In this paper, we are interested in solving an elliptic inverse Cauchy problem. As it is well known this problem is one of highly ill posed problem in Hadamard’s sense (Hadamard, 1953). We first establish formally a relationship between the Cauchy problem and an interface problem illustrated in a rectangular structure divided into two domains. This relationship allows us to use classical methods of non-overlapping domain decomposition to develop some regularizing and stable algorithms for solving elliptic inverse Cauchy problem. Taking advantage of this relationship we reformulate this inverse problem into a fixed point one, based on Steklov–Poincaré operator. Thus, using the topological degree of Leray–Schauder we show an existence result. Finally, the efficiency and the accuracy of the developed algorithms are discussed.

在本文中，我们有兴趣解决椭圆逆柯西问题。众所周知，从哈达玛的角度来看，这个问题是病态严重的问题之一（哈达玛，1953年）。我们首先正式建立柯西问题和以矩形结构（分为两个区域）所示的界面问题之间的关系。这种关系使我们能够使用经典的非重叠域分解方法来开发一些正则化和稳定的算法来解决椭圆逆柯西问题。利用这种关系，我们根据Steklov–Poincaré算子将该逆问题重新构造为一个固定点。因此，利用Leray–Schauder的拓扑度，我们证明了存在的结果。最后，讨论了所开发算法的效率和准确性。