to δε\delta_\varepsilon. The first term drives concentration along a minimal surface; the second ensures that the support of the interface, as ε0\varepsilon\to 0, separates and spans the prescribed curves in the sense of Reifenberg’s formulation or its generalizations.

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

The paper presents rigorous Γ\Gamma-convergence analysis for the model in the setting of a single closed curve Γ\Gamma on the boundary of a cylinder. Let C0C_0 be a cylindrical domain and Γ\Gamma a closed Lipschitz curve on its side. The classical Plateau problem is to find a minimal-area surface SS in C0C_0 with boundary Γ\Gamma.

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

  • Liminf inequality: For any sequence uεuu_\varepsilon \to u (in L1L^1) with bounded energy,

lim infε0Fε(uε)H2(S)\liminf_{\varepsilon\to 0} F_\varepsilon(u_\varepsilon) \geq \mathcal{H}^2(S)

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

  • Limsup inequality (recovery sequence): For any admissible SS, there exists a sequence uεu_\varepsilon approximating SS with

lim supε0Fε(uε)H2(S)\limsup_{\varepsilon\to 0} F_\varepsilon(u_\varepsilon) \leq \mathcal{H}^2(S)

realized by phase fields with sharp transition across SS.

The proof leverages the co-area formula, compactness results in BVBV, 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 (γ0,γi)(\gamma_0, \gamma_i), the penalty function

du,εp(γ0,γi)=inflHom(γ0,γi)Sl(up+δε)dH2d_{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|u|-“weight” do not exist between prescribed boundary components. As ε0\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 FεF_\varepsilon:

  • Phase field evolution: uu 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 (γ0,γi)(\gamma_0, \gamma_i), minimal uu-weighted geodesic homotopies are (approximately) computed using Fast Marching or Eikonal solvers, treating (14u)2+δε(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 ε\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 Γ\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 Γ\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
ATε(u)\mathrm{AT}_\varepsilon(u) Approximates area of interface Standard Modica–Mortola/Ambrosio–Tortorelli
Geodesic penalty du,εpd_{u,\varepsilon}^p Enforces topological spanning constraint Fast Marching, reduced-space Eikonal solvers
pp parameter, δε\delta_\varepsilon Regularization, flexibility in topology Must vanish suitably as ε0\varepsilon \to 0
WiLLMore-Cahn-Hilliard variant Improved regularity for computations Higher-order regularization of uu
Alternating optimization scheme Efficient numerical minimization Time-splitting: update uu and geodesics

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.


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 Γ\Gamma-convergence analysis and practical realization in challenging geometric settings.