类退化椭圆型方程的边值问题

在由光滑Jordan曲线Ω:H(x,y)=0围成的区域Ω中,考虑方程(u)≡A_0H~au_(xx)+2B_0H~βu_(xy)+C_0H~yu_(yy)+au_x+bu_y+cu=0, (1) 对0<λ<1,我们规定以C~(m+λ)(Ω)表示C~m(Ω)中其m阶导数在Ω上满足具有指数λ的Holder条件的那些函数所成的类。我们假设:方程(1)的系数A_0、B_0、C_0、a、b、c∈CC~λ(Ω),H∈C~(2+λ)(Ω),并且c0,常数a、β、r>0。不妨设在Ω中H(x,y)>0,而在Ω上A_0、B_0、C_0均不恒为零。 假设在Ω中B_0~2H~(2β)-A_0C_0H~(a+γ)<0,即方程是椭圆型的。由假设可知2βα+γ。显然,方程(1)在整个边界Ω上呈退化。而以往的许多工作,如丁正中和他的文献中指出的     

Boundary Value Problems for Degenerate Hyperbolic Systems of First Order Equations

In this paper we discuss some boundary value problems for degenerate hyperbolic complex equations of first order in a simply connected domain, in which the boundary value problems include the Riemann-Hilbert problem and the Cauchy problem. We first give the representation of solutions of the boundary value problems for the equations, and then prove the uniqueness and existence of solutions for the problems. In [A.V. Bitsadze (1988). Some Classes of Partial Differential Equations . Gordon and Breach, New York; A.V. Bitsadze and A.N. Nakhushev (1972). Theory of degenerating hyperbolic equations. Dokl. Akad. Nauk, SSSR , 204 , 1289-1291 (Russian); M.H. Protter (1954). The Cauchy problem for a hyperbolic second order equation. Can. J. Math ., 6 , 542-553], the authors discussed some boundary value problems for hyperbolic equations of second order.     

On the Boundary Behaviour,Including Second Order Effects,of Solutions to Singular Elliptic Problems

For γ≥1 we consider the solution u=u（x） of the Dirichlet boundary value problem Δu ＋ u^-γ=0 in Ω, u=0 on δΩ. For γ= 1 we find the estimate u（x）=p（δ（x））[1＋A（x）（log 1/δ（x）^-6], where p（r） ≈ r r√2 log（1/r） near r = 0,δ（x） denotes the distance from x to δΩ, 0 〈ε 〈 1/2, and A（x） is a bounded function. For 1 〈 γ 〈 3 we find u（x）=（γ＋1/√2（γ-1）δ（x））^2/γ＋[1＋A（x）（δ（x））2γ-1/γ＋1] For γ3= we prove that u（x）=（2δ（x））^1/2[1＋A（x）δ（x）log 1/δ（x）]     

Boundary Behavior of Semigroups of Holomorphic Mappings on the Unit Ball in C n

We study the asymptotic behavior of semigroups generated by holomorphic mappings by using an infinitesimal version of the boundary Schwarz-Wolff Lemma. In particular, the best rate of exponential convergence is obtained. In addition, we establish a geometrical version of the implicit function theorem.     

单位复超球上多复变解析函数的边值问题

讨论了二元复变解析函数在单位复超球区域上的某些边值问题,包括Dirichlet问题和Riemann-Hilbert问题,利用Cauchy公式、Plemelj公式以及级数展开的方法,我们对不同标数的情形,给出了所提问题可解的充分必要条件.     

Regularity of Keldys-Fichera Boundary Value Problem for Degenerate Elliptic Equations

The authors discuss the W1,p-solutions and the interior regularity of weak solutions for the Keldys-Fichera boundary value problem using the acute angle principle,the reversed Hlder inequality and the generalized poincar'e inequalities.     

在边界附近蜕化的椭圆型Monge-Ampère方程解的正则性

我们证明了在边界附近蜕化的椭圆型Monge-Ampère方程Dirichlet问题解的整体C1,1正则性,并举例说明了我们的结果是最佳的.     

