The asymptotic shift for the principal eigenvalue for second order elliptic operators in the presence of small obstacles
The asymptotic shift for the principal eigenvalue for second order elliptic operators in the presence of small obstacles
复制标题
存在小障碍物时二阶椭圆算子主特征值的渐近平移
DOI：
10.1007/s11856-009-0009-x
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发表时间：
2008
影响因子：
1
通讯作者：
Iddo Ben
中科院分区：
文献类型：
--
作者：
Iddo Ben
设L是一个二阶一致椭圆微分算子，定义在d≥ 2的有界光滑区域上，具有Dirichlet边界条件或Dirichlet-导数边界条件.本文研究了L在硬障碍和软障碍扰动下的主特征值的渐近性。通过在有限个孔的边界上设置Dirichlet边界条件，得到了L的硬障碍摄动。主要结果给出了主特征值随孔洞收缩为点的渐近位移。速率表示的牛顿容量的孔和未扰动的运营商及其正式伴随的主要特征函数。软障碍对应于有限个紧支撑的有限势威尔斯阱。这里我们只考虑拉普拉斯算子的导数。与硬障碍问题的主要区别在于，由于各种缩放可能性，相变发生。我们的结果推广了自伴算子相似扰动的已知结果。我们的方法是概率性的。
Let L be a uniformly elliptic second order differential operator with nice coefficients, defined on a smooth, bounded domain in ℝd,d≥ 2, with either the Dirichlet or an oblique-derivative boundary condition. In this work we study the asymptotics for the principal eigenvalue of L under hard and soft obstacle perturbations. The hard obstacle perturbation of L is obtained by making a finite number of holes with the Dirichlet boundary condition on their boundaries. The main result gives the asymptotic shift of the principal eigenvalue as the holes shrink to points. The rates are expressed in terms of the Newtonian capacity of the holes and the principal eigenfunctions for the unperturbed operator and its formal adjoint. The soft obstacle corresponds to a finite number of compactly supported finite potential wells. Here we only consider the oblique-derivative Laplacian. The main difference from the hard obstacle problem is that phase transitions occur, due to the various scaling possibilities. Our results generalize known results on similar perturbations for selfadjoint operators. Our approach is probabilistic.