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Calibration for minimal surfaces with free boundary and Cheeger-type problems

Guy Bouchitté, Minh Phan

TL;DR

The paper addresses minimal surfaces with free boundary by formulating a nonconvex variational problem β(λ) and its BV relaxation, linking it to a Cheeger-type shape problem. It develops a dual calibration framework in dimension N+1 and derives explicit convex duals, enabling optimality conditions via calibrations; a 2D cut-locus potential ρ is constructed to obtain explicit calibrations and to identify the optimal region Ω_λ as the union of all balls of radius 1/λ contained in D. It proves a strict inequality β(λ) < β0(λ) when trivial minimizers are not optimal and characterizes when equality can occur through Cheeger self-ness, defining λ0 and λ1 as transition thresholds. The cut-locus potential yields a concrete geometric method in 2D to calibrate the unique minimizer Ω_λ for λ ≥ h_D, and the calibration field is extendable to all of D, with explicit results for squares and ellipsoids and potential extensions to higher dimensions.

Abstract

We study a problem of minimal surfaces with free boundary written in the form of a non convex minimization problem. Our aim is to characterize optimal solutions by finding a suitable calibration field. A natural upper bound of the infimum is given by a variant of the Cheeger problem that we solve explicitly proving the optimality thanks to the construction of a cut-locus potential. The comparison with the original problem is then discussed in detail.

Calibration for minimal surfaces with free boundary and Cheeger-type problems

TL;DR

The paper addresses minimal surfaces with free boundary by formulating a nonconvex variational problem β(λ) and its BV relaxation, linking it to a Cheeger-type shape problem. It develops a dual calibration framework in dimension N+1 and derives explicit convex duals, enabling optimality conditions via calibrations; a 2D cut-locus potential ρ is constructed to obtain explicit calibrations and to identify the optimal region Ω_λ as the union of all balls of radius 1/λ contained in D. It proves a strict inequality β(λ) < β0(λ) when trivial minimizers are not optimal and characterizes when equality can occur through Cheeger self-ness, defining λ0 and λ1 as transition thresholds. The cut-locus potential yields a concrete geometric method in 2D to calibrate the unique minimizer Ω_λ for λ ≥ h_D, and the calibration field is extendable to all of D, with explicit results for squares and ellipsoids and potential extensions to higher dimensions.

Abstract

We study a problem of minimal surfaces with free boundary written in the form of a non convex minimization problem. Our aim is to characterize optimal solutions by finding a suitable calibration field. A natural upper bound of the infimum is given by a variant of the Cheeger problem that we solve explicitly proving the optimality thanks to the construction of a cut-locus potential. The comparison with the original problem is then discussed in detail.

Paper Structure

This paper contains 11 sections, 22 theorems, 189 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Dual problems and calibrations
  3. Dual problem of \ref{['pbbeta']}
  4. Dual problem of beta0
  5. Relations with Cheeger problem and calibrability notions
  6. Comparison results
  7. Cut-locus potential and construction of a two-dimensional calibration
  8. Cut-locus potential
  9. The set $\Omega_\lambda$ is calibrable and solves $m(\lambda,D)$
  10. Extending the calibration field to all $D$
  11. Some techical lemmas in $\mathbb{R}^2$

Key Result

Theorem 2.1

The duality principle given in bouphan2025 leads to the following no-gap equality

Figures (13)

  • Figure 1: A surface with prescribed boundary $u=0$ on $\partial D$ and free boundary $\partial\{u=1\}$.
  • Figure 2: Illustration for an optimal $u$ and optimality conditions.
  • Figure 3: Critical values of $\lambda$ in term of the radius $R$ of a disk $D\subset\mathbb{R}^2$.
  • Figure 4: Dependence in $\lambda$ for $R \in \{0.5, 0.8, 1\}$.
  • Figure 5: Dependence in $\lambda$ for $R \in \{1.2, 1.5, 1.8, 2,3,5\}$.
  • ...and 8 more figures

Theorems & Definitions (57)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • Remark 2.5
  • proof
  • Proposition 2.6
  • proof
  • Definition 2.7
  • Remark 2.8
  • ...and 47 more