带有Lipschitz右端项的标量曲率方程的半凸解的内部曲率估计

中文摘要 AI 辅助

本文中，设 $u\in C^4(B_{10})$ 满足 $D^2u\geq -KI$，定义2-容许图 $M=\{(x,u(x)):x\in B_{10}\}\subset\mathbb{R}^{n+1}$，满足 $\sigma_2(\kappa[u])=f(x)$。我们证明了一个依赖于右端项Lipschitz范数的内部曲率估计。证明结合了 $b=\log(H+J_0)$ 的移位Jacobi不等式、使牛顿张量一致椭圆的平行超曲面变换，局部有界性估计随后将点态界归约为加权 $L^1$ 估计，该估计通过Jacobi能量不等式和分部积分完成。

英文摘要

In this paper, let $u\in C^4(B_{10})$ with $D^2u\geq -KI$ define a $2$-admissible graph $M=\{(x,u(x)):x\in B_{10}\}\subset\R^{n+1}$ satisfying \[ σ_2(κ[u])=f(x). \] We prove an interior curvature estimate depending on the Lipschitz norm of the right-hand sides. The proof combines a shifted Jacobi inequality for \(b=\log(H+J_0)\), a parallel hypersurface transformation that makes the Newton tensor uniformly elliptic and local boundedness estimate then reduces the pointwise bound to a weighted $L^1$ estimate, which is completed using the Jacobi energy inequality and integration by parts.

发表机构

• School of Mathematical Sciences, Chongqing Normal University（重庆师范大学数学科学学院）

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