A STUDY ON GRADIENT BLOW UP FOR VISCOSITY SOLUTIONS OF FULLY NONLINEAR,UNIFORMLY ELLIPTIC EQUATIONS

We investigate sharp conditions for boundary and interior gradient estimates of continuous viscosity solutions to fully nonlinear, uniformly elliptic equations under Dirichlet boundary conditions. When these conditions are violated, there can be blow up of the gradient in the interior or on the boundary of the domain. In particular we de- rive sharp results on local and global Lipschitz continuity of continuous viscosity solutions under more general growth conditions than before. Lipschitz regularity near the boundary allows us to predict when the Dirichlet condition is satisfied in a classical and not just in a viscosity sense, where detachment can occur. Another consequence is this： if interior gra- dient blow up occurs, Perron-type solutions can in general become discontinuous, so that the Dirichlet problem can become unsolvable in the class of continuous viscosity solutions.

SINGULAR LIMITS FOR INHOMOGENEOUS EQUATIONS OF ELASTICITY

Based on the framework introduced in [4] or [5], the singular limits of stiff relaxation and dominant diffusion for the Cauchy problem of inhomogeneous equations of elasticity is studied. We are able to reach equilibrium even though the nonlinear stress term is not strictly increasing.

The Unitary Group in Its Strong Topology

The goal of this paper is to confirm that the unitary group U(H) on an infinite dimensional complex Hilbert space is a topological group in its strong topology, and to emphasize the importance of this property for applications in topology. In addition, it is shown that U(H) in its strong topology is metrizable and contractible if H is separable. As an application Hilbert bundles are classified by homotopy.

Cosmology with Bounce by Flat Space-Time Theory of Gravitation and a New Interpretation

General relativity predicts a singularity in the beginning of the universe being called big bang. Recent developments in loop quantum cosmology avoid the singularity and the big bang is replaced by a big bounce. A classical theory of gravitation in flat space-time also avoids the singularity under natural conditions on the density parameters. The universe contracts to a positive minimum and then it expands during all times. It is not symmetric with regard to its minimum implying a finite age measured with proper time of the universe. The space of the universe is flat and the total energy is conserved. Under the assumption that the sum of the density parameters is a little bit bigger than one the universe is very hot in early times. Later on, the cosmological model agrees with the one of general relativity. A new interpretation of a non-expanding universe may be given by virtue of flat space-time theory of gravitation.

Minimal Legendrian surfaces in the tangent sphere bundle of R^(3)

In this paper,we study minimal Legendrian surfacesΣimmersed in the tangent sphere bundle T_(1)R^(3).We classify(1)totally geodesic Legendrian surfaces,(2)closed minimal Legendrian surfaces of genus smaller than or equal to 1 and complete minimal Legendrian surfaces with the non-negative Gauss curvature,and(3)complete stable minimal Legendrian surfaces.

相对Alexandrov-Fenchel型不等式

Alexandrov-Fenchel不等式是凸几何中的重要不等式之一,它的特殊形式可以看作是高阶均质积分之间的等周型不等式.近年来,带自由边界或毛细边界超曲面获得了很多关注.本文从微分几何角度研究球体内或半空间中带有自由边界或毛细边界凸超曲面上的一类相对Alexandrov-Fenchel型不等式,并综述它的最新进展.首先,介绍单位球体或半空间中毛细边界超曲面的高阶Minkowski型积分公式.从第一变分角度给出单位球体或半空间中毛细边界超曲面的均质积分的定义.然后,本文利用Minkowski型公式构造局部限制型的超曲面曲率流,通过分析流的长时间存在性和收敛性,证明单位球体和半空间中的一类相对Alexandrov-Fenchel型不等式.

Interpolation of Morrey-Campanato and related smoothness spaces

We study the interpolation of Morrey-Campanato spaces and some smoothness spaces based on Morrey spaces, e. g., Besov-type and Triebel-Lizorkin-type spaces. Various interpolation methods, including the complex method, the ±-method and the Peetre-Gagliardo method, are studied in such a framework. Special emphasis is given to the quasi-Banach case and to the interpolation property.

NOTE ON GROUND STATES OF NONLINEAR SCHRODINGER SYSTEMS

We give sufficient and necessary conditions for the existence and nonexistence of positive ground state solutions of a class of coupled nonlinear Schroedinger equations.

The Characterization of PSL(2,q) by Its Element Orders

The well-known theorem of Dickson provides information on the element ordersof the groups PSL(2, q). In search of a converse to this, Refs. [1] and [2] proved thatif π_e.(G)=π_e(PSL(2, q)) and if q=2~m or q=3~m (m≥2, q≠9), then G is isomorphic toPSL(2, q), where π_e(G) denotes the set of all orders of elements in G.

On the sum of two integral squares in the imaginary quadratic field Q(-2p)^(1/2)

We determine the sum of two integral squares over imaginary quadratic fields Q(√-2p),where p≡1 mod 8 is a prime satisfying 2p=r2+s2with r,s≡±3 mod 8.

