AI 中文总结
针对带有Lipschitz右端项的σ₂/σ₁方程的半凸解，证明了依赖于右端项Lipschitz范数的内部Hessian估计，应用于二维凸粘性解的正则性且确定右端项Lipschitz正则性要求最优。
AI 中文摘要
设n≥2，u是方程σ₂(D²u)/σ₁(D²u)=f(x)的光滑2-凸且半凸解。我们证明了依赖于f的Lipschitz范数的内部Hessian估计。证明结合了Chen-Jian-Zhou的积分方法，以及将商方程代数约化为σ₂结构的步骤。主要创新点是一个移位代数不等式，它对log(Δu+a)产生了散度形式的移位迹雅可比不等式。我们处理等价方程σ₂(D²u)=fΔu的线性化算子G=(Δu-f)I-D²u。几乎散度自由的恒等式∂iGij=-fj使我们能够仅通过分部积分，用f的Lipschitz范数控制Δf项。勒让德-刘易斯变换将得到的退化散度形式方程转化为一致椭圆型方程。随后通过均值不等式结合加权能量论证得到估计。作为应用，在二维情形下，我们得到右端项为正Lipschitz的凸粘性解的内部C²正则性。此外，我们的反例表明右端项所需的Lipschitz正则性是最优的。
英文摘要
Let $n\ge2$ and let $u$ be a smooth 2-convex and semi-convex solution of \[ \frac{σ_2(D^2u)}{σ_1(D^2u)}=f(x). \] We prove an interior Hessian estimate depending on the Lipschitz norm of $f$. The proof combines the integral approach of Chen--Jian--Zhou with the algebraic reduction of the quotient equation to a $σ_2$ structure. The main new point is a shifted algebraic inequality that yields a shifted trace Jacobi inequality in divergence form for $\log(Δu+a)$. We work with the linearized operator $G=(Δu-f)I-D^2u$ of the equivalent equation $σ_2(D^2u)=f Δu$. The almost divergence-free identity \(\partial_iG_{ij}=-f_j\) enables us to control the \(Δf\) term by integration by parts solely in terms of the Lipschitz norm of \(f\). A Legendre--Lewy transformation converts the resulting degenerate divergence-form equation into a uniformly elliptic one. The estimate then follows from a mean-value inequality together with a weighted energy argument. As an application, in dimension two we obtain interior $C^2$ regularity for convex viscosity solutions with positive Lipschitz right-hand side. Moreover, our counterexamples show that the Lipschitz regularity required of the right-hand side is optimal.
