Affine maximal hypersurfaces

This is a brief survey of recent works by Neil Trudinger and myself on the Bernstein problem and Plateau problem for affine maximal hypersurfaces

A priori estimates and existence for a class of fully nonlinear elliptic equations in conformal geometry

In this paper we prove the interior gradient and second derivative estimates for a class of fully nonlinear elliptic equations determined by symmetric functions of eigenvalues of the Ricci or Schouten tensors. As an application we prove the existence of solutions to the equations when the manifold is locally conformally flat or the Ricci curvature is positive

Answer to the questions of Yanyan Li and Luc Nguyen in arXiv:1302.1603

In this note we answer the two questions raised by Y.Y Li and L. Nguyen in their note [LN2] below

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 $\kappa$-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

Boundary regularity for the Monge-Ampere and affine maximal surface equations

In this paper, we prove global second derivative estimates for solutions of the Dirichlet problem for the Monge-Ampere equation when the inhomogeneous term is only assumed to be Holder continuous. As a consequence of our approach, we also establish the existence and uniqueness of globally smooth solutions to the second boundary value problem for the affine maximal surface equation and affine mean curvature equation.Comment: 3

Blind separation of rotor vibration signals in high-noise environments

During the operation of the engine rotor, the vibration signal measured by the sensor is the mixed signal of each vibration source, and contains strong noise at the same time. In this paper, a new separation method for mixed vibration signals in strong noise environment(SNR=-5) is proposed. Firstly, the time-delay auto-correlation de-noising method is used to de-noise the mixed signals, and then the common blind separation algorithm (MSNR algorithm is used here) is used to separate the mixed vibration signals, which improves the separation performance. The simulation results verify the validity of the method. The proposed method provides a new idea for health monitoring and fault diagnosis of engine rotor vibration signals.Comment: 8 pages, 4 figure

On Harnack inequalities and singularities of admissible metrics in the Yamabe problem

In this paper we study the local behaviour of admissible metrics in the k-Yamabe problem on compact Riemannian manifolds $(M, g_0)$ of dimension $n\ge 3$. For $n/2 <k<n$, we prove a sharp Harnack inequality for admissible metrics when $(M,g_0)$ is not conformally equivalent to the unit sphere $S^n$ and that the set of all such metrics is compact. When $(M,g_0)$ is the unit sphere we prove there is a unique admissible metric with singularity. As a consequence we prove an existence theorem for equations of Yamabe type, thereby recovering a recent result of Gursky and Viaclovski on the solvability of the $k$-Yamabe problem for $k>n/2$.Comment: 22 page

On the second boundary value problem for Monge-Ampere type equations and optimal transportation

This paper is concerned with the existence of globally smooth solutions for the second boundary value problem for Monge-Ampere equations and the application to regularity of potentials in optimal transportation. The cost functions satisfy a weak form of our condition A3, under which we proved interior regularity in a recent paper with Xi-nan Ma. Consequently they include the quadratic cost function case of Caffarelli and Urbas as well as the various examples in the earlier work. The approach is through the derivation of global estimates for second derivatives of solutions.Comment: In this version, we remove a hypothesis,used previously for the continuity method, through direct construction of a uniformly c-convex function, approximately satisfying the prescribed image conditio

On strict convexity and C1C^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.Comment: 13page

The affine Plateau problem

In this paper, we study a second order variational problem for locally convex hypersurfaces, which is the affine invariant analogue of the classical Plateau problem for minimal surfaces. We prove existence, regularity and uniqueness results for hypersurfaces maximizing affine area under appropriate boundary conditions.Comment: 48 page