Clifforrd分析中广义双正则函数的(线性)非线性边值问题

A (Linear) Nonlinear Boundary Value Problem for Generalized Biregular Functions in Clifford Analysis

本文研究Ｃｌｉｆｏｒｄ分析中广义双正则函数的一个非线性边值问题：Ａ（ｔ１，ｔ２）Ｗ＋＋（ｔ１，ｔ２）＋Ｂ（ｔ１，ｔ２）Ｗ＋－（ｔ１，ｔ２）＋Ｃ（ｔ１，ｔ２）Ｗ－＋（ｔ１，ｔ２）＋Ｄ（ｔ１，ｔ２）Ｗ－－（ｔ１，ｔ２）＝ｇ（ｔ１，ｔ２）ｆｔ１，ｔ２，Ｗ＋＋（ｔ１，ｔ２），Ｗ＋－（ｔ１，ｔ２），Ｗ－＋（ｔ１，ｔ２），Ｗ－－（ｔ１，ｔ２）［］．先讨论解的积分表示式，再研究几个奇异算子，最后用Ｓｃｈａｕｄｅｒ不动点原理（压缩映射定理）证明了解的存在性（唯一性）．目前还没有见到其它国内外学者研究广义双正则函数的非线性边值问题．本文推广了Ｆ．Ｂｒａｃｋｓ，Ｗ．Ｐｉｎｃｋｅｔ［１０］，ＬｅＨｕａｎｇＳｏｎ［１１］，Ｒ．Ｐ．ＧｉｌｂｅｒｔａｎｄＪ．Ｌ．

In this paper, the author discusses generalized biregular function on bi domains in Clifford analysis. Firstly the Plemelj's formula is obtained and then considers the nonlinear boundary value problem: A(t 1, t 2)W ++ (t 1,t 2) +B(t 1, t 2) W +- (t 1,t 2) +C (t 1, t 2) W -+ (t 1,t 2) +D(t 1, t 2)W -- (t 1,t 2)=g(t 1,t 2) f t 1, t 2 , W ++ (t 1,t 2) ,W +- (t 1,t 2), W -+ (t 1,t 2) , W -- (t 1,t 2)).Applying the method of integral equations and Schauder fixed point theorem, the existence of solution for above problem is proved.