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Variants of Bernstein's theorem for variational integrals with linear and nearly linear growth

Michael Bildhauer, Martin Fuchs

Abstract

Using a Caccioppoli-type inequality involving negative exponents for a directional weight we establish variants of Bernstein's theorem for variational integrals with linear and nearly linear growth. We give some mild conditions for entire solutions of the equation \[ {\rm div} \Big[Df(\nabla u)\Big] = 0 \, , \] under which solutions have to be affine functions. Here $f$ is a smooth energy density satisfying $D^2 f>0$ together with a natural growth condition for $D^2 f$.

Variants of Bernstein's theorem for variational integrals with linear and nearly linear growth

Abstract

Using a Caccioppoli-type inequality involving negative exponents for a directional weight we establish variants of Bernstein's theorem for variational integrals with linear and nearly linear growth. We give some mild conditions for entire solutions of the equation \[ {\rm div} \Big[Df(\nabla u)\Big] = 0 \, , \] under which solutions have to be affine functions. Here is a smooth energy density satisfying together with a natural growth condition for .
Paper Structure (4 sections, 4 theorems, 61 equations)

This paper contains 4 sections, 4 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['in theo 1']}
  3. Proof of Theorem \ref{['in theo 2']}
  4. Appendix

Key Result

Theorem 1.1

Let $f$ satisfy in 5 and in 6 and consider an entire solution $u \in C^2(\mathbb{R}^2)$ of equation in 3. Suppose that with numbers $0\leq m < 1$, $K >0$ the solution satisfies or Then $u$ is affine.

Theorems & Definitions (8)

  • Remark 1.1
  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 3.1
  • Remark 4.1