multiply connected圆形域的Hilbert问题及板的霍尔效应分析
Hilbert problem for a multiply connected circular domain and the analysis of the Hall effect in a plate
摘要：
本文分析了$(N+1)$连通圆形域上解析函数理论的Hilbert边值问题。对于连续系数的情况，已经导出了一个精确的级数形式的解。出于对多连接板中霍尔效应的研究，我们通过研究不连续系数的情况来扩展这些结果。希尔伯特问题映射到对称分段亚纯函数关于对称肖特基群不变的黎曼-希尔伯特问题。这个问题的解决方案是来自柯西核，拟自守和拟乘法内核的两个类似物。前一个核对于任何对称肖特基群都是已知的。我们证明了存在定理的第二，quasimultiplicative，内核的任何肖特基组（其系列表示是已知的第一类群体）。我们还表明，使用的自守内核需要解决相关的真正的模拟的Jacobi反演问题，可以绕过，如果我们采用的quasiautomorphic和quasimultiplicative内核。我们应用这一理论的模型稳态问题上的运动的带电电子在一个板与$N+1$圆形孔与电极和电介质的墙壁上时，导体被放置在直角施加的磁场。
英文摘要：
In this paper we analyze the Hilbert boundary-value problem of the theory of analytic functions for an $(N+1)$-connected circular domain. An exact series-form solution has already been derived for the case of continuous coefficients. Motivated by the study of the Hall effect in a multiply connected plate we extend these results by examining the case of discontinuous coefficients. The Hilbert problem maps into the Riemann-Hilbert problem for symmetric piece-wise meromorphic functions invariant with respect to a symmetric Schottky group. The solution to this problem is derived in terms of two analogues of the Cauchy kernel, quasiautomorphic and quasimultiplicative kernels. The former kernel is known for any symmetry Schottky group. We prove the existence theorem for the second, quasimultiplicative, kernel for any Schottky group (its series representation is known for the first class groups only). We also show that the use of an automorphic kernel requires the solution to the associated real analogue of the Jacobi inversion problem which can be bypassed if we employ the quasiautomorphic and quasimultiplicative kernels. We apply this theory to a model steady-state problem on the motion of charged electrons in a plate with $N+1$ circular holes with electrodes and dielectrics on the walls when the conductor is placed at right angle to the applied magnetic field.
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