椭圆方程的边值问题是数学物理中一类特别重要的问题。本项目拟用李群胚的C*-代数方法来研究椭圆方程在不光滑区域或者带有奇点区域上的边值问题。当用传统的位势方法求解椭圆方程的边值问题时，会产生两个经典的算子：单位势算子S和双位势算子K，这两个算子都是边界上的积分算子。同时我们会得到一些边界上的积分方程，需要研究算子1/2+K和S在边界上的某些函数空间上的可逆性。本项目从边界的不光滑性或者奇性出发，通过对边界的几何性质的研究，构造出一个李群胚，使得S和K包含在这个李群胚的C*-代数中。进而对算子 1/2+K和S进行谱分析，得到这两个算子在边界上的某些加权Sobolev空间上的Fredholm性质。

Boundary value problems for elliptic equations are very important in mathematics and physics. This project is to use Lie groupoid C*-algebras to study elliptic boundary value problems on non-smooth domains. When solving elliptic boundary value problems via the method of layer potentials, we obtain two classical operators: single layer potential operator S and double layer potential operator K, where S and K are boundary integral operators. The virtue of layer potential method is to reduce differential equations to integral equations on the boundary. We need to invert the operators 1/2+K and S on some appropriate function spaces on the boundary. In this project, we associate a canonical Lie groupoid to the non-smooth or singular boundary, such that S and K belong to the C*-algebra of this Lie groupoid, and study the spectra of 1/2+K and S to obtain the Fredholmness of these two operators between suitable weighted Sobolev spaces.