一类非散度型椭圆方程解的梯度估计

得到了一类非散度型二阶椭圆方程解的梯度在 Lp中的局部估计 ,其中 p >0 .方程形式为 :L0 u+ b .　Δu -vu =f,L0 为具 H lder连续系数的非散度型椭圆算子 ,f有界可测 ,| b| 2 与 v均属于 Kato类     

一类非散度型椭圆方程的正则性

本文得到了一类带奇异低阶项椭圆方程的非负解的Harnack不等式.方程的形式为Lou+biuxi=0,其中L0为一具Holder连续系数的非散度型椭圆算子,|b|2属于Kato类.     

判断直线与椭圆位置关系的两种新方法

命题对任意的椭圆c:x2/a2 y2/b2=1,直线L:Ax By C=0,设椭圆c的两焦点为F1,F2,F1关于L的对称点为F1’. 当|F1'F2|<2a时,直线L与椭圆c相交; 当|F1'F2|=2a时,直线L与椭圆c相切; 当|F1'F2|>2n时,直线L与椭圆c相离.     

一类带奇异低阶项椭圆型方程的正则性

本文得到了一类带奇异低阶项椭圆方程的一个正则性结果.方程形式为Lu+ vu=μ,其中L为二阶散度型椭圆算子,v属于某一Morrey空间,μ为一非负Radon 测度.     

关于椭圆内接四边形和三角形的最大面积

(一) 椭圆x~2/a~2 y~2/b~2=1(n>b>0)内接四边形的最大面积为2ab。 (一) 内接平行四边形的最大面积为2ab [证明一] 设ABGD是椭圆x~2/a~2 y~2/b~2=1的内接平行四边形(图1).由于对角线AC、BD互相平分,即有共同的中点.则以椭圆内定点(非中心)为中点的弦(简称中点弦)是唯一的。(设定点为M(x_0,y_0),则中点弦方程为x_0x/a~2 y_0y/b~2=x_0~2/a~2 y_0~2/b~2).因而,AC,BD相交于椭圆的中心(即为椭圆的两     

对2011年山东卷文科压轴题的探究

2011年高考山东卷文科压轴题：在平面直角坐标系xOy中,已知椭圆C：x^2/3＋y^2=1.如图1所示,斜率为k（k〉0）且不过原点的直线L交椭圆C于A,B两点,线段AB的中点为E,射线0E交椭圆C于点G,交直线X=-3于点D（-3,m）．     

椭圆切线的尺规作法

在研究椭圆问题时 ,得到以下椭圆切线的一个尺规作法 :已知椭圆方程为x2a2 + y2b2 =1 (a>b >0 ) ,过椭圆上一点Q(x0 ,y0 )的切线方程为x0 xa2 + y0 yb2 =1 .设Q(x0 ,y0 )为椭圆上任一点 ,下面给出切线的作法 .作法 :( 1 )若Q为椭圆的顶点 ,则切线垂直于所在的轴 ;( 2 )若Q在任一非顶点处如图 ,过Q作QA ⊥x轴 ,垂足为A ,反向延长QA ,①以O为圆心 ,a为半径画弧交射线AQ的延长线于P点②过P点作OP的垂线PN交x轴于N点③连结NQ ,即为过Q点的切线 .　　证明　不妨设Q在第一象限 ,Q(x0 ,y0 ) ,则A为 (x0 ,0 )因为OP =a ,x0 2a2 + y0 2b2…     

判断直线与椭圆位置关系的两种新方法

“判别式”法是判断直线与椭圆位置关系的常用方法,笔者在进行“研究性学习”教学时发现了两种判断直线与椭圆位置关系的新方法.1　提出问题已知直线L :x -y + 9=0 ,椭圆E :x21 2 +y23=1 ,E的两焦点为F1,F2 ,求以F1,F2 为焦点,且与L有公共点M的椭圆中,长轴最短的椭圆E′的方程.经过学生探索讨论,一般可得下面两种解法.方法1　F1( - 3,0 ) ,F2 ( 3,0 ) ,设椭圆E′的方程为x2m+ y2m - 9=1 (m >9) ,原题转化为求m最小时E′的方程.由x2m+ y2m - 9=1 ,x -y + 9=0得( 2m - 9)x2 + 1 8mx + 90m -m2 =0 .由Δ=8m3- 432m2 =32 4 0m≥0得m≥4 5…     

一道高考题的反思——用平面向量来研究二次曲线的切线与割线

它短轴长为2√2,相应于焦点F(c,0)(c＞0)的准线L与x轴相交于A,|OF|=2|FA|,过点A的直线与椭圆相交于P,Q两点.     

本刊外文版8卷3期文章简介(1992)

<正> 具临界 Sobolev 指数的非线性椭圆方程的正解存在性汪徐家本文将 Brezis 和 Nirenberg 的结果推广到问题(A)(?)其中 L 为一致椭圆算子,b(x)(?)0,f(x,u)为 u~p 在无穷远点的低阶扰动项.问题(A)的解的存在性强烈地依赖于 α_(ij)(x),b(x)和 f(x,u)的性状.例如对任何有界光滑区域Ω都可找到a_ij(x)∈C(?)使 Lu=u~p 在 H_0~1(Ω)中具有一正解.作者还对一类 f(x,u)证明了下面问题非径向解的存在性:-△u=f(|x|,u),u>0于Ω,u=0于(?)Ω,Ω=B(0,1).     

