Moisil-Theodorsco方程组的一个非线性边值问题

A Nonlinear Boundary Value Problem in Moisil-Theodorsco System

考虑了在R3空间中的非齐次Moisil-Theodorsco方程组的一个非线性边值问题,首先讨论Moisil-Theodorsco方程组的Cauchy型积分,Plemelj公式,进而得到了非齐次Moisil-Theodorsco方程组解的积分表示式和它的Plemelj公式,在此基础上还讨论了它的一个非线性边值问题A(η)F+(η)+B(η)F-(η)=g(η)f(η,F+(η),F-(η)),η∈Γ。为了证明以上非线性边值问题解的存在性,利用已得到的Plemelj公式,将非线性边值问题转化为与它等价的积分方程(A+B)(-2φ+Kφ)+(A+1)φ+(A+B)T^f=gf,其中(Kφ)(η)=41π∫Γ(--ηη)3n()φ()dS,η∈Γ,T^f=-41π∫D(--ηη)3f()dV,最后运用Schauder不动点原理证明了该边值问题解的存在性,同时也给出了其解的积分表示式F()=1∫(ζ-)n(ζ)φ(ζ)dS+~Tf(),Γ。

It is well known that function theory methods become,in recent years,powerfull mathematical tool for the treatment of various boundary value problems which have a lots of application in mathematical physics and engineering in domains over Eulidean spaces of higher dimension.In this paper the nonlinear boundary value problem is investigated in inhomogeneous Moisil-Theodorsco system F=f in the domain D which is a bounded domain of R3 with a piece-wise smooth Liapunov surface Γ.Find a solution F（ ） of equation F（ ）=f（ ）in the domain D, which is continuous in D, such that F（∞）=0and the nonlinear boundary condition A （η）F^＋（η）＋B（η）F^-（η）=g（η）f（η）f（η,F^＋（η）,F^-（η））η∈Г,where A（η）,（η）,g（η）∈H（Г,β）are given complex value matrix function on Г. To deal with the nonlinear boundary value problem, first of all, this article studies some properties of the Cauchy type integral and the Plemelj formulae in Moisil-Theodorsco system, and then gives the integral expression of its solution as F（ ）=Ф（ ）＋Tf（ ）,∩D,where T^f=-41π∫D（--ηη）3f（）dVФ（ ）is a regular function in D, and the Plemelj formulae in inhomogeneous Moisil-Theodorsco system are {F^＋（η）=1/2φ（η）＋（Kφ）（η）＋Tf（η）F^-（η）=1/2φ（η）＋（Kφ）（η）＋Tf（η）η=Г,where （Kφ）（η）=41π∫Γ（--ηη）3n（）φ（）dS,η∈Γ,T^f=-41π∫D（--ηη）3f（）dV,η∈Г .Then in the use of the above obtained Plemelj formulae, the nonlinear boundary value problem should be translated correspond-ingly the equivalent integral equation（A＋B）（-2φ＋Kφ）＋（A＋1）φ＋（A＋B）T^f=gf,Finally, applying the method of integral equations and Schauder fixed point theorem, existence of the solution to the above mentioned boundary value problem has already be proved. At the same time the precisely integral expression of its solution is also obrained F（）=1∫（ζ-）n（ζ）φ（ζ）dS＋~Tf（）,Γ.