ca.analysis and odes - Transformation from domains to half-spaces

The half spaces R^n are the inward-pointing halves of the tangent spaces to the points on the boundary.
Starting with a local boundary value problem you end up with a model equation on the half space associated to each point on the boundary by "localizing" at the point.
A solution to the elliptic boundary value problem would give you also a solution to each of these model problems.
Conversely, you can construct a formal solution to the elliptic boundary value problem at the boundary by using these model problems.
Then you can improve this formal solution to an `honest' solution by solving away the error; this is easier because you no longer have to worry about the boundary.

This system of ODEs is used to state the Lopatinski-Shapiro condition, and searching for this will yield many references. My favorite is Hormander's analysis of partial differential operators -- this topic in particular is best explained in the first edition.