L^(2)-范数与L^(∞)-范数及其联合管制下的微分方程求解  

L^(2)-Norm and L^(∞)-Norm and Their Joint Control to the Solution of Some Differential Equations

作　　者：杨秋花 黄晴 吕玉兰 卢卫君[2] YANG Qiu-hua;HUANG Qing;LV Yu-lan;LU Wei-jun(College of Mathematics and Electronic Information Engineering,Guangxi Minzu Normal University,Chongzuo 532200,China;College of Mathematics and Physics,Guangxi Minzu University,Nanning 530006,China;College of Science,Beibu Gulf University,Qinzhou 53501l,China)

机构地区：[1]广西民族师范学院数理与电子信息工程学院,广西崇左532200 [2]广西民族大学数学与物理学院,广西南宁530006 [3]北部湾大学理学院,广西钦州535011

出　　处：《数学的实践与认识》2022年第12期169-179,共11页Mathematics in Practice and Theory

基　　金：国家自然科研基金项目(12061014);广西自然科学基金项目(2019GXNSFAA245043)。

摘　　要：主要简述Lebesgue积分意义下的L^(2)-范数与L^(k)-范数及L^(∞)-范数,分别从三个方面即环面、一般紧致流形以及向量丛,来探讨Sobolev空间中弱解的存在性问题.特别地,受到Liu Zhu利用Hodge理论求解微分方程的方法的启发,引进两个新的Green算子■*G_(■)■与■*G_(■)■并证明它们具有拟等距性质,着重讨论在L^(2)-范数与L^(∞)-范数联合限制下算子■*G_(■)■在■-微分方程■Ω+■(i_(φ)Ω)=0的全纯可解性.This paper mainly discusses the L^(2)-norm,L^(∞)-norm and L-norm in the sense of Lebesgue integral.We consider the existence of weak solutions in the Sobolev space with three levels:torus,general compact manifold and vector bundle,respectively.In particular,motivated by solving equations with Hodge theory from Liu and Zhu's work,we introduce two extended Green operators■*G_(■)■and■*G_(■)■and show that they are of quasi-isometry.We study the holomorphic solvability of the 3-differential equation■Ω+■(i_(φ)Ω)=0 with the extended Green operator■*G_(■)■under two joint restricted conditions of L^(2)-norm and L^(∞)-norm.

关 键 词：L^(2)-范数 L^(∞)-范数 SOBOLEV空间 微分方程的弱解 拟等距 

分 类 号：O175[理学—数学]