In this paper we prove that a process we call tropical relaxation gives a convergent method for numerically approximating the Monge--Amp\`ere 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.
We extend the quantitative Dirichlet theorem of Gui, Hu, and Li \cite{GuiHuLi2026} to Robin torsional rigidity. For every fixed $\beta>0$, we prove that the regular $N$-gon uniquely maximizes Robin torsional rigidity among tangential $N$-gons of prescribed area. Our explicit perimeter deficit bound contains the Dirichl...
Chang-Feng Gui, Ye-Yao Hu, Qin-Feng Li et al.· 0 citations
We determine the Leray--Schauder degree of the singular Dirichlet $N$-mean field equation on a smooth bounded domain in $\mathbb{R}^N$, $N\geq2$, for every positive nonresonant parameter and singular exponents $-1<\alpha_i\leq0$. The formula depends only on the Euler characteristic of the domain and the weights $(1+\al...
We prove an interior $C^{1,\alpha}$ estimate for solutions of the homogeneous linearized Monge--Amp\`ere equation in dimension two under the assumption \[ 0<\lambda\leq \det D^2\varphi\leq\Lambda<+\infty. \] No continuity assumption on the Monge--Amp\`ere density is required. Our result is an affine-invariant analogue...
We study the uniqueness and boundary behavior of nonzero convex Aleksandrov solutions to $\det D^2 u=M|u|^p\nu$ with zero boundary values on bounded convex domains in $\mathbb{R}^n (n \geq 2)$. For $0<p<n$, we prove the uniqueness of nonzero convex solutions in the finite-energy class when $\nu$ is a locally finite Bor...
We solve the asymptotic Plateau problem in hyperbolic space for the curvature given by the normalized geometric mean of the $p$-fold sums of the principal curvatures. For every $n\ge3$, $2\le p\le n-1$, and $\sigma\in(0,1)$, each bounded domain $\Omega\subset\mathbb{R}^n$ with smooth mean-convex boundary admits a uniqu...
We establish interior and global H\"older estimates for linearized Monge-Amp\`ere equations in divergence form in all dimensions $n\geq 3$, when the Hessian determinant of the convex Monge-Amp\`ere potential is bounded above and below by positive constants and the vector field on the right-hand side is bounded. The est...
Guo-Qing Cui, Chong Gu, N. Q. Le et al.· 3 citations
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