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Xu-Jia Wang

Publications and source records attributed to Xu-Jia Wang.

At least 19 recordsLinked to original sources
arXiv2025

A new minimax principle and application to the p-Laplace equation

We introduce a new minimax principle to prove the existence of multi-peak solutions to the Neumann problem of the $p$-Laplace equation $$ -\varepsilon^p Δ_p u = u^{q-1} - u^{p-1} \ \ \text{in}\ Ω,$$ where $\Om$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary, $1<p<n$ and $p<q< \frac{np}{n-p}$. The minimax principle will be applied to the set of peak functions, which is a subset of the Sobolev space $W^{1,p} (Ω)$. The argument is based on a combination of variational method, topological degree theory, and gradient flow.

math.AP↗
arXiv2025

Limiting shape of the $L_p$-Minkowski problem

Ben Andrews classified the limiting shape for isotropic curvature flow corresponding to the solutions of the $L_p$-Minkowski problem as $p\to-\infty$ in the planar case. In this paper, we use the group-invariant method to study the asymptotic shape of solutions to the $L_p$-Minkowski problem as $p\to-\infty$ in high dimensions. For any regular polytope $T$, we establish the existence of a solution ${Ω^{(p)}}$ to the $L_p$-Minkowski problem that converges to $T$ as $p\to-\infty$, thereby revealing the intricate geometric structure underlying this limiting behavior. We also extend the result to the dual Minkowski problem.

math.AP↗
arXiv2024

Regularity of the $p$-Gauss curvature flow with flat side

We study the regularity of the $p$-Gauss curvature flow with flat side. In our previous paper(arxiv:2403.12292), we obtained the regularity of the interface, namely the boundary of the flat part. In this paper, we study the regularity of the convex hypersurface near the interface.

math.DG↗
arXiv2023

$C^{2,α}$ regularity of free boundaries in optimal transportation

The regularity of the free boundary in optimal transportation is equivalent to that of the potential function along the free boundary. By establishing new geometric estimates of the free boundary and studying the second boundary value problem of the Monge-Ampère equation, we obtain the $C^{2,α}$ regularity of the potential function as well as that of the free boundary, thereby resolve an open problem raised by Caffarelli and McCann in \cite{CM}.

math.AP↗
arXiv2022

The $L_p$-Minkowski problem with super-critical exponents

The $L_p$-Minkowski problem deals with the existence of closed convex hypersurfaces in $\mathbb{R}^{n+1}$ with prescribed $p$-area measures. It extends the classical Minkowski problem and embraces several important geometric and physical applications. The Existence of solutions has been obtained in the sub-critical case $p>-n-1$, but the problem remains widely open in the super-critical case $p<-n-1$. In this paper, we introduce new ideas to solve the problem for all the super-critical exponents. A crucial ingredient in our proof is a topological method based on the calculation of the homology of a topological space of ellipsoids.

math.AP↗
arXiv2021

Regularity of free boundary for the Monge-Ampère obstacle problem

In this paper, we prove the regularity of the free boundary in the Monge-Ampère obstacle problem $\det D^2 v= f(y)χ_{\{v>0\}}. $ By duality, the regularity of the free boundary is equivalent to that of the asymptotic cone of the solution to the singular Monge-Ampère equation $\det D^2 u = 1/f (Du)+δ_0$ at the origin. We first establish an asymptotic estimate for the solution $u$ near the singular point, then use a partial Legendre transform to change the Monge-Ampère equation to a singular, fully nonlinear elliptic equation, and establish the regularity of solutions to the singular elliptic equation.

math.AP↗
arXiv2021

Global regularity for the Monge-Ampère equation with natural boundary condition

In this paper, we establish the global $C^{2,α}$ and $W^{2,p}$ regularity for the Monge-Ampère equation $\det\,D^2u = f$ subject to boundary condition $Du(Ω) = Ω^*$, where $Ω$ and $Ω^*$ are bounded convex domains in the Euclidean space $\mathbb{R}^n$ with $C^{1,1}$ boundaries, and $f$ is a Hölder continuous function. This boundary value problem arises naturally in optimal transportation and many other applications.

