论文标题
一种解决椭圆形库奇问题的迭代方法
An iterative method for solving elliptic Cauchy problems
论文作者
论文摘要
我们在常规套装$ω\子集r^2 $的情况下研究了椭圆运算符的凯奇问题，这是一个经典的问题的经典示例。 Cauchy数据是在子集$γ\ subset \partialΩ$上给出的，我们的目标是重建椭圆方程的$ h^1（ω）$溶液的痕迹，该方程在$ \ partialω/γ$上。此处描述的方法是Maz'ya等人开发的算法的概括。 [MA]对于Laplace操作员，他提出了一种基于连续良好的混合边界价值问题（BVP）的方法，该方法是用作边界数据的一部分。在情况下，我们提供了一个带有$ C^\ infty $ cefficients的线性椭圆操作员的算法的替代收敛证明。我们还为特殊的非线性问题提供了一些数值实验，并且获得的结果非常有利。
We investigate the Cauchy problem for elliptic operators with $C^\infty$-coefficients at a regular set $Ω\subset R^2$, which is a classical example of an ill-posed problem. The Cauchy data are given at the subset $Γ\subset \partialΩ$ and our objective is to reconstruct the trace of the $H^1(Ω)$ solution of an elliptic equation at $\partial Ω/ Γ$. The method described here is a generalization of the algorithm developed by Maz'ya et al. [Ma] for the Laplace operator, who proposed a method based on solving successive well-posed mixed boundary value problems (BVP) using the given Cauchy data as part of the boundary data. We give an alternative convergence proof for the algorithm in the case we have a linear elliptic operator with $C^\infty$-coefficients. We also present some numerical experiments for a special non linear problem and the obtained results are very promisive.
