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Asymptotic behaviour of the weak inverse anisotropic mean curvature flow

Chaoqun Gao, Yong Wei, Rong Zhou

TL;DR

This work extends the isotropic inverse mean curvature flow to the anisotropic setting by establishing a local gradient estimate for anisotropic p-harmonic functions that remains uniform as p →1+. This uniform bound enables a rigorous analysis of the weak inverse anisotropic mean curvature flow, culminating in a precise asymptotic description: the flow converges to expanding Wulff shapes at infinity, with an explicit logarithmic correction determined by anisotropic perimeters. The results rely on a Bochner-type framework for the anisotropic p-Laplacian and a key third-derivative bound for the Minkowski norm, bridging gradient control with geometric asymptotics. Overall, the paper generalizes prior isotropic results to the anisotropic, providing foundational insights for large-scale IAMCF behavior in Minkowski geometries.

Abstract

We first establish a local gradient estimate for anisotropic $p$-harmonic functions. A key feature of our estimate is that the constant remains bounded as $p\to 1$; consequently, in the limit $p\to 1$, this estimate yields the local gradient estimate for weak solutions of the inverse anisotropic mean curvature flow (IAMCF). As an application, we show that the weak IAMCF is asymptotic to the expanding Wulff shape solution at the infinity, thereby extending the result of Huisken and Ilmanen in [8] to the anisotropic case.

Asymptotic behaviour of the weak inverse anisotropic mean curvature flow

TL;DR

This work extends the isotropic inverse mean curvature flow to the anisotropic setting by establishing a local gradient estimate for anisotropic p-harmonic functions that remains uniform as p →1+. This uniform bound enables a rigorous analysis of the weak inverse anisotropic mean curvature flow, culminating in a precise asymptotic description: the flow converges to expanding Wulff shapes at infinity, with an explicit logarithmic correction determined by anisotropic perimeters. The results rely on a Bochner-type framework for the anisotropic p-Laplacian and a key third-derivative bound for the Minkowski norm, bridging gradient control with geometric asymptotics. Overall, the paper generalizes prior isotropic results to the anisotropic, providing foundational insights for large-scale IAMCF behavior in Minkowski geometries.

Abstract

We first establish a local gradient estimate for anisotropic -harmonic functions. A key feature of our estimate is that the constant remains bounded as ; consequently, in the limit , this estimate yields the local gradient estimate for weak solutions of the inverse anisotropic mean curvature flow (IAMCF). As an application, we show that the weak IAMCF is asymptotic to the expanding Wulff shape solution at the infinity, thereby extending the result of Huisken and Ilmanen in [8] to the anisotropic case.

Paper Structure

This paper contains 8 sections, 10 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Local gradient estimate
  3. Asymptotical behaviour of weak IAMCF
  4. Preliminaries
  5. Minkowski norm and Wulff shape
  6. Weak inverse anisotropic mean curvature flow
  7. Local gradient estimate of anisotropic $p$-harmonic functions
  8. Asymptotic behaviour of weak IAMCF

Key Result

Theorem 1.1

The weak inverse mean curvature flow starting from an open bounded domain $\Omega\subset\mathbb{R}^{n}$ with smooth boundary satisfies where $\Omega^*$ denotes the strictly outward minimizing hull of $\Omega$. Moreover, the expanding sphere solution is the only solution to eq:WIMCF on $\mathbb{R}^n\backslash\{0\}$ with compact level sets.

Theorems & Definitions (16)

  • Theorem 1.1: BFM24 and HI01
  • Theorem 1.2: CMS24DGX23
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Remark 1.6
  • Lemma 2.1
  • Definition 2.2
  • Lemma 2.3: see DGX23
  • Lemma 2.4: Compactness
  • ...and 6 more