论文标题
通过几何扰动在二维域中通过几何扰动的近似
Approximation of point interactions by geometric perturbations in two-dimensional domains
论文作者
论文摘要
我们提出了具有点相互作用的平面域中二阶椭圆算子的新型近似值。它具有几何性质，近似家庭由具有相同符号和规则系数的操作员组成，域上有一个小孔。在它的边界处，罗宾条件是用系数施加的，该系数取决于孔的线性大小。我们表明，随着孔缩小到一个点，边界条件中的参数以适当的方式，非线性和单数缩放时，指示的家族以范围内的意义收敛到操作员，并通过点相互作用收敛。相对于几个运算符规范和订单 - 转换率的估计，建立了这种分解收敛。
We present a new type of approximation of a second-order elliptic operator in a planar domain with a point interaction. It is of a geometric nature, the approximating family consists of operators with the same symbol and regular coefficients on the domain with a small hole. At the boundary of it Robin condition is imposed with the coefficient which depends on the linear size of a hole. We show that as the hole shrinks to a point and the parameter in the boundary condition is scaled in a suitable way, nonlinear and singular, the indicated family converges in the norm-resolvent sense to the operator with the point interaction. This resolvent convergence is established with respect to several operator norms and order-sharp estimates of the convergence rates are provided.
