一类椭圆方程很弱解关于区域的连续性

本文讨论了二阶椭圆型方程-△u=f(x,u),x∈Ω的Dirichlet问题u | Ω=0的很弱解u∈W ,r(Ω)(1＜r＜2)关于区域Ω的连续性及很弱边值问题的很弱解的唯一性.     

非线性次椭圆方程障碍问题很弱解的高阶可积性

该文首先证明了一类由满足Hrmander条件的向量场构成的次椭圆方程K_(φ,u_0)~T-障碍问题很弱解的局部高阶可积性,进而说明了其很弱解即为经典意义下的弱解.作为其应用,得到了障碍问题很弱解的紧性结果.此外,在当区域Ω满足某容度条件假设时,证明了上述障碍问题很弱解的全局高阶可积性.     

一类非齐次A-调和方程很弱解的正则性

研究形如divA(x,▽u)=f(x)的非齐次A-调和方程,就方程右端是非散度的情况下,定义了非齐次A-调和方程的很弱解,并运用扰动向量场的Hodge分解理论,证明了很弱解的正则性问题.     

回归函数核估计的强相合性

设(X_i,Y_i),i=1,…,n是从取值于R~d×R~1的随机向量(X,Y)中抽取的iid.样本。设E|Y|<∞,而以m(x)=E(|Y|X=x)表示回归函数。在本文中,我们考虑m(x)的通常的和递推形式的核估计:其中K(x)假定是R~d上的概率密度,而h_n>0。我们在K(x)很弱的条件下建立了m_n~((i))(x)的a.s.收敛性,i=1,2,3,但是要求X的边际分布具有密度,这种情况曾在Schuster和Yakowitz中讨论过,那里,更要求(X,Y)的联合分布有概率密度。     

FC-空间上的鞍点定理及应用

在不具有任何凸性结构和线性结构的有限连续空间(简称FC-空间)中给出了一个鞍点定理.并应用此定理,在很弱的条件下证明了一些非空交定理及截口定理,推广了近期文献中的一些相关的结果.     

强近邻相依性

引入了一类新的相依随机变量序列, 它可以看做近邻相依序列的子类, 但是仍然可以用混合序列逼近, 称这类序列为强近邻相依序列. 在很弱的相依性大小的条件下, 证明了一个p (>2)阶(极大)矩不等式.     

Banach空间中算子的秩定理

设E和F是Banach空间,B(E,F)表示映E到F的有界线性算子全体.记T+0 ∈ B(F,E)为T0 ∈ B(E,F)的一个广义逆.本文证明,每一个具有‖T+0(T-T0)‖＜1的算子T ∈ B(E,F),B≡(I+T+0(T-T0))-1T+0是T的广义逆当且仅当(I-T+0T0)N(T)=N(T0),其中N(·)表示括弧中算子的零空间.这一结果改进了Nashed和Cheng的一个有用的定理,并进一步证明Nashed和Cheng的一个引理对半-Fredholm算子有效但一般未必成立.     

一类非线性椭圆组很弱解的局部正则性

讨论了Rn(n≥2)中有界开集Ω上二阶非线性椭圆组一divA(x,u,Du)=B(x,u,Du),当A(x,u,Du)满足强制与增长条件,B(x,u,Du)满足控制增长条件时,其很弱解u(x)∈W1,4loc(Ω,Rn)的正则性.其中max{1,p-1}＜r＜p,p出现在A与B的强制与增长假设中.本文采用Hodge分解的方法建立适当的检验函数,借助一些引理,对椭圆组的很弱解得到了逆Holder不等式,从而改进了其很弱解偏微商的可积性,使其成为经典意义下的弱解.     

A-调和方程很弱解的比较原理

应用McShane扩张定理证明了A-调和方程很弱解的比较原理.这个结果可以看作经典弱解比较原理的一个扩展.     

A-调和方程很弱解的比较原理

应用McShane扩张定理证明了A-调和方程很弱解的比较原理.这个结果可以看作经典弱解比较原理的一个扩展.     

Perturbation of semi-linear evolution equations under weak assumptions at initial time

We prove perturbation results for abstract semi-linear evolution equations in a Banach space. The main feature is that only very weak assumptions are needed at initial time. This allows to prove weak continuity properties and to deal with rather general domain perturbation problems for semi-linear parabolic and hyperbolic boundary value problems with various boundary conditions. The theory also implies the well known theory on parameter dependent equations.     

