Asymptotic Analysis of Localized Solutions to Some Linear and Nonlinear Biharmonic Eigenvalue Problems
Asymptotic Analysis of Localized Solutions to Some Linear and Nonlinear Biharmonic Eigenvalue Problems
复制标题
一些线性和非线性双调和特征值问题局部解的渐近分析
DOI：
10.1111/j.1467-9590.2010.00507.x
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发表时间：
2011
影响因子：
2.7
通讯作者：
M. Ward
中科院分区：
文献类型：
--
作者：
M. Kropinski;A. Lindsay;M. Ward
在任意有界的二维区域中，利用奇异摄动方法分析了几个双调和线性和非线性特征值问题的渐近行为，这些问题的解要么由于区域中的空洞，要么由于区域中某些局部区域的非线性不可忽略而表现出集中行为.双调和非线性特征值问题的具体形式是由微机电系统电容器中两个表面之一的稳态偏转的研究激发的。线性本征值问题被认为是计算的频谱的双调和算子在一个区域的内部孔的渐近小半径。在我们的奇摄动双调和问题的分析中，一个关键的新特征，这是在相关的二阶椭圆问题中所不存在的，是必须对首阶外解施加点约束，以渐近匹配解的内外表示。我们的渐近分析也严重依赖于使用对数之字形，臭名昭著的低雷诺数流体流动的研究，和详细的双调和绿色的功能和相关的经常性的奇异性附近的一部分。对于几个简单区域，计算了双调和问题的全数值解，以验证分析所得的渐近结果。
In an arbitrary bounded 2‐D domain, a singular perturbation approach is developed to analyze the asymptotic behavior of several biharmonic linear and nonlinear eigenvalue problems for which the solution exhibits a concentration behavior either due to a hole in the domain, or as a result of a nonlinearity that is nonnegligible only in some localized region in the domain. The specific form for the biharmonic nonlinear eigenvalue problem is motivated by the study of the steady‐state deflection of one of the two surfaces in a Micro‐Electro‐Mechanical System capacitor. The linear eigenvalue problem that is considered is to calculate the spectrum of the biharmonic operator in a domain with an interior hole of asymptotically small radius. One key novel feature in the analysis of our singularly perturbed biharmonic problems, which is absent in related second‐order elliptic problems, is that a point constraint must be imposed on the leading order outer solution to asymptotically match inner and outer representations of the solution. Our asymptotic analysis also relies heavily on the use of logarithmic switchback terms, notorious in the study of Low Reynolds number fluid flow, and on detailed properties of the biharmonic Green’s function and its associated regular part near the singularity. For a few simple domains, full numerical solutions to the biharmonic problems are computed to verify the asymptotic results obtained from the analysis.