线性增长径向结构变分积分的伯恩斯坦定理
Bernstein's theorem for variational integrals of linear growth and radial structure
AI总结：
该研究针对线性增长径向结构变分积分的欧拉-拉格朗日方程整体解，证明积分∫₀^∞ t g''(t)dt < ∞时解必为仿射函数，弱化条件仍有部分结果，拓展了伯恩斯坦定理的适用范围。
AI中文摘要：
我们考虑从ℝ²到ℝ的整体解u，其对应于变分积分∫_Ω g(|∇u|)dx的欧拉-拉格朗日方程，其中密度g: [0,∞)→ℝ严格凸且呈线性增长。我们证明积分∫₀^∞ t g''(t)dt < ∞的条件蕴含伯恩斯坦性质，即u必为仿射函数；若弱化g的该条件，仍可得到部分伯恩斯坦结果。
英文摘要：
We consider entire solutions $u: \mathbb{R}^2 \rightarrow \mathbb{R}$ of the Euler-Lagrange equation associated to the variational integral $\int_Ω g(|\nabla u|)\,dx$ with a strictly convex density $g: [0,\infty)\rightarrow \mathbb{R}$ being of linear growth. We show that the condition $\int_{0}^{\infty} t\,g''(t)\,dt < \infty$ implies the Bernstein property, which means that $u$ must be an affine function. If this condition on g is weakened, we still have some partial Bernstein results.
