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In this paper we establish an asymptotic expansion near the boundary for solutions to the Dirichlet problem of elliptic equations with singularities near the boundary. This expansion formula shows the singularity profile of solutions at the boundary. We deal with both linear and nonlinear elliptic equations, including fully nonlinear elliptic equations and equations of Monge-Ampere type.

In this paper, we use the Sobolev type inequality in Wang et al. (Moser–Trudinger inequality for the complex Monge–Ampere equation, arXiv:2003.06056v1 (2020)) to establish the uniform estimate and the Holder continuity for solutions to the complex Monge–Ampere equation with the right-hand side in $$L^p$$ for any given $$p>1$$ . Our proof uses various PDE techniques but not the pluripotential theory.

The Christoffel problem is equivalent to the existence of convex solutions to the Laplace equation on the unit sphere $${{\mathbb S}^n}$$. Necessary and sufficient conditions have been found by Firey (1967) and Berg (1969), by using the Green function of the Laplacian on the sphere. Expressing the Christoffel problem as the Laplace equation on the entire space ℝn+1, we observe that the second derivatives of the solution can be given by the fundamental solutions of the Laplace equation. Therefore we find new and simpler necessary and sufficient conditions for the solvability of the Christoffel problem. We also study the Lp extension of the Christoffel problem and provide sufficient conditions for the problem, for the case p ⩾ 2.

In this paper we study a contracting flow of closed, convex hypersurfaces in the Euclidean space $\mathbb R^{n+1}$ with speed $f r^{\alpha} K$, 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 $\alpha \ge n+1$, 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 $q0$, 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., Comment: 34 pages

In this paper, we study a class of contracting flows of closed, convex hypersurfaces in the Euclidean space $${\mathbb {R}}^{n+1}$$ with speed $$r^{\alpha } \sigma _k$$, where $$\sigma _k$$ is the k-th elementary symmetric polynomial of the principal curvatures, $$\alpha \in {\mathbb {R}}^1$$, and r is the distance from the hypersurface to the origin. If $$\alpha \ge k+1$$, we prove that the flow exists for all time, preserves the convexity and converges smoothly after normalisation to a sphere centred at the origin. If $$\alpha

In this paper, we establish convergence rate estimates for convex solutions to the Dirichlet problem of the Monge--Ampere equation $\det D^2u=f$ in $\Omega$, where $f$ is a positive and continuous ...

By employing the Green function, in this paper we provide a new and elementary proof for the interior regularity of solutions to the Monge–Ampere equation. This proof also applies to the complex Monge–Ampere equation and the $k$-Hessian equation.

In the traditional fermentation process of strong-aroma Baijiu, a fermentation pit mud (FPM) provides many genera of microorganisms for fermentation. However, the functional microorganisms that have an important effect on the quality of Baijiu and their changes with the age of fermentation pit (FP) are poorly understood. Herein, the Roche 454 pyrosequencing technique and a phospholipid fatty-acid analysis were employed to reveal the structure and diversity of prokaryotic communities in FPM samples that have been aged for 5, 30, and 100 years. The results revealed an increase in total prokaryotic biomass with an FP age; however, Shannon's diversity index decreased significantly (p < 0.01). These results suggested that a unique microbial community structure evolved with uninterrupted use of the FP. The number of functional microorganisms, which could produce the flavor compounds of strong-aroma Baijiu, increased with the FP age. Among them, Clostridium and Ruminococcaceae are microorganisms that directly produce caproic acid. The increase of their relative abundance in the FPM might have improved the quality of strong-aroma Baijiu. Syntrophomonas, Methanobacterium, and Methanocorpusculum might also be beneficial to caproic acid production. They are not directly involved but provide possible environmental factors for caproic acid production. Overall, our study results indicated that an uninterrupted use of the FP shapes the particular microbial community structure in the FPM. This research provides scientific support for the concept that the aged FP yields a high-quality Baijiu. In the traditional fermentation process of strong-aroma Baijiu, a fermentation pit mud (FPM) provides many genera of microorganisms for fermentation. However, the functional microorganisms that have an important effect on the quality of Baijiu and their changes with the age of fermentation pit (FP) are poorly understood. Herein, the Roche 454 pyrosequencing technique and a phospholipid fatty-acid analysis were employed to reveal the structure and diversity of prokaryotic communities in FPM samples that have been aged for 5, 30, and 100 years. The results revealed an increase in total prokaryotic biomass with an FP age; however, Shannon’s diversity index decreased significantly (p < 0.01). These results suggested that a unique microbial community structure evolved with uninterrupted use of the FP. The number of functional microorganisms, which could produce the flavor compounds of strong-aroma Baijiu, increased with the FP age. Among them, Clostridium and Ruminococcaceae are microorganisms that directly produce caproic acid. The increase of their relative abundance in the FPM might have improved the quality of strong-aroma Baijiu. Syntrophomonas, Methanobacterium, and Methanocorpusculum might also be beneficial to caproic acid production. They are not directly involved but provide possible environmental factors for caproic acid production. Overall, our study results indicated that an uninterrupted use of the FP shapes the particular microbial community structure in the FPM. This research provides scientific support for the concept that the aged FP yields a high-quality Baijiu.