The Dichotomy Between Traces on d-sets Г in R^n and the Density of D(R^n\Г) in Function Spaces

A space Apq^s （R^n） with A ： B or A = F and s ∈R, 0 〈 p, q 〈 ∞ either has a trace in Lp（Г）, where Г is a compact d-set in R^n with 0 〈 d 〈 n, or D（R^n／Г） is dense in it. Related dichotomy numbers are introduced and calculated.

Complex Interpolation of Weighted Besov and Lizorkin–Triebel Spaces

We study complex interpolation of weighted Besov and Lizorkin–Triebel spaces.The used weights w0,w1 are local Muckenhoupt weights in the sense of Rychkov.As a first step we calculate the Calder′on products of associated sequence spaces.Finally,as a corollary of these investigations,we obtain results on complex interpolation of radial subspaces of Besov and Lizorkin–Triebel spaces on Rd.

Lagrangian surfaces with Legendrian boundary

In this paper,we first introduce a boundary problem for Lagrangian submanifolds,analogous to the problem for free boundary hypersurfaces and capillary hypersurfaces.Then we present several interesting examples of Lagrangian submanifolds satisfying this boundary condition and we prove a Lagrangian version of the Nitsche(or Hopf)type theorem.Some problems are proposed at the end of this paper.

Phase Separation in PCDTBT:PCBM Blends:from Flory-Huggins Interaction Parameters to Ternary Phase Diagrams

The substantial increase in the efficiency of organic solar cells achieved in recent years would not have been possible without work on the synthesis of new materials and understanding the relationship between the morphology and performance of organic photovoltaic devices.The structure of solvent-cast active layers is a result of phase separation in mixtures of donor and acceptor components.To a large extent,this process depends on the interactions between the components of the mixture.Here,we present a systematic analysis of the morphology of poly[N-9'-heptadecanyl-2,7-carbazole-alt-5,5-(4',7'-di-2-thienyl-2',1',3'-benzothiadiazole)](PCDTBT)and[6,6]-phenyl-C71-butyric acid methyl ester(PC70BM)films in terms of the ternary phase diagram.The interaction parameters between PCDTBT and four different solvents,namely chloroform,chlorobenzene,o-dichlorobenzene,and toluene,were estimated based on swelling experiments.Based on these values,ternary phase diagrams of PCDTBT:PC70BM in different solvents were calculated.The morphology of spin-coated films with different blend ratios cast from different solvents is discussed in terms of the obtained phase diagrams.

双曲空间上的Alexandrov-Fenchel不等式

本文在双曲空间H^n(n≥5)中建立如下的双曲Alexandrov-Fenchel不等式:若超曲面为h-凸的,则有∫∑σ_4dμ≥C_(n-1)~4w_(n-1){(|Σ|/w_(n-1))~1/2)+(|Σ|/w_(n-1))^((1/2)(n-5)/(n-1)}~2等号成立当且仅当Σ为H^n中的测地球面.

Grade zero part of forced graded algebras

This paper concerns a certain subcategory of the category of representations for a semisimple algebraic group G in characteristic p, which arises from the semisimple modules for the corresponding quantum group at a p-th root of unity. The subcategory, thus, records the cohomological difference between quantum groups and algebraic groups. We define translation functors in this category and use them to obtain information on the irreducible characters for G when the Lusztig character formula does not hold.

Finite morphisms of projective and K■hler manifolds

In this paper we study finite morphisms of projective and compact K■hler manifolds, in particular,positivity properties of the associated vector bundle,deformation theory and ramified endomorphisms.

The maximal size of 6-and 7-arcs in projective Hjelmslev planes over chain rings of order 9 Dedicated to Professor Feng Keqin on the Occasion of his 70th Birthday

We complete the determination of the maximum sizes of （k,n）-arcs, n ≤ 12, in the projective gjelmslev planes over the two （proper） chain rings Z9 = Z/9Z and S3 = F3[X]/（X2） of order 9 by resolving the hitherto open cases n = 6 and n = 7. Parts of our proofs rely on decidedly geometric properties of the planes such as Desargues＇ theorem and the existence of certain subplanes.

Global regularity of three-dimensional density patches for inhomogeneous incompressible viscous flow

Toward Lions’ open question in Lions(1996) concerning the propagation of regularity for density patch, we prove that the boundary regularity of the 3-D density patch persists by time evolution for inhomogeneous incompressible viscous flows, with the initial density given by(1-η)1?0+ 1?0c for some small enough constant η and some Wk+2p,domain ?0, p ∈]3, ∞[, and with the initial velocity satisfying some smallness condition and appropriate conormal regularities.

The Formulas of Exact Calculation of the Discrepancy of Low-dimensional Finite Point Sets(Ⅰ)

Let d≥1,S_d={u_g(1≤k≤n}be a finite sequence of points in the d-dimensional unitcube G_d=[0,1)~d.We define the discrepancy of S_d byD_nbID_n(S_d)