积分方程PS-3与裤子模量
Integral equations PS-3 and moduli of pants
摘要：
100多年前，H.Poincare和V. A. Steklov考虑了一个在边界条件中带谱参数的Laplace方程的问题。今天，在各种情况下出现了类似的问题，为两个相邻的域的光谱参数的条件下的域的公共边界上：在理由和优化的区域分解方法，简单的二维模型的石油开采，（热）复合材料的导电性。将二维问题归结到区域的公共边界上，自然得到带谱参数的奇异一维积分Poincare-Steklov方程。我们提出了一个建设性的表示的特征值和本征函数的积分方程的显式构造裤子，一个最简单的Riemann曲面的边界的模量。将求解积分方程的问题归结为求解三个先验方程，其中三个先验方程的未知数为裤子的模量、未知数。方程的离散谱与B.Maskit（1969）、D.Hejhal（1975）和D.Sullivan- W.Thurston（1983）发明的某些外科手术程序（“移植”）有关。
英文摘要：
More than a hundred years ago H.Poincare and V.A.Steklov considered a problem for the Laplace equation with spectral parameter in the boundary conditions. Today similar problems for two adjacent domains with the spectral parameter in the conditions on the common boundary of the domains arises in a variety of situations: in justification and optimization of domain decomposition method, simple 2D models of oil extraction, (thermo)conductivity of composite materials. Singular 1D integral Poincare-Steklov equation with spectral parameter naturally emerges after reducing this 2D problem to the common boundary of the domains. We present a constructive representation for the eigenvalues and eigenfunctions of this integral equation in terms of moduli of explicitly constructed pants, one of the simplest Riemann surfaces with boundary. Essentially the solution of integral equation is reduced to the solution of three transcendent equations with three unknown numbers, moduli of pants. The discreet spectrum of the equation is related to certain surgery procedure ('grafting') invented by B.Maskit (1969), D.Hejhal (1975) and D.Sullivan- W.Thurston (1983).
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