具有频繁非周期交换边界条件的Laplacian特征元的渐近性与估计
Asymptotics and Estimates for Eigenelements of Laplacian with Frequent Nonperiodic Interchange of Boundary Conditions
摘要：
本文考虑二维区域上Laplace算子的奇异摄动特征值问题。在边界中，我们选择一个集合，它取决于一个字符小参数，并且由大量不相交的小部分组成。在这一组的Dirichlet边界条件的实施，而对其余部分的边界，我们施加诺依曼条件。对于均匀化Neumann或Robin边值问题的情况下，我们得到了非常弱的限制的分布和边界Dirichlet部分的边界下，我们设法得到的领先项的渐近展开的扰动特征元的长度。我们为这些项提供了明确的公式。在较弱的假设下，我们估计扰动特征值的收敛程度。
英文摘要：
We consider singular perturbed eigenvalue problem for Laplace operator in a two-dimensional domain. In the boundary we select a set depending on a character small parameter and consisting of a great number of small disjoint parts. On this set the Dirichlet boundary condition is imposed while on the rest part of the boundary we impose the Neumann condition. For the case of homogenized Neumann or Robin boundary value problem we obtain highly weak restrictions for distribution and lengths of boundary Dirichlet parts of the boundary under those we manage to get the leading terms of asymptotics expansions for perturbed eigenelements. We provide explicit formulae for these terms. Under weaker assumptions we estimate the degrees of convergence for perturbed eigenvalues.
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