零理想边界的素端上某类椭圆型方程的非负解  

NONNEGATIVE SOLUTIONS OF SOME KIND OF ELLIPTIC EQUATIONS ON ENDS WITH NULL IDEAL BOUNDARY

作　　者：邱曙熙[1] 曾建武[1] 

机构地区：[1]厦门大学数学系,福建厦门361005

出　　处：《数学年刊（A辑）》2001年第6期743-750,共8页Chinese Annals of Mathematics

基　　金：国家自然科学基金(No19871068)资助的项目.

摘　　要：考虑具有零理想边界的非紧镶边Ｒｉｅｍａｎｎ曲面Ω＝Ω∪　Ω及其上的Ｄｉｒｉｃｈｌｅｔ积分有限的非负局部Ｈｏｌｄｅｒ连续的二重共变量Ｐ．用Ｆ表示方程上Δｕ＝Ｐｕ在 Ω取极限值０的非负连续解全体．本文讨论拟Ｐｉｃａｒｄ原理成立的充要条件并证明：若Ω的每一理想边界点都有端邻域满足广义Ｈｅｉｎｓ条件，则Ｍａｒｔｉｎ函数全体所成之集是Ｆ中的极小正解全体所支撑的子半线性空间Ｐ的一个Ｈａｍｅｌ基，而且Ｆ可表示为与Ｐ相关的直和形式．Consider a non-compact bordered Riemann surface Ω = Ω∪ Ω with compact border Ω and null ideal boundary in Kerekjato-Stoilow' sense. Let F be the cone of all the nonnegative solutions of the elliptic equation Δu = Pu, which vanish on Ω and are continuous on Ω, where the density P is a nonnegative locally Holder continuous covariant bivector on Ω with a finite Dirichlet integral. In this article, the authors give a necessary and sufficient condition that the Picard priciple is valid. Moreover, it is shown that if each ideal boundary point of Ω satisfies so called generalized Heins' conditions, then the collection of all Martin functions on Ω is a Hamel base of the sub-cone P of F, spanned by the extremal positive solutions of the equation, and F is a direct sum with respect to P.

关 键 词：二阶椭圆型方程 椭圆调和维数 拟Picard原理 Hamel基 RIEMANN曲面 Jordam曲线 Picard原理 非负解 

分 类 号：O174.51[理学—数学] O174.3[理学—基础数学]