In a paper I read, an elliptic boundary value problem
on a bounded domain D x (0,T) is solved by first transforming
it in a set of equations on half-spaces R^n and then applying
partial Fourier transform. Eventually one single ODE is obtained.
It is not explained how this transformation is done and
it seems very mysterious to me. Can someone please give
references for this approach?
I am also interested if it is possible in general
for a single evolution equation to be transformed 
in system of equations on half spaces.
Thank you for all replies.