ON BOUNDARY VALUE PROBLEMS FOROVERDETERMINED ELLIPTIC SYSTEMS OF TWO COMPLEX VARIABLES

In this paper,based on previous results,the Riemann-Hilbert boundary valueproblems with general forms for overdetermined elliptic equations of first order are consi-dered.The characteristic of modified function space is given.It is proved that there existsa unique solution for modified problem of the problem which we discuss.By the way,itis pointed out that there are great differences between overdetermined elliptic systems andfirst order elliptei systems in the plane.     

On the Continuity of Solutions to Degenerate Elliptic Equations in Two Dimensions

We study second order degenerate linear elliptic equations in divergence form in the plane. Under an exponential integrability assumption on the degenerate function we study the modulus of continuity of certain finite energy weak solutions. An application to mappings of finite distortion is also provided.     

On the Boundedness of Solutions of Degenerate Anisotropic Elliptic Variational Inequalities

In this article, for a class of degenerate anisotropic elliptic second-order variational inequalities we give conditions on the right-hand side and the set of constraints under which solutions of the variational inequalities are bounded. Our conditions on the set of constraints admit the consideration of a sufficiently large class of problems with pointwise constraints, and in particular, unilateral and bilateral problems. They also admit the consideration of the Dirichlet problem for the corresponding equations. We provide a series of examples which demonstrate the essentiality of the imposed conditions. In particular, we show that the condition assumed for the right-hand side of the variational inequalities in general is unimprovable in the scale of Lebesgue spaces.     

对角型蜕化椭圆组广义解的有界性和可积性

考虑一类对角型蜕化椭圆组,证明广义解的有界性和L_Ф-可积性.     

On L 2 Solvability of BVPs for Elliptic Systems

In this article we prove solvability results for L ² boundary value problems of some elliptic systems Lu=0 on the upper half-space ${\mathbb{R}} ^{n+1}_{+}, n\ge1$ , with transversally independent coefficients. We use the first order formalism introduced by Auscher-Axelsson-McIntosh and further developed with a better understanding of the classes of solutions in the subsequent work of Auscher-Axelsson. The interesting fact is that we prove only half of the Rellich boundary inequality without knowing the other half.     

带Neumann 边界条件的$p(x)$-Kirchhoff 型椭圆系统的研究

本文研究一类带Neumann 边界条件的~$p(x)$-Kirchhoff 型系统解的存在性. 借助于Ekeland变分原理和变指数Sobolev空间理论, 我们给出使得该问题存在解的合适条件.由于Poincar\'{e} 不等式在$W^{1,p(x)}(\Omega)$ 中不再成立, 我们将在$W^{1,p(x)}(\Omega)$ 的某个子空间中证明Poincar\'{e}-Wirtinger 不等式.     

On the Integrability of Eigenfunctions of the Laplace-Beltrami Operator in the Unit Ball of C n

Let B denote the unit ball in C ⁿ , n1, and let , , and denote the volume measure, gradient, and Laplacian respectively, with respect to the Bergman metric on B. For R and 0<p<, we denote by L ^p the set of real, or complex-valued measurable functions f on B for which _B (1–|z|²)|f(z)| ^p d(z)<, and by D ^p the Dirichlet space of C ¹ functions f on B for which | f|L ^p . Also, for C, we denote by X the set of C ² real, or complex-valued functions f on B for which f=f. The main result of the paper is as follows: Let 0<p< and suppose R with –n ². Then L ^p X ={0}, and for 0, D ^p X ={0}(a) for all n+ when p1, and(b) for all when 0<p<1.By example it is shown that the result is best possible for all values of p with pn/(n+ .     

球外部区域上一类椭圆边值问题的正径向解

研究了一类椭圆边值问题在球外部区域上正径向解的存在性,当非线性项f（u）关于u超线性或次线性增长的情形,获得了该问题正径向解的存在性.     

C^n中单位球上的强拟凸映照

若记B^n为C^n中的单位球,本文研究B^n上的强拟凸映强,（1）强拟凸映照与星形映照及凸映照的关系,（2）强拟凸映照的二次项系;（3）单位多圆柱上的强拟凸映照。     

On the Dynamics of Homeomorphisms on the Unit Ball of Rn

Positivity - Consider a compact convex subset X of R n (n ≥ 2) with non-empty interior and let H(X) be the set of all homeomorphisms from X onto X endowed with the supremum metric. We are...