math.AP↗
arXiv2019

The Christoffel problem by fundamental solution of the Laplace equation

The Christoffel problem is equivalent to the existence of convex solutions to the Laplace equation on the unit sphere $S^n$. Necessary and sufficient conditions have been found by Firey and Berg, using the Green function of the Laplacian on the sphere. Expressing the Christoffel problem as the Laplace equation on the entire space $R^{n+1}$, we observe that the second derivatives of the solution can be given by the fundamental solutions of the Laplace equations. Therefore we find new and simpler necessary and sufficient conditions for the solvability of the Christoffel problem. We also study the $L_p$ extension of the Christoffel problem and provide sufficient conditions for the problem, for the case $p\geq 2$.

math.AP↗
arXiv2019

Global regularity of optimal mappings in non-convex domains

In this paper, we establish a global regularity result for the optimal transport problem with the quadratic cost, where the domains may not be convex. This result is obtained by a perturbation argument, using a recent global regularity of optimal transportation in convex domains by the authors.

math.AP↗
arXiv2018

Boundary regularity for the second boundary-value problem of Monge-Ampère equations in dimension two

In this paper, we introduce an iteration argument to prove that a convex solution to the Monge-Ampère equation $\mbox{det } D^2 u =f $ in dimension two subject to the natural boundary condition $Du(Ω) = Ω^*$ is $C^{2,α}$ smooth up to the boundary. We establish the estimate under the sharp conditions that the inhomogeneous term $f\in C^α$ and the domains are convex and $C^{1,α}$ smooth. When $f\in C^0$ (resp. $1/C<f<C$ for some positive constant $C$), we also obtain the global $W^{2,p}$ (resp. $W^{2,1+ε}$) regularity.

math.AP↗
arXiv2017

Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems

In this paper we study a contracting flow of closed, convex hypersurfaces in the Euclidean space $\mathbb R^{n+1}$ with speed $f r^α K$, where $K$ is the Gauss curvature, $r$ is the distance from the hypersurface to the origin, and $f$ is a positive and smooth function. If $α\ge n+1$, we prove that the flow exists for all time and converges smoothly after normalisation to a soliton, which is a sphere centred at the origin if $f \equiv 1$. Our argument provides a parabolic proof in the smooth category for the classical Aleksandrov problem, and resolves the dual q-Minkowski problem introduced by Huang, Lutwak, Yang and Zhang (Acta Math. 216 (2016): 325-388), for the case $q<0$. If $α< n+1$, corresponding to the case $q>0$, we also establish the same results for even function $f$ and origin-symmetric initial condition, but for non-symmetric $f$, counterexample is given for the above smooth convergence.

math.AP↗
arXiv2010

Convex solutions to the mean curvature flow

In this paper we study the classification of ancient convex solutions to the mean curvature flow in $\R^{n+1}$. An open problem related to the classification of type II singularities is whether a convex translating solution is $k$-rotationally symmetric for some integer $2\le k\le n$, namely whether its level set is a sphere or cylinder $S^{k-1}\times \R^{n-k}$. In this paper we give an affirmative answer for entire solutions in dimension 2. In high dimensions we prove that there exist non-rotationally symmetric, entire convex translating solutions, but the blow-down in space of any entire convex translating solution is $k$-rotationally symmetric. We also prove that the blow-down in space-time of an ancient convex solution which sweeps the whole space $\R^{n+1}$ is a shrinking sphere or cylinder.

math.DG↗
arXiv2009

Singularity Profile in the Mean Curvature Flow

In this paper we study the geometry of first time singularities of the mean curvature flow. By the curvature pinching estimate of Huisken and Sinestrari, we prove that a mean curvature flow of hypersurfaces in the Euclidean space $\R^{n+1}$ with positive mean curvature is $κ$-noncollapsing, and a blow-up sequence converges locally smoothly along a subsequence to a smooth, convex blow-up solution. As a consequence we obtain a local Harnack inequality for the mean convex flow.

math.DG↗
arXiv2007

On strict convexity and $C^1$ regularity of potential functions in optimal transportation

This note concerns the relationship between conditions on cost functions and domains and the convexity properties of potentials in optimal transportation and the continuity of the associated optimal mappings. In particular, we prove that if the cost function satisfies the condition (A3), introduced in our previous work with Xinan Ma, the densities and their reciprocals are bounded and the target domain is convex with respect to the cost function, then the potential is continuously differentiable and its dual potential strictly concave with respect to the cost function. Our result extends, by different and more direct proof, similar results of Loeper proved by approximation from our earlier work on global regularity.

math.AP↗