带位移Markushevich问题的求解

Some solutions of Markushevich problem with shift

研究退化情形下一般单连通域的带位移的Markushevich问题的求解过程 ,指出了其可解条件及解的个数 ,给出了问题解的表达式 ,并在一些给定条件下 ,得出上述问题的封闭形式解 .所得结论包含了G .S .Litvinchuk的相关工作 。

In this paper,the Markushevich problem with shiftΦ +=G 1(t)Φ -(t)+G 2(t) Φ -(t) +f(t), t∈Γ,is investigated in the class of piecewise analytic functions. The boundary Γ is a simple closed Lyapunov curve in complex plane C , let D + be the interior domain, and let D -=C\ D + ,α(t) is a homeomorphism onto itself which preserves or changes the orientation of Γ .The coefficients G 1(t),G 2(t),f(t) belong to H μ(Γ) .When the degenerate condition |G 1(t)| =|G 2(t)|≠0 is satisfied,the solvable condition of the Markushevich problem and the number of its solution are discussed. Meanwhile, the author established the closed form of the solution of problem above, if some conditions are satisfied. Several well known important conclusions on Markushevichs problem, such as Noether theorem in the degeneration, Wang Chuan rong result etc, can also be obtained as an immediate consequence of our results.