arXiv subjects

Xianduo Wang

Publications and source records attributed to Xianduo Wang.

6 recordsLinked to original sources

arXiv2026

Global $W^{2,p}$ Regularity in Optimal Transport

In this paper we establish global $W^{2,p}$ estimates for the convex potentials of quadratic optimal transport between bounded convex domains with continuous positive densities. All the assumptions are optimal. The main new ideas include a blow-up analysis that allows for lower-dimensional collapse of the limiting source measure and reduces the limiting problem to a transport problem on its affine hull, and a good-bad scale decomposition in which rigidity controls the good scales while a counting argument shows that the proportion of bad scales tends to zero.

math.AP↗

arXiv2026

On the monotonicity of affine quermassintegrals

Lutwak's affine quermassintegral theory is a foundational component of modern affine Brunn--Minkowski theory. Developed in the 1980s, it provides affine analogues of the classical quermassintegrals and has led to a rich family of sharp affine isoperimetric inequalities. A central question in this program, going back to Lutwak's 1988 work, is an Alexandrov--Fenchel-type monotonicity principle for the normalized $L^{-n}$-moment quermassintegrals $I_{k,-n}$. In one form, this principle predicts that \[ I_{m,-n}(K)^{1/m}\ge I_{k,-n}(K)^{1/k}, \qquad 1\le m (m+2)(k+2)-2$, there exists an origin-symmetric $C^2_+$ convex body $K\subset\mathbb R^n$ such that \[ I_{m,-n}(K)^{1/m} < I_{k,-n}(K)^{1/k}. \] The example is obtained from the Euclidean ball by an arbitrarily small degree-four spherical harmonic perturbation. On the positive side, we prove that the endpoint chain is true in dimension three: for every convex body $K\subset\mathbb R^3$, \[ I_{1,-3}(K)\ge I_{2,-3}(K)^{1/2}\ge I_{3,-3}(K)^{1/3}=1. \] The equality cases in both non-trivial inequalities are exactly ellipsoids, up to translation and nonsingular affine transformations.

math.AP↗

arXiv2026

Uniqueness of Blow-ups for the Superconductivity Free Boundary Problem

We study the free-boundary equation \[ \Delta u=\chi_{\{|\nabla u|>0\}} \] near the origin. We prove that, at a singular point of $\partial\{|\nabla u|>0\}$, the quadratic blow-up is unique. As noted in \cite[Notes to Chapter 7]{PSU2012}, little is known about the singular set for this problem. The usual Weiss--Monneau monotonicity argument does not seem to apply directly, because the inactive set is determined by the vanishing of $\nabla u$, rather than by a sign condition on $u$. The proof follows the quadratic part of the rescalings. Projecting onto the trace-free quadratic harmonics yields a finite-dimensional differential equation for the quadratic coefficient. Together with a Lyapunov identity and estimates on dyadic annuli, this implies convergence of the quadratic coefficient, and hence uniqueness of the blow-up.

math.AP↗

arXiv2023

Global regularity in the Monge-Ampère obstacle problem

In this paper, we establish the global $W^{2,p}$ estimate for the Monge-Ampère obstacle problem: $(Du)_{\sharp}fχ{_{\{u>\frac{1}{2}|x|^2\}}}=g$, where $f$ and $g$ are positive continuous functions supported in disjoint bounded $C^2$ uniformly convex domains $\overlineΩ$ and $\overline{Ω^*}$, respectively. Furthermore, we assume that $\int_Ωf\geq \int_{Ω^*}g$. The main result shows that $Du:\overline U\rightarrow\overline{Ω^*}$, where $ U=\{u>\frac{1}{2}|x|^2\}$, is a $W^{1, p}$ diffeomorphism for any $p\in(1,\infty)$. Previously, it was only known to be a continuous homeomorphism according to Caffarelli and McCann \cite{CM}. It is worth noting that our result is sharp, as we can construct examples showing that even with the additional assumption of smooth densities, the optimal map $Du$ is not Lipschitz. This obstacle problem arises naturally in optimal partial transportation.

math.AP↗

arXiv2023

Global analyticity of affine hyperbolic spheres in the even dimensional space

A number of geometric problems, including affine hyperbolic spheres, Hilbert metrics and Minkowski type problems, are reduced to a singular Monge-Ampère equation which can be written locally as a class of Monge-Ampère equations with singularity at boundary. We estimate the boundary derivatives of all orders for the solutions to the class of singular equations as possible as optimal and prove that the solution to the equation is globally analytic if the dimension of the space is even. As a corollary, we obtain the global analyticity of affine hyperbolic spheres in the even dimensional space.

math.AP↗

arXiv2023

Sharp boundary regularity for some degenerate-singular Monge-Ampère Equations on k-convex domain

We introduce the concept of k-strictly convexity to describe the accurate convexity of convex domains some directions of which boundary may be flat. Basing this accurate convexity, we construct sub-solutions the Dirichlet problem for some degenerate-singular Monge-Ampère type equations and prove the sharp boundary estimates for convex viscosity solutions of the problem. As a result, we obtain the optimal global Hölder regularity of the convex viscosity solutions.

math.AP↗