UNIQUENESS OF STATIONARY SOLUTIONS WITH VACUUM OF EULER-POISSON EQUATIONS

In this paper, the uniqueness of stationary solutions with vacuum of Euler-Poisson equations is considered. Through a nonlineax transformation which is a functionof density and entropy the corresponding problem can be reduced to a semilinear ellipticequation with a nonlinear source term consisting of a power function, for which the classical     

Multiplicity of solutions for a class of quasilinear elliptic equation involving the critical Sobolev and Hardy exponents

In this paper, by using concentration-compactness principle and a new version of the symmetric mountain-pass lemma due to Kajikiya (J Funct Anal 225:352–370, 2005), infinitely many small solutions are obtained for a class of quasilinear elliptic equation with singular potential $$- \Delta_p u - \mu \frac{|u|^{p-2}u}{|x|^p} =\frac{|u|^{p^\ast(s)-2}u}{|x|^s} + \lambda f(x, u),\quad u\in H_0^{1,p}(\Omega).$$      

Scattering theory for elliptic operators of arbitrary order

We prove the main theorems of scattering theory for selfadjoint elliptic partial differential operators of arbitrary order. Under various hypotheses we show that the wave operators exist and are complete, that the intertwining relations hold, and that the invariance principle holds. One of our main hypotheses is that each lower order coefficientq(x) satisfies. $$(1 + \left| x \right|)^\alpha \int\limits_{\left| {x - y} \right|< a} {\left| {q(y)} \right|dy \in L^p (E^n )}$$ for some α≥0,a>0 and forp≤∞ such that $$\alpha > 1 - \frac{{2n}}{{(n + 1)p}}$$      

A Contribution to the Theory of Quasilinear Elliptic Equations and Application to the Minimization of Integral Functionals

The aim of this work is to study the existence of solutions of quasilinear elliptic problems of the type

On the Solution of a Second-Order Nonlinear Equation in the Exterior of a Compact Set

In this paper, we study the behavior of solutions of a semilinear elliptic equation in the exterior of a compact set as $|x| \to \infty $ . Such equations were considered by many authors (for example, Kondrat'ev, Landis, Oleinik, Veron, etc.). In the present paper, we study the case in which in the equation contains lower terms. The coefficients of the lower terms are arbitrary bounded measurable functions. It is shown that the solutions of the equation tend to zero as $|x| \to \infty $ .     

非线性椭圆方程自由边值问题球对称解的存在性

给出了如下的非线性椭圆方程自由边值问题-Δu=λu+(1+ε)u+p,x∈B Rn,u|Ω=μ,∫Ωnu=-M(1)在C[0,1]中的球对称解的存在性.并得到比上述问题更一般的非线性椭圆方程自由边值问题-Δu=h(u),x∈B Rn,u|Ω=μ,∫Ωun=-M,在C[0,1]中的球对称解的存在性,其中B为Rn中的单位球,p>1,λ>0,μ<0,M>0,ε>0;λ,μ,M,ε均为常数,n为正整数.     

Partial regularity for quasilinear nonuniformly elliptic systems of the general type

We establish the partial C^1,α-regularity of weak solutions of nonhomogeneous nonuniformly elliptic systems of the type $$ - \frac{\partial }{{\partial x_\alpha }}A_\alpha ^i (x,u,u_x ) = B^i (x,u,u_x ),{\text{ }}i = 1,...,n$$ . The system of Euler equations of the variational problem of finding a minimum of the integral $\int\limits_\Omega {\mathcal{F}(u_x )dx} $ with an integrand of the type $$\mathcal{F}(p) = a|p|^2 + b|p|^m + \sqrt {1 + \det ^2 p,} {\text{ }}a > 0,{\text{ }}b > 0$$ , for b large enough, is a typical example of systems under consideration. Bibliography: 11 titles.     

Solvability of the Dirichlet problem for degenerate quasilinear elliptic equations

In a bounded domain of the n -dimensional (n?2) space one considers a class of degenerate quasilinear elliptic equations, whose model is the equation $$\sum\limits_{i = 1}^n {\frac{{\partial F}}{{\partial x_i }}} (a^{\ell _i } (u)\left| {u_{x_i } } \right|^{m_i - 2} u_{x_i } ) = f(x),$$ where x =(x₁,..., x_r), l_i?0, m_i>1, the function f is summable with some power, the nonnegative continuous function a(u) vanishes at a finite number of points and satisfies \(\frac{{lim}}{{\left| u \right| \to \infty }}a(u) > 0\) . One proves the existence of bounded generalized solutions with a finite integral $$\int\limits_\Omega {\sum\limits_{i = 1}^n {a^{\ell _i } (u)\left| {u_{x_i } } \right|^{m_i } dx} }$$ of the Dirichlet problem with zero boundary conditions.     

Nonexistence of Solutions of a Semilinear Biharmonic Equation with Singular Potential

The nonexistence of a global solution of the semilinear elliptic equation Δ²u ? (C/|x|⁴)u ? |x|^σ|u|^q = 0 in the exterior of a ball is studied. A sufficient condition for the nonexistence of a global solution is established. The proof is based on the test function method.     

Zur lokalen Werteverteilung der Lösungen linearer Differentialgleichungen

Every solution w of the linear differential equation (*) $$L_n (w) = w^{(n)} + a_{n - 1^{w^{(n - 1)} } } + \ldots + a_0 w = 0$$ with polynomial coefficients a_j is a polynomial or an entire function of finite order λ>0. In this paper we prove the following theorem: Let w be a solution of (*) and no polynomial. Let further λ be the order of w and n_a (R, 1/(w?c)) the number of the zeros in the disc |z?a|4^1/λ $$n_a \left( {L|a|^{1 - \lambda } ,1/(w - c)} \right) \leqq N.$$ It is also shown, that for certain solutions of (*) there exists a constant r₀>0 such that we can replace N by n+α for |a|> r₀. α₀ is the degree of the polynomial a₀. An important tool for the proofs is the index of an entire function.