When I started giving talks on regularity theory for degenerate and sin- lar parabolic
equations a ?xed-point in the conversation during the co?- break that usually followed the
seminar was the apparent contrast between the beauty of the subject and its technical di?culty.
I could not agree more on the beauty part but most of the times overwhelmingly failed to
convince my audience that the technicalities were not all that hard to follow. As in many other
instances it was the fact that the results in the literature were eventually stated and proved
in their most possible generality that made the whole subject seem inexpugnable. So when I had
the chance of preparing a short course on the method of intrinsic scaling I decided to present
the theory from scratch for the simplest model case of the degenerate p-Laplace equation and to
leave aside technical re?nements needed to deal with more general situations. The ?rst part of
the notes you are about to read is the result of that e?ort: an introductory and
self-containedapproachtointrinsicscaling aimingatbringingtolightwhatis really essential in this
powerful tool in the analysis of degenerate and singular equations. As another striking feature
of the method is its pervasiveness in terms of the applications in the second part of the book
intrinsic scaling is applied to several models arising from ?ows in porous media chemotaxis
and phase transitions.
