arXiv subjects

Shibing Chen

Publications and source records attributed to Shibing Chen.

35 records · Page 2Linked to original sources

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↗

arXiv2023

On the planar Gaussian-Minkowski problem

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.

math.MG↗

arXiv2022

A note on the singular set of the no-sign obstacle problem

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

math.AP↗

arXiv2022

Uniqueness of solutions to the logarithmic Minkowski problem in $\mathbb{R}^3$

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 $C^α$ 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 $C^α$ close to a positive constant.

math.AP↗

arXiv2022

Mean Curvature Rigidity and Non-rigidity Results on Spherical Caps

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).

math.DG↗

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↗

arXiv2020

Regularity of free boundaries in optimal transportation

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

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

$L_p$-Brunn-Minkowski inequality for $p\in (1-\frac{c}{n^{\frac{3}{2}}}, 1)$

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 $C^α$ density, and the second one is for the Logarithmic Minkowski problem when the density of measure is a small $C^α$ even perturbation of the uniform density.

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↗

arXiv2016

Partial $W^{2,p}$ regularity for optimal transport maps

We prove that, in the optimal transportation problem with general costs and positive continuous densities, the potential function is always of class $W^{2,p}_{loc}$ for any $p \geq 1$ outside of a closed singular set of measure zero. We also establish global $W^{2,p}$ estimates when the cost is a small perturbation of the quadratic cost. The latter result is new even when the cost is exactly the quadratic cost.

math.AP↗

arXiv2014

Boundary $\varepsilon$-regularity in optimal transportation

We develop an $\e$-regularity theory at the boundary for a general class of Monge-Ampère type equations arising in optimal transportation. As a corollary we deduce that optimal transport maps between Hölder densities supported on $C^2$ uniformly convex domains are $C^{1,α}$ up to the boundary, provided that the cost function is a sufficient small perturbation of the quadratic cost $-x\cdot y$.

math.AP↗

arXiv2013

On the regularity of the free boundary in the optimal partial transport problem

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 $C_{loc}^{1,α}$ hypersurface. Furthermore, we show that a locally Lipschitz cost implies a rectifiable free boundary and initiate a corresponding regularity theory in the Riemannian setting.

math.AP↗

arXiv2013

Classifying convex compact ancient solutions to the affine curve shortening flow

In this paper we classify convex compact ancient solutions to the affine curve shortening flow: namely, any convex compact ancient solution to the affine curve shortening flow must be a shrinking ellipse. The method combines a rescaling argument inspired by \cite{Wang}, affine invariance of the equation and monotonicity of the affine isoperimetric ratio. We will give two proofs. The essential ideas are related, but the first one uses level set represenation of the evolution. The second proof employs Schauder's estimates, and it also provides a new simple proof for the corresponding classification result to the higher dimensional affine normal flow.

math.DG↗

arXiv2012

Convex solutions to the power-of-mean curvature flow

We prove some estimates for convex ancient solutions (the existence time for the solution starts from $-\infty$) to the power-of-mean curvature flow, when the power is strictly greater than 1/2. As an application, we prove that in two dimension, the blow-down of the entire convex translating solution, namely $u_{h}=\frac{1}{h}u(h^{\frac{1}{1+α}}x),$ locally uniformly converges to $\frac{1}{1+α}|x|^{1+α}$ as $h\rightarrow\infty$. Another application is that for generalized curve shortening flow (convex curve evolving in its normal direction with speed equal to a power of its curvature), if the convex compact ancient solution sweeps $\textbf{R}^{2}$, it it has to be a shrinking circle. Otherwise the solution is defined in a strip region.

math.AP↗

arXiv2012

Remainder terms in the fractional Sobolev inequality

We show that the fractional Sobolev inequality for the embedding $\H \hookrightarrow L^{\frac{2N}{N-s}}(\R^N)$, $s \in (0,N)$ can be sharpened by adding a remainder term proportional to the distance to the set of optimizers. As a corollary, we derive the existence of a remainder term in the weak $L^{\frac{N}{N-s}}$-norm for functions supported in a domain of finite measure. Our results generalize earlier work for the non-fractional case where $s$ is an even integer.

math.AP↗

arXiv2012

A new family of sharp conformally invariant integral inequalities

We prove a one-parameter family of sharp integral inequalities for functions on the $n$-dimensional unit ball. The inequalities are conformally invariant, and the sharp constants are attained for functions that are equivalent to a constant function under conformal transformations. As a limiting case, we obtain an inequality that generalizes Carleman's inequality for harmonic functions in the plane to poly-harmonic functions in higher dimensions.

math.FA↗