The Einstein field equations have been subject of research in theoretical physics, and differential geometry, apparently with methods from classical analysis and geometry. In particular, solutions in closed form have been of interest.

It seems that the classical programme of the PDE community, i.e., (i) existence (ii) uniqueness (iii) regularity, heavily employing concepts from functional analysis, has not found prominent application in general relativity.

Why is this the case? Is it simple due to the Sobolev theory on manifolds still being rather fresh, or do serious technical obstacles exist? Or have I overlooked something?