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Quantitative estimates for the relative isoperimetric problem and its gradient flow outside convex bodies in the plane

Elena Mäder-Baumdicker, Robin Neumayer, Jiewon Park, Melanie Rupflin

TL;DR

This paper analyzes the exterior planar isoperimetric problem relative to a convex body Ω by leveraging a two-dimensional reduction to families of circular arcs Circ_η. Under a simple nondegeneracy condition, it furnishes sharp quantitative results: (i) a Łojasiewicz-type rigidity estimate for critical points, (ii) exponential convergence of the free boundary area-preserving curve shortening flow to minimizers, and (iii) a flow-based, quantitative stability result for minimizers. The authors provide explicit constants and optimal rates, and crucially integrate a flow approach to establish stability in the isoperimetric context for the first time. The framework accommodates the special case when Σ is a circle and yields transparent geometric criteria via the Hessian of the 2D reduced problem, with precise control of the area-constraint and end-point angles.

Abstract

We prove three related quantitative results for the relative isoperimetric problem outside a convex body $Ω$ in the plane: (1) Łojasiewicz estimates and quantitative rigidity for critical points, (2) rates of convergence for the gradient flow, and (3) quantitative stability for minimizers. These results come with explicit constants and optimal exponents/rates, and hold whenever a simple two-dimensional auxiliary variational problem for circular arcs outside of $Ω$ is nondegenerate. The proofs are inter-related, and in particular, for the first time in the context of isoperimetric problems, a flow approach is used to prove quantitative stability for minimizers.

Quantitative estimates for the relative isoperimetric problem and its gradient flow outside convex bodies in the plane

TL;DR

This paper analyzes the exterior planar isoperimetric problem relative to a convex body Ω by leveraging a two-dimensional reduction to families of circular arcs Circ_η. Under a simple nondegeneracy condition, it furnishes sharp quantitative results: (i) a Łojasiewicz-type rigidity estimate for critical points, (ii) exponential convergence of the free boundary area-preserving curve shortening flow to minimizers, and (iii) a flow-based, quantitative stability result for minimizers. The authors provide explicit constants and optimal rates, and crucially integrate a flow approach to establish stability in the isoperimetric context for the first time. The framework accommodates the special case when Σ is a circle and yields transparent geometric criteria via the Hessian of the 2D reduced problem, with precise control of the area-constraint and end-point angles.

Abstract

We prove three related quantitative results for the relative isoperimetric problem outside a convex body in the plane: (1) Łojasiewicz estimates and quantitative rigidity for critical points, (2) rates of convergence for the gradient flow, and (3) quantitative stability for minimizers. These results come with explicit constants and optimal exponents/rates, and hold whenever a simple two-dimensional auxiliary variational problem for circular arcs outside of is nondegenerate. The proofs are inter-related, and in particular, for the first time in the context of isoperimetric problems, a flow approach is used to prove quantitative stability for minimizers.

Paper Structure

This paper contains 18 sections, 22 theorems, 174 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The isoperimetric profile
  4. Notation and basic properties of oriented curves
  5. Proof of Theorem \ref{['thm:loj']}
  6. Reduction to circular arcs
  7. Analysis of circular arcs
  8. Proof of Theorem \ref{['thm:loj']}
  9. Exponential convergence of the gradient flow
  10. Background on the flow
  11. Proof of \ref{['mainthm-flow']}
  12. Quantitative stability
  13. Proof of \ref{['lem:distances']}
  14. Reduction to a set bounded by a rectifiable curve
  15. Reduction to a set bounded by a convex curve
  16. ...and 3 more sections

Key Result

Theorem 1.2

Let $\Omega\subset{\mathbb{R}}^2$ be a convex body with $C^2$ boundary $\Sigma =\partial \Omega$, fix $\eta>0$ and assume that either $\Sigma$ is a circle or the pair $(\Sigma,\eta)$ satisfies Assumption ass:LojassIntro. Then for each $\bar{L}>0$ and $\bar{\phi}>0$, there exist constants $C_{0,1} =C and where $\alpha_{1,2}\in [0,\pi]$ denote the intersection angles at which $\gamma$ intersects $\

Figures (2)

  • Figure 1: A critical arc $c^*$ with relevant points and directions
  • Figure 2: A solution of the flow $\gamma_t$ can cross at the endpoints $\gamma_t(a_1)$, $\gamma_t(a_2)$ at some time $t>0$. If we naively close the curves $\gamma_t$ for each $t$ by connecting $\gamma_t(a_2)$ with $\gamma_t(a_2)$ via the positive orientation of $\Sigma$, then the enclosed area jumps by $|\Omega|$ or $-|\Omega|$. Our choice of boundary curves closing up $\gamma_t$, Definition \ref{['def: sigma_gamma']}, ensures that the algebraic enclosed area is continuous with respect to $t$ along the flow.

Theorems & Definitions (52)

  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Lemma 3.1
  • ...and 42 more