Asymptotic Behavior of Stekloff Eigenvalues and Eigenfunctions
Asymptotic Behavior of Stekloff Eigenvalues and Eigenfunctions
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Stekloff 特征值和特征函数的渐近行为
DOI：
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发表时间：
1971
期刊：
影响因子：
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通讯作者：
S. Shamma
中科院分区：
文献类型：
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作者：
S. Shamma
我们考虑拉普拉斯方程的解 $u_n (x,y)$，该解在光滑闭合平面曲线 c 的内部是正则的，边界条件 $partial u_n /partial v = lambda _n gu_n $，其中 g 足够光滑、正、周期且为弧长的指定函数，$u_n $、$lambda _n $ 是特征函数和特征值 待定。对于大的 $lambda _n $，我们证明 $lambda _n = O(n)$，n 是一个大整数，并且 $u_n $ 是渐近三角函数。所采用的方法是将问题简化为边界积分方程并研究该方程。结果在通过 $g(s)$ 和曲线 c 的特殊选择而产生的可分离情况下得到证实。
We consider solutions $u_n (x,y)$ of Laplace’s equation which are regular in the interior of a smooth closed plane curve c, and the boundary conditions $partial u_n /partial v = lambda _n gu_n $, where g is sufficiently smooth, positive, periodic and a prescribed function of arclength, and $u_n $, $lambda _n $ are eigenfunctions and eigenvalues to be determined. For large $lambda _n $ we show that $lambda _n = O(n)$, n a large integer, and that $u_n $ is trigonometric, asymptotically. The method employed is the reduction of the problem to a boundary integral equation and the studying of that equation. The results are confirmed in a separable case which arises by a special choice of $g(s)$ and the curve c.