解析函数带位移的复合边值问题

ON COMPOUND BOUNDARY PROBLEM OF ANALYTIC FUNCTION WITH SHIFTS

县1引言设区域D是、+,十1连通区域,边界由简单封闭曲线L0,Ll,…,L。,rlI’,组成,L。把其他曲线围在其内部.记L=U i=0单封闭曲线l:,…,几,儿,…,f,.记l=L,,,一日,1八 j=1在D内有若干条互相外离的简f一日几.L,r,l,f都是Lja- 友=punov曲线.

Let D be a m+n+1 connected domain bounded by closed Ljapunov curves L= Li,T=,and in the interior of D there are some disjoint closed Ljapunov curves l=.The domain D is partitioned by l,f into two parts:D^+ and D^-. In this paper,the Haseman,Haseman-type,Carleman,Carleman-type compound boundary problem is considered: Find a piecewise analytic ruction Φ(z)in D(i.e,Φ(z)is analytic in D^+,D^- and continuous to f,l on each side of them)such that Φ^+[η(t)]=G(t)Φ^-(t)+g(t),for t∈f, Φ^++[γ(t)]=G(t)+g(t),for t∈l, simultaneously Φ(z)be continuous to L,Γfrom the interior of D and satisfy Φ+[β(t)]=G(t)^+(t)+g(t),for t∈Γ, Φ^+[α(t)]=G(t),for t∈L, where G(t),g(t)EH and G(t)(?)0,for t∈lUf, G(t)G[β(t)]=1,G(t)g[β(t)]+g(t)=0,for t∈Γ, G(t)G[α(t)=1,G(t)g[α(t)]+g(t)=0,for t∈L, γ(t),β(t)are reverse shifts on l,Γ respectively and η(t),α(t)are positive shifts on f,L respectively,α[α(t)]≡t,β[β(t)]≡t,α'(t)0,β'(t)0,η'(t) 0,γ~'(t)0 and α'(t),β'(t),η'(t),γ'(t)∈H. This problem may be solved by conformal mapping and elimination: first this problem is transformed into R,C-type problem by conformal mappi- ng,then R,C-type problem is reduced to C-type problem by elinination,and finally the solvability theorem of thisproblem according to the solvability theorem of C-type problem is obtained.