Semismall perturbations, semi-intrinsic ultracontractivity, and integral representations of nonnegative solutions for parabolic equations

We consider nonnegative solutions of a parabolic equation in a cylinder D×I, where D is a noncompact domain of a Riemannian manifold and I=(0,T) with 0<T?∞ or I=(−∞,0). Under the assumption [SSP] (i.e., the constant function 1 is a semismall perturbation of the associated elliptic operator on D), we establish an integral representation theorem of nonnegative solutions: In the case I=(0,T), any nonnegative solution is represented uniquely by an integral on (D×{0})∪(M_∂D×[0,T)), where M_∂D is the Martin boundary of D for the elliptic operator; and in the case I=(−∞,0), any nonnegative solution is represented uniquely by the sum of an integral on M_∂D×(−∞,0) and a constant multiple of a particular solution. We also show that [SSP] implies the condition [SIU] (i.e., the associated heat kernel is semi-intrinsically ultracontractive).     

Positive Solutions of Nonlinear Elliptic Equations in Unbounded Domains in R 2

We study the existence of positive solutions of the nonlinear equation u+f(,u)=0, in D with u=0 on D, where D is an unbounded domain in R ² with a compact nonempty boundary D consisting of finitely many Jordan curves. The aim is to prove an existence result for the above equation in a general setting by using potential theory.     

Integral representations of nonnegative solutions for parabolic equations and elliptic Martin boundaries

We consider nonnegative solutions of a parabolic equation in a cylinder D×(0,T), where D is a noncompact domain of a Riemannian manifold. Under the assumption [IU] (i.e., the associated heat kernel is intrinsically ultracontractive), we establish an integral representation theorem: any nonnegative solution is represented uniquely by an integral on (D×{0})∪(M_∂D×[0,T)), where M_∂D is the Martin boundary of D for the associated elliptic operator. We apply it in a unified way to several concrete examples to explicitly represent nonnegative solutions. We also show that [IU] implies the condition [SP] (i.e., the constant function 1 is a small perturbation of the elliptic operator on D).     

拟线性抛物方程的弱解

该文给出了拟线性退化抛物方程pa_t{u}+pa_x{f(u)}=pa_xx{A(u(x,t))}∈R^2_+×(0,+∞) ,u(x,0)=u_0(x),x∈R 一种弱解的新定义, 利用Div Curl引理证明了解的存在性．     

Asymptotic Behavior of Nodal Solutions for Semilinear Elliptic Equationsin R~n

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＲｅｃｅｎｔｌｙ，ｔｈｅｓｅｍｉｌｉｎｅａｒｅｌｌｉｐｔｉｃｅｑｕａｔｉｏｎｓ△ｕ＋ｆ（ｕ）＝０（１．１）ｕ（ｘ）→０ａｓ｜ｘ｜→∞（１．２）ｉｎＲｎｗｅｒｅｃｏｎｓｉｄｅｒｅｄｗｉｄｅｌｙ（ｓｅｅ［１］－［７］）．Ｉｎｔｈｅｎｉｃｅｐａｐｅｒｓ［１］ａｎｄ［２］，ｉｔｗａｓｐｒｏｖｅｄｔｈａｔａｎｙｐｏｓｉｔｉｖｅｓｏｌｕｔｉｏｎｏｆ（１．１）ｍｕｓｔｂｅｒａｄｉａｌｉｎｃａｓｅｆ（ｕ）∈Ｃ１＋δ（δ＞０）．Ｔｈｅｒｅｆｏｒｅ，ａｎｙｐｏｓｉｔｉｖｅｓｏｌｕｔｉ…     

环域上一类拟线性椭圆方程正的径向解的多重性

研究了方程-div(‖Du‖^-2Du)=λf(u)在R^n,n≥2中环域上的正的径向解的多重性。当f在正区域上有多个峰的情况下,我们获得了多个解。     

On the Existence of Positive Solutions of Nonlinear Elliptic Equations

We study the existence of positive solutions of the nonlinear elliptic problem in D with u=0 on D, where and are two Randon's measures belonging to a Kato subclass and D is an unbounded smouth domain in ^d(d3). When g is superlinear at 0 and 0f(t)t for t(0,b), then probabilistic methods and fixed point argument are used to prove the existence of infinitely many bounded continuous solutions of this problem.     

