Approximation of Dirichlet Eigenvalues on Domains with Small Holes
Approximation of Dirichlet Eigenvalues on Domains with Small Holes
复制标题
小孔域上狄利克雷特征值的逼近
DOI：
10.1006/jmaa.1995.1228
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发表时间：
1995
影响因子：
1.3
通讯作者：
M. Flucher
中科院分区：
文献类型：
--
作者：
M. Flucher
摘要导出并讨论了具有Dirichlet边界条件的拉普拉斯算子在小孔区域上的特征值的近似公式。自由振动板和支承振动板的相应结果如下。基于Courant最小-最大原理给出了一个概念上简单的证明。作者对小球容量的近似得到了具有球孔的区域的高精度特征值近似公式。一般的空穴用调和校正法、容量与体积的等周不等式和容量势的庞加莱不等式来处理。另外，我们给出了特征函数的L∞界。
Abstract Approximation formulas for the eigenvalues of the Laplacian with Dirichlet boundary conditions on domains with small holes are derived and discussed. The corresponding results for the free and the supported vibrating plate follow. A conceptually simple proof is given based on Courant′s min-max principle. The author′s approximation for the capacity of small balls leads to a highly accurate eigenvalue approximation formula for domains with spherical holes. General holes are treated by means of a harmonic correction method, an isoperimetric inequality relating capacity to volume, and a Poincare inequality for capacity potentials. In addition we provide L∞-bounds for the eigenfunctions.