Analysis on Singular Spaces: Applications of Operator Algebras to Boundary Value Problems

The method of layer potentials is one of the classical approaches to solving boundary value problems for (strongly) elliptic equations. This method reduces the original problem to that of inverting an operator of the form "1/2 +K" on appropriate boundary function spaces. Recently, this method has attacted considerable attention both in the theoretical and the applied mathematics. This dissertation deals with the method of layer potentials on domains with conical points from a groupoid point of view. By a desingularization process and integration of a Lie algebroid, we can construct a Lie groupoid that encodes the geometry and singularities of this domain. Then we identify the double layer potential operator K with an invariant family of that Lie groupoid. From this, we can obtain the Fredholmness of the operator "1/2+K".