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The current work focuses on the Gaussian-Minkowski problem in dimension 2. In particular, we show that if the Gaussian surface area measure is proportional to the spherical Lebesgue measure, then the corresponding convex body has to be a centered disk. As an application, this ``uniqueness'' result is used to prove the existence of smooth small solutions to the Gaussian-Minkowski problem via a degree-theoretic approach.

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.

We consider the optimal transport problem when the {target} consists of two disjoint convex domains. This is a basic model where singularities of optimal transport maps arise. When the domains are $C^2$, uniformly convex and the densities are $C^\alpha$ smooth, we show that the singular set of the optimal transport map is an $(n-1)$-dimensional $C^{2,\alpha}$ regular sub-manifold of $\mathbb{R}^n.$

In this note, we prove the uniqueness of blowups at singular points of the no-sign obstacle problem $\Delta u=\chi_{_{B_1\backslash \{u=|Du|=0\}}}\ \text{in}\ B_1,$ thus give a positive answer to a problem raised in \cite[Notes of Chaper 7, page 149]{PSU12}.

In this paper, we prove the uniqueness of solutions to the logarithmic Minkowski problem in $\mathbb{R}^3$ without symmetry condition, provided the density of the measure is close to $1$ in $C^{\alpha}$ norm. This result also implies the uniqueness of self-similar solutions to the anisotropic Gauss curvature flow in $\mathbb{R}^3$ when the speed function is $C^{\alpha}$ close to a positive constant.

We prove that a hemisphere in the Euclidean space $R^{n+1}$, viewed as the graph of a function, admits no smooth perturbations as graphs with mean curvature $H\ge 1$ whose boundary equator is fixed up to $C^2$. This is an extension of the \emph{Mean Curvature Rigidity} phenomenon discovered by Gromov and Souam on non-compact totally umbilic hypersurfaces in space forms. The proof uses a Tangency Principle. On the other hand, we show that there exist nontrivial smooth perturbations with $H\ge 1$ on a great spherical cap whose boundary is fixed up to $C^2$. Similar results hold true for perturbations decreasing $H$, and for the $r$ mean curvature function $H_r$. This contrast between rigidity and non-rigidity is even true in the 1-dimensional case for circles and for discrete objects (polygons inscribed in a circle)., 21 pages, 12 figures, comments welcome

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-Amp\`ere equation, we obtain the $C^{2,\alpha}$ 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}., Comment: The paper is accepted by Communications on Pure and Applied Mathematics

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}., The paper is accepted by Communications on Pure and Applied Mathematics

In this paper, we obtain some regularities of the free boundary in optimal transportation with the quadratic cost. Our first result is about the $C^{1,\alpha}$ regularity of the free boundary for optimal partial transport between convex domains for densities $f, g$ bounded from below and above. When $f, g \in C^\alpha$, and $\partial\Omega, \partial\Omega^*\in C^{1,1}$ are far apart, by adopting our recent results on boundary regularity of Monge-Amp\`ere equations \cite{CLW1}, our second result shows that the free boundaries are $C^{2,\alpha}$. As an application, in the last we also obtain these regularities of the free boundary in an optimal transport problem with two separate targets., Comment: updated version; submitted

In this paper, we introduce an iteration argument to prove that a convex solution to the Monge-Amp\`ere equation $\mbox{det } D^2 u =f $ in dimension two subject to the natural boundary condition $Du(\Omega) = \Omega^*$ is $C^{2,\alpha}$ smooth up to the boundary. We establish the estimate under the sharp conditions that the inhomogeneous term $f\in C^{\alpha}$ and the domains are convex and $C^{1,\alpha}$ smooth. When $f\in C^0$ (resp. $1/C

In this paper, we establish the global $C^{2,\alpha}$ and $W^{2,p}$ regularity for the Monge-Amp\`ere equation $\det\,D^2u = f$ subject to boundary condition $Du(\Omega) = \Omega^*$, where $\Omega$ and $\Omega^*$ are bounded convex domains in the Euclidean space $\mathbb{R}^n$ with $C^{1,1}$ boundaries, and $f$ is a H\"older continuous function. This boundary value problem arises naturally in optimal transportation and many other applications., Comment: The exposition is polished and references are updated

In this paper, we introduce an iteration argument to prove that a convex solution to the Monge-Amp��re equation $\mbox{det } D^2 u =f $ in dimension two subject to the natural boundary condition $Du(��) = ��^*$ is $C^{2,��}$ smooth up to the boundary. We establish the estimate under the sharp conditions that the inhomogeneous term $f\in C^��$ and the domains are convex and $C^{1,��}$ smooth. When $f\in C^0$ (resp. $1/C

Kolesnikov-Milman [9] established a local $L_p$-Brunn-Minkowski inequality for $p\in(1-c/n^{\frac{3}{2}},1).$ Based on their local uniqueness results for the $L_p$-Minkowski problem, we prove in this paper the (global) $L_p$-Brunn-Minkowski inequality. Two uniqueness results are also obtained: the first one is for the $L_p$-Minkowski problem when $p\in (1-c/n^{\frac{3}{2}}, 1)$ for general measure with even positive $C^{\alpha}$ density, and the second one is for the Logarithmic Minkowski problem when the density of measure is a small $C^{\alpha}$ even perturbation of the uniform density.

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

We develop an $\e$-regularity theory at the boundary for a general class of Monge-Amp\`ere type equations arising in optimal transportation. As a corollary we deduce that optimal transport maps between H\"older densities supported on $C^2$ uniformly convex domains are $C^{1,\alpha}$ up to the boundary, provided that the cost function is a sufficient small perturbation of the quadratic cost $-x\cdot y$., Comment: 24 pages, accepted by Advances in Mathematics

This paper concerns the regularity and geometry of the free boundary in the optimal partial transport problem for general cost functions. More specifically, we prove that a $C^1$ cost implies a locally Lipschitz free boundary. As an application, we address a problem discussed by Caffarelli and McCann \cite{CM} regarding cost functions satisfying the Ma-Trudinger-Wang condition (A3): if the non-negative source density is in some $L^p(\mathbb{R}^n)$ space for $p \in (\frac{n+1}{2},\infty]$ and the positive target density is bounded away from zero, then the free boundary is a semiconvex $C_{loc}^{1,\alpha}$ hypersurface. Furthermore, we show that a locally Lipschitz cost implies a rectifiable free boundary and initiate a corresponding regularity theory in the Riemannian setting., Comment: Submitted