非齐次调和发展方程弱解的正则性和紧性

在满足抛物单调不等式和能量不等式的条件下，证明了一类映入球面的非齐次调和发展方程弱解除了一个ｎ－维Ｈａｕｓｄｏｒｆ测度（对抛物度量）为零的闭集外是光滑的此外，还证明了弱解序列的紧性     

算子方程AX-XA=C的可解性

本文在一般Ｂａｎａｃｈ空间研究带无界算子Ａ的算子方程ＡＸ－ＸＡ＝Ｃ的可解性,利用单参数积分双半群方法,通过在算子代数Ｌ（Ｅ）上考虑间断问题弱解,证明了当算子Ａ在Ｌ（Ｅ）上诱导的算子Ａ是弱积分双半群的母元时,只要Ｃ满足一定条件,上述算子方程可解．     

双特征的Beltrami方程和拟正则映射

设Ω为Ｒｎ上的一个区域，ｎ２，对于具有双特征矩阵Ｇ（ｘ），Ｈ（ｘ）∈Ｃｋ，α（Ω，Ｒｎ），ｋ１，０＜α＜１的Ｂｅｌｔｒａｍｉ方程（１．４），建立了在Ｓｏｂｏｌｅｖ空间Ｗ１，ｎｌｏｃ（Ω，Ｒｎ）上广义解的正则性：ｆ（ｘ）∈Ｃｋ＋１，δｌｏｃ（Ω），对某一δ：０＜δ＜１．     

关于型Ⅲ对称空间的热核构造

引进了第Ⅲ类典型城上的内切超圆坐标;计算出了矩阵极坐标下相应于不变度量的体积元素;利用积分变换构造出了不变度量的Ｌａｐｌａｃｅ－Ｂｅｌｔｒａｍｉ算子的热核．     

偶数维空间上广义Beltrami方程组及高维空间的Beltrami方程组

本文在ｎ＝２ｌ维偶数空间上得到广义Ｂｅｌｔｒａｍｉ方程组Ｄｔｆ（ｘ）Ｈ（ｘ）Ｄｆ（ｘ）＝Ｊ（ｘ，ｆ）２／ｎＧ（ｘ）的伸张公式，并在Ｈ（ｘ）为对角阵的条件下，给出广义Ｂｅｌｔｒａｍｉ方程组的正则性定理，Ｃａｃｉｏｐｏｌｉ型不等式和可去性定理．此外，还将所有维数的Ｂｅｌｔｒａｍｉ方程组Ｄｔｆ（ｘ）Ｄｆ（ｘ）＝Ｊ（ｘ，ｆ）２／ｎＧ（ｘ）化为一个“Ｂｅｌｔｒａｍｉ方程”     

算子方程AX+X~*B=C的解

利用算子的广义逆及相关投影,研究了一类算子方程的可解性,得到了方程可解的若干条件,并给出了解的一般表示.最后利用算子的矩阵表示,得到了此类算子方程可解的又一充要条件,进而丰富了这方面的研究.     

微分形式中的非齐次Dirac-调和方程解的若干不等式（英文）

本文研究了与微分形式中一类非齐次的Dirac-调和方程解相关的不等式问题.利用非齐次的Dirac-调和方程的条件和Dirac-调和算子D的运算法则,获得了Poincare不等式,Caccioppoli不等式和弱逆H?lder不等式.作为相关不等式的应用,证明了Poincare不等式赋特殊权和在L~s(μ)平均域上的形式.本文的研究将齐次Dirac-调和方程解的相关不等式推广到了对应该方程非齐次的情形.     

(G,H)-拟正则映照和B-调和方程

本文将双特征拟正则映照化为变化问题,得到其Euler-Lasranse方程,利用它得到B-调和方程的先验估计、拟正则映照的Caccippoli型不等式、正则性和可去性结果．     

Couette-Taylor流的谱Galerkin逼近

利用谱方法对轴对称的旋转圆柱问的Couette—Taulor流进行数值模拟．首先给出Navier-Stokes方程流函数形式，利用Couette流把边界条件齐次化．其次给出Stokes算子的特征函数的解析表达式，证明其正交性，并对特征值进行估计．最后利用Stokes算子的特征函数作为逼近子空间的基函数，给出谱Galerkin逼近方程的表达式．证明了Navier-Stokes方程非奇异解的谱Galerkin逼近的存在性、唯一性和收敛性，给出了解谱Galerkin逼近的误差估计，并展示了数值计算结果。     

Banach空间非线性混合单调Hammerstein型积分方程的迭代解

在Ｂａｎａｃｈ空间中，建立了“非线性混合单调型算子”不动点和最大最小耦合不动点的存在与迭代逼近定理，并应用到序Ｂａｎａｃｈ空间非线性混合单调Ｈａｍｍｅｒｓｔｅｉｎ型积分方程．     

Uniqueness of Solutions of the Derrida-Lebowitz-Speer-Spohn Equation and Quantum Drift-Diffusion Models

We prove uniqueness of solutions of the DLSS equation in a class of sufficiently regular functions. The global weak solutions of the DLSS equation constructed by Jüngel and Matthes belong to this class of uniqueness. We also show uniqueness of solutions for the quantum drift-diffusion equation, which contains additional drift and second-order diffusion terms. The results hold in case of periodic or Dirichlet-Neumann boundary conditions. Our proof is based on a monotonicity property of the DLSS operator and sophisticated approximation arguments; we derive a PDE satisfied by the pointwise square root of the solution, which enables us to exploit the monotonicity property of the operator.     