In this paper, we prove the continuity of the monotone optimal mapping of the Monge mass transfer problem in two dimensions under certain conditions on the domains and the mass distributions.

In this paper we study the prescribed centroaffine curvature problem in the Euclidean space R n + 1 . This problem is equivalent to solving a Monge–Ampere equation on the unit sphere. It corresponds to the critical case of the Blaschke–Santalo inequality. By approximation from the subcritical case, and using an obstruction condition and a blow-up analysis, we obtain sufficient conditions for the a priori estimates, and the existence of solutions up to a Lagrange multiplier.

The global C n + 2 , α -regularity for the graph of solutions to a singular Monge–Ampere equation was established in [7] . This equation arises in the study of hyperbolic affine spheres and is singular at the boundary. A compatibility condition for higher regularity was found in [7] . In this paper, we verify the compatibility condition and obtain the global C ∞ -regularity in the even dimensional case.

In this paper, we prove a Moser-Trudinger type inequality for pluri-subharmonic functions vanishing on the boundary. Our proof uses a descent gradient flow for the complex Monge-Ampere functional.

In this paper we prove the strict c-convexity and the C 1 , α regularity for potential functions in optimal transportation under condition (A3w). These results were obtained by Caffarelli [1] , [3] , [4] for the cost c ( x , y ) = | x − y | 2 , by Liu [11] , Loeper [15] , Trudinger and Wang [20] for costs satisfying the condition (A3). For costs satisfying the condition (A3w), the results have also been proved by Figalli, Kim, and McCann [6] , assuming that the initial and target domains are uniformly c-convex, see also [21] ; and by Guillen and Kitagawa [8] , assuming the cost function satisfies A3w in larger domains. In this paper we prove the strict c-convexity and the C 1 , α regularity assuming either the support of source density is compactly contained in a larger domain where the cost function satisfies A3w, or the dimension 2 ≤ n ≤ 4 .

In this paper, we introduce a potential theory for the k-curvature equation, which can also be seen as a PDE approach to curvature measures. We assign a measure to a bounded, upper semicontinuous function which is strictly subharmonic with respect to the k-curvature operator, and establish the weak continuity of the measure.

In this paper we show by example that there is no uniform estimate for the L p -Minkowski problem. As a result we obtain the nonuniqueness of solutions to the problem.

In this paper, we study the regularity and analyticity of solutions to linear elliptic equations with measurable or continuous coefficients. We prove that if the coefficients and inhomogeneous term are Holder-continuous in a direction, then the second-order derivative in this direction of the solution is Holder-continuous, with a different Holder exponent. We also prove that if the coefficients and the inhomogeneous term are analytic in a direction, then the solution is analytic in that direction.

In this paper, we study the regularity of convex solutions to the Dirichlet problem of the homogeneous Monge-Ampere equation $\det D^2 u=0$. We prove that if the domain is a strip region and the boundary functions are locally uniformly convex and $C^{k+2,\alpha}$ smooth, then the solution is $C^{k+2,\alpha}$ smooth up to boundary. By an example, we show the solution may fail to be $C^{2}$ smooth if boundary functions are not locally uniformly convex. Similar results have also been obtained for the Dirichlet problem on bounded convex domains.

In this paper we study local properties of cost and potential functions in optimal transportation. We prove that in a proper normalization process, the cost function is uniformly smooth and converges locally smoothly to a quadratic cost x · y, while the potential function converges to a quadratic function. As applications we obtain the interior W2, p estimates and sharp C1, α estimates for the potentials, which satisfy a Monge–Ampere type equation. The W2, p estimate was previously proved by Caffarelli for the quadratic transport cost and the associated standard Monge–Ampere equation.

