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A half-space Liouville theorem for anisotropic minimal graph with free boundary

Guofang Wang, Wei Wei, Chao Xia, Xuwen Zhang

TL;DR

The paper proves a half-space Liouville-type theorem for anisotropic minimal graphs with free boundary: any $C^2$ solution $u$ to the anisotropic minimal surface equation in $\mathbb{R}^n_+$ with free boundary, whose negative part has at most linear growth, must be affine. The authors develop an integral-analytic approach based on the graphical area element $W_f$ and its anisotropic gradient, establishing a boundary-tangency identity that cancels boundary terms, a mean-value inequality for $\log W_f$, and a robust gradient bound via Sobolev inequalities and De Giorgi–Nash–Moser iteration. This gradient estimate holds in full generality for any uniformly elliptic integrand $F$ and all dimensions, enabling the affine rigidity conclusion. Moreover, the method extends to anisotropic capillary graphs in the half-space, removing angle restrictions present in prior works and highlighting the sharp linear-growth condition through known counterexamples.

Abstract

In this paper we prove the following Liouville-type theorem: any anisotropic minimal graph with free boundary in the half-space must be flat, provided that the graph function has at most one-sided linear growth. This extends the classical results of Bombieri-De Giorgi-Miranda and Simon to an appropriate free boundary setting.

A half-space Liouville theorem for anisotropic minimal graph with free boundary

TL;DR

The paper proves a half-space Liouville-type theorem for anisotropic minimal graphs with free boundary: any solution to the anisotropic minimal surface equation in with free boundary, whose negative part has at most linear growth, must be affine. The authors develop an integral-analytic approach based on the graphical area element and its anisotropic gradient, establishing a boundary-tangency identity that cancels boundary terms, a mean-value inequality for , and a robust gradient bound via Sobolev inequalities and De Giorgi–Nash–Moser iteration. This gradient estimate holds in full generality for any uniformly elliptic integrand and all dimensions, enabling the affine rigidity conclusion. Moreover, the method extends to anisotropic capillary graphs in the half-space, removing angle restrictions present in prior works and highlighting the sharp linear-growth condition through known counterexamples.

Abstract

In this paper we prove the following Liouville-type theorem: any anisotropic minimal graph with free boundary in the half-space must be flat, provided that the graph function has at most one-sided linear growth. This extends the classical results of Bombieri-De Giorgi-Miranda and Simon to an appropriate free boundary setting.
Paper Structure (11 sections, 23 theorems, 172 equations)

This paper contains 11 sections, 23 theorems, 172 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Elliptic integrand
  4. Anisotropic free boundary minimal hypersurfaces
  5. Anisotropic minimal graph
  6. Integral estimates
  7. Mean value inequality
  8. Area growth estimates
  9. Sobolev inequality
  10. Iteration argument
  11. Proof of the main results

Key Result

Theorem 1.1

Let $F:\mathbb{R}^{n+1}\rightarrow\mathbb{R}_+$ be a uniformly $C^2$-elliptic integrand, $u$ be a $C^2$-solution to defn:AMSE on $\mathbb{R}^n_+$ satisfying the free boundary condition in the anisotropic sense defn:anisotropic-free-bdry-Intro. If the negative part of $u$ has at most linear growth on for some $\beta\geq0$, then $u$ must be affine.

Theorems & Definitions (44)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Proposition 2.1
  • proof
  • Definition 2.2
  • Lemma 2.3: GX25
  • proof
  • Definition 2.4
  • ...and 34 more