Hyperbolic Eisenstein series on n-dimensional hyperbolic spaces

We define a generalized hyperbolic Eisenstein series for a pair of a hyperbolic manifold of finite volume and its submanifold. We prove the convergence, the differential equation and the precise spectral expansion associated to the Laplace–Beltrami operator. We also derive the analytic continuation with the location of the possible poles and their residues from the spectral expansion.     

Lagrange multiplier and singular limit of double obstacle problems for the Allen–Cahn equation with constraint

We study the properties of the Lagrange multiplier for an Allen–Cahn equation with a double obstacle potential. Here, the dynamic boundary condition, including the Laplace–Beltrami operator on the boundary, is investigated. We then establish the singular limit of our system and clarify the limit of the solution and the Lagrange multiplier of our problem. We present remarks on a trace problem as well as on the Neumann boundary condition. Moreover, we describe a numerical experiment for a problem with Neumann boundary condition using the Lagrange multiplier. Copyright © 2016 John Wiley & Sons, Ltd.     

The Poisson bracket compatible with the classical reflection equation algebra

We introduce a family of compatible Poisson brackets on the space of 2 × 2 polynomial matrices, which contains the reflection equation algebra bracket. Then we use it to derive a multi-Hamiltonian structure for a set of integrable systems that includes the XXX Heisenberg magnet with boundary conditions, the generalized Toda lattices and the Kowalevski top.      

Non-stretch mappings for a sharp estimate of the Beurling–Ahlfors operator

In this paper we identify certain classes of non-stretch mappings that enjoy a sharp estimate of the Beurling–Ahlfors operator. We first make use of a property of subharmonic functions to prove that the Bañuelos–Wang conjecture and the Iwaniec conjecture are true for a class of mappings that satisfy a quasilinear conjugate Beltrami equation. By utilizing the principal solutions of Beltrami equations, we further explicitly construct some classes of non-stretch mappings for which the Bañuelos–Wang conjecture and the Iwaniec conjecture are true.     

Liouville-type results for stationary maps of a class of functional related to pullback metrics

We study a generalized functional related to the pullback metrics (3). We derive the first variation formula which yield stationary maps. We introduce the stress–energy tensor which is naturally linked to conservation law and yield the monotonicity formula via the coarea formula and the comparison theorem in Riemannian geometry. A version of this monotonicity inequalities enables us to derive some Liouville type results. Also, we investigate the constant Dirichlet boundary value problems and the generalized Chern type results for tension field equation with respect to this functional.     

Convexity with respect to a differential equation

The concepts of convexity of a set, convexity of a function and monotonicity of an operator with respect to a second-order ordinary differential equation are introduced in this paper. Several well-known properties of usual convexity are derived in this context, in particular, a characterization of convexity of function and monotonicity of an operator. A sufficient optimality condition for a optimization problem is obtained as an application. A number of examples of convex sets, convex functions and monotone operators with respect to a differential equation are presented.     

On a fractional integral equation of periodic functions involving Weyl-Riesz operator in Banach algebras

In this paper, we study the existence of periodic solutions for a nonlinear integral equation of periodic functions involving Weyl-Riesz fractional integral operator under the mixed generalized Lipschitz, Carathéodory and monotonicity conditions. The fixed point theorems due to Dhage are the main tool in carrying out our proofs.     

Use of the hyperbolic differential operator to find a scalar statistic which has constant regression on the mean of a sample of Wishart matrices

Scalar polynomial statistics are found which have constant regression on the mean of a sample of Wishart matrices. The method used is to differentiate the characteristic function associated with the Wishart distribution, thus expressing the constant regression condition as a differential equation which is satisfied by the Wishart characteristic function. In this respect, use is made of the hyperbolic differential operator.     

Generalized quasilinearization method for reaction-diffusion equations under nonlinear and nonlocal flux conditions

In this paper we consider an initial boundary value problem for a reaction-diffusion equation under nonlinear and nonlocal Robin type boundary condition. Assuming the existence of an ordered pair of upper and lower solutions we establish a generalized quasilinearization method for the problem under consideration whose characteristic feature consists in the construction of monotone sequences converging to the unique solution within the interval of upper and lower solutions, and whose convergence rate is quadratic. Thus this method provides an efficient iteration technique that produces not only improved approximations due to the monotonicity of its iterates, but yields also a measure of the convergence rate.