AI 中文总结
研究平面亚历山德罗夫蒙日 - 安培方程，构建关联驱动的热带逼近，通过结合多种方法，得到\(N^{-1/2}F_N\)的收敛性、曲率估计等结果，还包括随机点云极限、协方差及对角线相关结论。
AI 中文摘要
我们构建了平面亚历山德罗夫蒙日 - 安培方程的一种由关联驱动的热带逼近。设\(\Omega\subset\mathbb R^2\)为有界开凸域，\(K\Subset\Omega\)，\(F_N = G_{P_N}0_\Omega\)是具有整数斜率、零边界值且角轨迹包含\(N\)点集\(P_N\subset K\)的最小非负凹热带级数。对于经验测度收敛到\(\mu\)的通用一般配置，\(N^{-1/2}F_N\)在\(\overline\Omega\)上一致收敛到\(F_{\mu,\Omega}\)，其中\(F_{\mu,\Omega}\)是\(\mathrm{MA}(F)=\mu\)具有零边界值的唯一连续凹亚历山德罗夫解。在每个紧集\(L\Subset\Omega\)上，我们证明了曲率差异的\(O(N^{-1/2})\)有界 - 利普希茨型估计。不对\(\partial\Omega\)施加正则性或严格凸性假设。对于有理多边形，在一个开稠密全测度轨迹上强一般性成立。热带曲线恰好有\(N\)个有界单元，标记的对偶边形成一棵生成树，且每个紧内部边权重为一。这些有限陈述产生全局弱曲率收敛以及精确等式\(\mathrm{MA}(F_N)(\Omega^\circ)=N - 1+\frac{1}{2}D_{\mathrm{term}}(F_N)\)，其中\(D_{\mathrm{term}}(F_N)=O(\sqrt N)\)。证明结合了热带插值、半线性标记拓扑、欧拉 - 皮克曲率公式、最小系数变形、切向余面积以及在有理多边形穷竭上一致的加权克罗夫顿估计。我们还得到了随机点云的几乎必然极限、连续解的完全仿射协方差以及与配置相关的阿贝尔沙堆对角线。
英文摘要
In this paper we prove that a process we call tropical relaxation gives a convergent method for numerically approximating the Monge--Ampère Dirichlet problem on bounded convex planar domains. We approximate the prescribed probability measure by point clouds and construct the least concave tropical roof with zero boundary values and a corner at each point, using repeated updates of affine planes with integral gradients. For universally generic clouds of $N$ points in a fixed compact subset of the interior, convergence of the empirical measures implies uniform convergence of the roofs, divided by $\sqrt{N}$, to the Aleksandrov solution. The limiting measure may be singular. Euler's and Pick's formulas relate curvature to point counts, with an $O(N^{-1/2})$ discrepancy against compactly supported $C^1$ tests. A removal estimate gives almost-sure uniform convergence for independent samples from any absolutely continuous probability law on the whole domain. For generic clouds in a fixed interior compact set of a rational-slope polygon, suitably rescaled sandpile odometers converge to the same solution along meshes chosen sufficiently fine for each cloud. Their deficit measures converge to its negative Laplacian.
Comments81 pages, 9 figures. Expanded exposition and proofs; includes a removal estimate, sampling throughout the domain, and numerical examples