The transfer of a contralateral healthy seventh cervical spinal nerve root (cC7) to the recipient nerve in the injured side is considered a reliable and effective procedure for restoration of the physiological functions of an injured hand after brachial plexus root avulsion injury (BPAI). Growing evidence shows that the transhemispheric cortical reorganization is induced after cC7 nerve transfer surgery. However, little is known about the underlying molecular mechanism. Proinflammatory cytokines reportedly play an important role in the neural plasticity. We hypothesize that proinflammatory cytokines are involved in the transhemispheric functional reorganization after cC7 transfer. In the present study, we investigated the level of tumor necrosis factor-α (TNF-α), interleukin-6 (IL-6), and interleukin-1β (IL-1β) in the rat primary motor cortex after cC7 transfer following BPAI by enzyme-linked immunosorbent assay. The results showed that, in the sham group, no statistical significance was observed between the level of TNF-α, IL-6, and IL-1β at each time point after the operation compared with that at day 0, respectively. However, in the unrepaired and repaired groups, the level of TNF-α, IL-6, and IL-1β changed dynamically. The study is the first to provide evidence for the involvement of proinflammatory cytokines in transhemispheric functional reorganization after cC7 transfer following BPAI, which are useful for understanding the underlying mechanism.

In this paper we prove the optimal boundary regularity under natural structural conditions for a large class of nonlinear elliptic equations with singular terms near the boundary. By a careful construction of super- and sub-solutions, we obtain precise growth estimates for solutions at the boundary and reduce the boundary regularity to the interior one by a rescaling argument.

We prove the existence of infinitely many entire convex solutions to the Monge-Amp\`ere equation ${\rm det} D^2 u=f$ in ${\Bbb R}^n$, assuming that the inhomogeneous term $f\ge 0$ and is of polynomial growth at infinity. When $f$ satisfies the doubling condition, we show that solution is of polynomial growth. As an application, we resolve the existence of translating solutions to a class of Gauss curvature flow.

In this paper, we study the existence of multiple solutions to the \(L_{p}\)-Minkowski problem. We prove if \(p 0\), there exists a smooth positive function f on \(\mathbb S^n\) such that the \(L_{p}\)-Minkowski problem admits at least N different smooth solutions. We also construct nonsmooth, positive function f for which the \(L_{p}\)-Minkowski problem has infinitely many \(C^{1,1}\) solutions.

In this paper we study the L p -Minkowski problem for p = − n − 1 , which corresponds to the critical exponent in the Blaschke–Santalo inequality. We first obtain volume estimates for general solutions, then establish a priori estimates for rotationally symmetric solutions by using a Kazdan–Warner type obstruction. Finally we give sufficient conditions for the existence of rotationally symmetric solutions by a blow-up analysis. We also include an existence result for the L p -Minkowski problem which corresponds to the super-critical case of the Blaschke–Santalo inequality.

In this paper, we prove the Holder continuity of the second derivatives D-x(2) for fully nonlinear, uniformly parabolic equations. We assume the inhomogeneous term f is Holder continuous with respect to the spatial variables x and bounded and measurable with respect to the time t. As an application, we obtain higher regularity for curvature flows with volume constraints.

1.1. The k-Yamabe problem. The k-Yamabe problem is a higher order extension of the celebrated Yamabe problem for scalar curvature. It was initially proposed by Viaclovsky [72] and also arose in the study of Q-curvatures in [11]. Viaclovsky found that in a conformal metric, the resultant k-curvature equation can be expressed as an equation similar to the k-Hessian equation, which has been studied by many authors [77]. The k-Yamabe problem can be formulated as follows. Let (Mn, g) be a closed Riemannian manifold of dimension n ≥ 3. It is well-known that there is an orthonormal decomposition of the Riemannian curvature tensor

We assign a measure to an upper semicontinuous function which is subharmonic with respect to the mean curvature operator, so that it agrees with the mean curvature of its graph when the function is smooth. We prove that the measure is weakly continuous with respect to almost everywhere convergence. We also establish a sharp Harnack inequality for the minimal surface equation, which is crucial for our proof of the weak continuity. As an application we prove the existence of weak solutions to the corresponding Dirichlet problem when the inhomogeneous term is a measure.

Given two smooth and positive densities ρ 0 , ρ 1 on two compact convex sets K 0 , K 1 , respectively, we consider the question whether the support of the measure ρ t obtained as the geodesic interpolant of ρ 0 and ρ 1 in the Wasserstein space W 2 ( R d ) is necessarily convex or not. We prove that this is not the case, even when ρ 0 and ρ 1 are uniform measures.

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., 47 pages

The first author was supported by NSFC Grant 10871199. The second author was supported by the Australian Research Council.