Nirmalendu Chaudhuri, Neil S. Trudinger
In this note we show that $k$-convex functions on $\Bbb R^n$ are twice differentiable almost everywhere for every positive integer $k>n/2$. This generalizes the classical Alexsandrov's theorem for convex functions.
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A note on the honeycomb optimality among periodic convex tilings
本文证明在成本泛函满足Hausdorff收敛下半连续且对Steiner对称化递减时，六边形蜂窝是平面凸周期铺砌中的最优形状。
A note on the a.e. second-order differentiability of rank-one convex functions
本文借鉴全非线性椭圆方程的粘性技术，将凸函数几乎处处二阶可微的亚历山德罗夫定理推广至秩一凸函数，还将原定理简化为一维单调函数的几乎处处可微性。
It has recently been shown that for a Cauchy stress response induced by a strictly rank-one convex hyperelastic energy potential, a homogeneous Cauchy stress tensor field cannot correspond to a non-homogeneous deformation if the deformation gradient has discrete values, i.e. if the deformation is piecewise affine linear and satisfies the Hadamard jump condition. In this note, we expand upon these results and show that they do not hold for arbitrary deformations by explicitly giving an example of a strictly rank-one convex energy and a non-homogeneous deformation such that the induced Cauchy st…
We show that a class of Poincaré-Wirtinger inequalities on bounded convex sets can be obtained by means of the dynamical formulation of Optimal Transport. This is a consequence of a more general result valid for convex sets, possibly unbounded.
We give a new proof of a classical uniqueness theorem of Alexandrov using the weak uniqueness continuation theorem of Bers-Nirenberg. We prove a version of this theorem with the minimal regularity assumption: the spherical hessians of the corresponding convex bodies as Radon measures are nonsingular.
Thanks to the test function of Bian-Guan[2], we successfully obtain a constant rank theorem for partial convex solutions of a class partial differential equations. This is the microscopic version of the macroscopic partial convexity principle in [1], and also is a generalization of the result in [2].
We establish an interior $C^2$ estimate for $k+1$ convex solutions to Dirichlet problems of $k$-Hessian equations. We also use such estimate to obtain a rigidity theorem for $k+1$ convex entire solutions of $k$-Hessian equations in Euclidean space.
A short note on the Schiffer's conjecture for a class of centrally symmetric convex domains in $\mathbb{R}^2$
本文研究了 $\mathbb{R}^2$ 中有界中心对称区域的 Schiffer 猜想，通过定义边界上法线通过中心的点数 $τ$，证明了当 $τ<8$ 时该区域满足 Schiffer 猜想。
A note on the distribution of Neumann eigenvalues of the Laplacian on a Euclidean convex domain
本文研究了欧几里得凸域上拉普拉斯算子Neumann特征值的分布，建立了两个通用不等式。
A note on interior curvature estimates for strictly convex solutions to the equation of prescribed curvature quotient
本文针对椭圆型Weingarten曲率商方程的严格凸解，借助方程算子的凹性不等式与新颖辅助函数，通过初等最大值原理证明其先验内部曲率界，还得到低维特殊拉格朗日曲率方程严格凸解的对应估计。
This expository paper presents elementary proofs of four basic results concerning derivatives of quasi-convex functions. They are combined into a fifth theorem which is simple to apply and adequate in many cases. Along the way we establish the equivalence of the basic lemmas of Jensen and Slodkowski.
A note on hot-spots free subregions of convex domains
本文研究了凸域中Neumann拉普拉斯算子第二个特征函数的临界点分布，证明了热点不可能出现在域中心附近，并明确描述了排除区域。
In this note we provide a new proof of the Tikhonov theorem for the infinite time interval and discuss some of its applications.
In this note we will present an extension of the Krein-Rutman theorem for an abstract nonlinear, compact, positively 1-homogeneous, monotone non-decreasing operators on a Banach space and apply the result to many nonlinear elliptic partial differential operators.
In this paper, we prove the existence of fixed points of mappings satisfying the condition (Da), a kind of generalized nonexpansive mappings, on a weakly compact convex subset in a Banach space satisfying Opial's condition. And we use Sahu([6]) and Thakur([10])'s iterative scheme to establish several convergence theorems in uniformly convex Banach spaces and give an example to show that this scheme converges faster than the scheme in [1]
A note on domain monotonicity for the Neumann eigenvalues of the Laplacian
本文证明了凸区域及其子区域上拉普拉斯算子诺伊曼特征值的区域单调性变体，并基于该方法给出了凸区域诺伊曼特征值的上界。
The main point of this paper is to prove the following useful result: If the almost everywhere 2-jet of a locally quasi-convex function u satisfies a degenerate elliptic constraint F, then u is F-subharmonic, i.e., u is a viscosity F-subsolution. This AE Theorem makes otherwise difficult results transparent. Some instances of this are presented, including two versions of addition, and a comparison theorem.
A note on relations between convexity and concavity of thermodynamic functions
本文证明了不同变量集合下热力学函数凸性、凹性等性质的等价关系，该关系不依赖状态方程，可用于验证Euler方程熵密度的凹性等，为热力学相关数学分析提供支撑。
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