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

IF 1.1 4区 数学 Q1 MATHEMATICS Ricerche di Matematica Pub Date : 2024-04-20 DOI:10.1007/s11587-024-00857-6
Michael Bildhauer, Martin Fuchs
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引用次数: 0

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

$$\begin{aligned} {\text {div}} \Big [Df(\nabla u)\Big ] = 0 \,, \end{aligned}$$

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 \(D^2 f\).

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具有线性和近似线性增长的变分积分的伯恩斯坦定理变式
利用涉及方向权负指数的 Caccioppoli- 型不等式,我们为线性和近似线性增长的变分积分建立了伯恩斯坦定理的变体。我们给出了方程 $$\begin{aligned} {\text {div} 的全解的一些温和条件。}\Big [Df(\nabla u)\Big ] = 0 \,,\end{aligned}$$在此条件下,解必须是仿射函数。这里,f是满足\(D^2 f>0\) 以及\(D^2 f\) 自然增长条件的平滑能量密度。
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来源期刊
Ricerche di Matematica
Ricerche di Matematica Mathematics-Applied Mathematics
CiteScore
3.00
自引率
8.30%
发文量
61
期刊介绍: “Ricerche di Matematica” publishes high-quality research articles in any field of pure and applied mathematics. Articles must be original and written in English. Details about article submission can be found online.
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