Existence and Asymptotic Behaviour of Large Solutions of Semilinear Elliptic Equations

In this paper we are interested in the semilinear elliptic equations of the type u=u(,u), on bounded smooth domain of R ⁿ . We also treat existence of positive solution of u=p(x)f(u), which explodes near the boundary of (called large solutions). Our approach is based on potential theory.     

一类强退化拟线性抛物方程的弱解

In this paper, we show the existence of the renormalized solutions and the entropy solutions of a class of strongly degenerate quasilinear parabolic equations.     

Global Regularity of Weak Solutions to Quasilinear Elliptic and Parabolic Equations with Controlled Growth

We establish global regularity for weak solutions to quasilinear divergence form elliptic and parabolic equations over Lipschitz domains with controlled growth conditions on low order terms. The leading coefficients belong to the class of BMO functions with small mean oscillations with respect to x.     

球外部区域上非线性椭圆型方程的正径向解

本文讨论球外部区域Ω={x∈RN||x|>R}上非线性椭圆边值问题正径向解的存在性,其中g(r),f(u)为非负连续函数.在g(r)满足条件0<∫R∞rg(r)dr<∞,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.     

General Uniqueness Results and Variation Speed for Blow-Up Solutions of Elliptic Equations

Let be a smooth bounded domain in R^N. We prove general uniquenessresults for equations of the form – u = au – b(x)f(u) in , subject to u = on . Our uniqueness theorem is establishedin a setting involving Karamata's theory on regularly varyingfunctions, which is used to relate the blow-up behavior of u(x)with f(u) and b(x), where b 0 on and a certain ratio involvingb is bounded near . A key step in our proof of uniqueness usesa modification of an iteration technique due to Safonov. 2000Mathematics Subject Classification 35J25 (primary), 35B40, 35J60(secondary).     

On a New Kato Class and Singular Solutions of a Nonlinear Elliptic Equation in Bounded Domains of $$\mathbb{R}^n$$

Using a new form of the 3G-Theorem for the Green function of a bounded domain Ω in , we introduce a new Kato class K(Ω) which contains properly the classical Kato class K_n(Ω). Next, we exploit the properties of this new class, to extend some results about the existence of positive singular solutions of nonlinear differential equations. Mathematics Subject classification (1991): 34B15, 34B27.     

拟线性椭圆型方程带有混合奇异项的正整体解

在本文中,研究了方程div(｜u｜p-2u) f(x,u)=0,x∈RN,N≥3的正整体解,其中f(x,u)在u=0未假定是正则的,且f(x,u)可以同时包含超线性,亚线性项和奇异项.     

All Meromorphic Solutions of Some Algebraic Differential Equations

In this article, we introduce some results with respect to the integrality and exact solutions of some 2nd order algebraic DEs. We obtain the sufficient and necessary conditions of integrable and the general meromorphic solutions of these equations by the complex method, which improves the corresponding results obtained by many authors. Our results show that the complex method provides a powerful mathematical tool for solving a large number of nonlinear partial differential equations in mathematical physics.     

On the Interior Smoothness of Solutions to Second-Order Elliptic Equations

We study the interior smoothness properties of solutions to a linear second-order uniformly elliptic equation in selfadjoint form without lower-order terms and with measurable bounded coefficients. In terms of membership in a special function space we combine and supplement some properties of solutions such as membership in the Sobolev space W _{2, loc} ¹ and Holder continuity. We show that the membership of solutions in the introduced space which we establish in this article gives some new properties that do not follow from Holder continuity and the membership in W _2,loc ¹ .     

一类非线性中立型抛物微分方程边值问题解的振动性

讨论了一类非线性中立型抛物微分方程,得到了该类方程的两类边值问题解振动的充分条件.     

具间断系数的N维拟线性椭圆方程弱解的正则性

讨论了具间断系数的N维拟线性椭圆方程. 利用估计和差分逼近方法,证明了弱解的一阶导数H\"{o}lder连续到方程系数间断的内边界.     

某些半线性椭圆方程在环域上的正对径解的存在性

利用锥拉伸与锥压缩型的Krasnosel'skii不动点定理讨论了某些二阶非线性椭圆方程在环域上关于Dirichlet边界条件的正对径解的存在性。通过考察非线性项在有界闭区间上的性质建立了若干正对径解的存在性结论。主要结论不涉及非线性项的超线性增长和次线性增长。当非线性项存在极值并满足适当条件时,主要结论是非常有效的。