Exact Boundary Controllability of Weak Solutions for a Kind of First Order Hyperbolic System — The Constructive Method*

In this paper the authors first present the definition and some properties of weak solutions to 1-D first order linear hyperbolic systems. Then they show that the constructive method with modular structure originally given in the framework of classical solutions is still very powerful and effective in the framework of weak solutions to prove the exact boundary (null) controllability and the exact boundary observability for first order hyperbolic systems.     

A free boundary problem of elliptic equation arising in supersonic flow past a conical body

We study free boundary value problems of elliptic equation caused by a supersonic flow past a non-symmetric conical body. The flow is described by the potential flow equation. In the self-similar coordinate system the problem can be reduced to a boundary value problem of second order nonlinear elliptic equation with a free boundary. Applying the partial hodograph transformation and the method of nonlinear alternative iteration we proved the existence of solution to this boundary value problem. Consequently, we also proved the conclusion that for the problem of supersonic flow past a conical body, if the conical body is slightly different from a circular cone with its vertex angle less than a given value determined by the parameters of the coming flow, then there exists a weak entropy solution with an attached conical shock.     

一个具有内部物质对流的两项连铸Stefan问题

考虑了一个具有内部物质对流和非线性边界热交换的多维连铸Stefan问题,并得到了这个问题整体弱解的存在性、唯一性和对初边界条件的连续依赖性。本项工作改进和推广了J.F.Fodri-gues＆F.Yi的结果,放宽了他们对内部流和边界条件的一些不太符合实际的限制。     

Some nonlinear elliptic systems with right-hand side integrable data with respect to the distance to the boundary

In this paper, we study the very weak solutions to some nonlinear elliptic systems with right-hand side integrable data with respect to the distance to the boundary. Firstly, we study the existence of the approximate solutions. Secondly, a priori estimates are given in the framework of weighted spaces. Finally, we prove the existence, uniqueness and regularity of the very weak solutions.     

Hydrostatic Stokes equations with non-smooth data for mixed boundary conditions

The main subject of this work is to study the concept of very weak solution for the hydrostatic Stokes system with mixed boundary conditions (non-smooth Neumann conditions on the rigid surface and homogeneous Dirichlet conditions elsewhere on the boundary). In the Stokes framework, this concept has been studied by Conca [Rev. Mat. Apl. 10 (1989)] imposing non-smooth Dirichlet boundary conditions.In this paper, we introduce the dual problem that turns out to be a hydrostatic Stokes system with non-free divergence condition. First, we obtain strong regularity for this dual problem (which can be viewed as a generalisation of the regularity results for the hydrostatic Stokes system with free divergence condition obtained by Ziane [Appl. Anal. 58 (1995)]). Afterwards, we prove existence and uniqueness of very weak solution for the (primal) problem.As a consequence of this result, the existence of strong solution for the non-stationary hydrostatic Navier-Stokes equations is proved, weakening the hypothesis over the time derivative of the wind stress tensor imposed by Guillén-González, Masmoudi and Rodríguez-Bellido [Differential Integral Equations 50 (2001)].     

On Very Weak Positive Solutions to Some Semilinear Elliptic Problems With Simultaneous Singular Nonlinear and Spatial Dependence Terms

We use recent results by Díaz and Rakotoson concerning very weak solutions to linear boundary value problems in order to improve previous work on existence and properties of weak positive solutions to a model example of semilinear singular elliptic problem.     

Integration over non-rectifiable curves and Riemann boundary value problems

In this paper we introduce an alternative way of defining the curvilinear Cauchy integral over non-rectifiable curves in the case of complex functions of one complex variable. Especially the jump behavior on the boundary is considered. As an application, solvability conditions of the Riemann boundary value problem are derived on very weak conditions to the boundary. Besides the complex case the consideration can be extended to the theory of Douglis algebra valued functions.     

Analysis of a Cahn–Hilliard–Brinkman model for tumour growth with chemotaxis

Phase field models recently gained a lot of interest in the context of tumour growth models. Typically Darcy-type flow models are coupled to Cahn–Hilliard equations. However, often Stokes or Brinkman flows are more appropriate flow models. We introduce and mathematically analyse a new Cahn–Hilliard–Brinkman model for tumour growth allowing for chemotaxis. Outflow boundary conditions are considered in order not to influence tumour growth by artificial boundary conditions. Existence of global-in-time weak solutions is shown in a very general setting.     

Mean lattice point discrepancy bounds,II: Convex domains in the plane

We consider planar curved strictly convex domains with very weak (or no) smoothness assumptions on the boundary and prove sharp bounds for square-functions associated to the lattice point discrepancy. Research supported in part by NSF grants.