AI 中文总结
该研究针对二维齐次线性化Monge–Ampère方程，在Monge–Ampère密度无连续性假设的条件下，利用部分Legendre变换等方法证明了内部$C^{1,α}$估计，并将其应用于得到具至多线性增长的整体解的Liouville定理。
AI 中文摘要
我们在满足$0<λ≤\text{det}D^2φ≤Λ<+∞$的条件下，证明了二维齐次线性化Monge–Ampère方程解的内部$C^{1,α}$估计，无需对Monge–Ampère密度施加连续性假设。该结果是经典Morrey–Nirenberg二维$C^{1,α}$估计的仿射不变类似物。证明核心是部分Legendre变换，变换后解的一阶导数为一致椭圆非散度型方程伴随解的商；Bauman的Harnack不等式给出该商的Hölder控制，而部分Legendre变换的雅可比恒等式与Caccioppoli估计给出其局部有界性。作为应用，我们证明了具有至多线性增长的整体解的Liouville定理。
英文摘要
We prove an interior $C^{1,α}$ estimate for solutions of the homogeneous linearized Monge--Ampère equation in dimension two under the assumption \[ 0<λ\leq \det D^2φ\leqΛ<+\infty. \] No continuity assumption on the Monge--Ampère density is required. Our result is an affine-invariant analogue of the classical Morrey--Nirenberg $C^{1,α}$ estimate in two dimensions. The core of the proof is the partial Legendre transform. After the transform, the first derivatives of the solution are quotients of adjoint solutions for a uniformly elliptic non-divergence form equation. Bauman's Harnack inequality gives the Hölder control of the quotient, while the Jacobian identity of the partial Legendre transform and a Caccioppoli estimate give its local boundedness. As an application, we prove a Liouville theorem for entire solutions with at most linear growth.
Comments20 pages; comments and suggestions are welcome
