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

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research 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

摘要

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

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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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.