CHAPTER 1
Results
We shall present here the detailed description of the results announced in the
Introduction - partial regularity, singular sets estimates and extended Calder´on-
Zygmund theory - together with detailed comments on technical points involved in
the proofs, and on how the given results fit in the current literature. For precise
definitions and general notation we refer the reader to Chapter 2.
1.1. Partial regularity
Partial regularity describes that the gradient of the solution Du is regular,
namely older continuous, but only outside a negligible closed subset of ΩT , in
fact called “the singular set”. As nowadays widely understood, when dealing with
general systems as the one appearing in (1.3), and its elliptic/stationary version
div a(x, t, u, Du) = 0 ,
solutions might exhibit singularities of various types and nature [22, 43, 45, 46,
73, 75], and therefore the general strategy consists of first proving the H¨older
continuity of the gradient outside the singular set, and then eventually obtaining
estimates for the size of such singular set. This phenomenon concerns general
systems, since also in the parabolic case everywhere regularity is typical only when
dealing with special structures as in the fundamental contributions of DiBenedetto
[25], who, when considering the vectorial case N 1, mainly dealt with the special
evolutionary p-Laplacean system
(1.1) ut div
(|Du|p−2Du)
ut pu = 0 .
See also the work of Wiegner [77], and that of Uhlenbeck [76] for the basic elliptic
case
pu = 0 .
In such case the systems has a special structure called “almost diagonal”, more
general cases are systems of the type
(1.2) ut div (g(|Du|)Du) = 0 ,
of which the p-Laplacean system (1.1) is the most important instance. In fact, the
results in [76, 25, 77] extend to systems as (1.2).
As mentioned, such everywhere regularity properties of solutions do not any
longer hold in the case of general parabolic as well as elliptic systems, and in
this paragraph we shall concentrate on the first point, that is partial regularity,
while the second one concerning the size of the singular sets will be addressed in
the following paragraph. A general and updated introductory reference at this
stage, for the elliptic case, could be [63] while classical treats are [39, 40]. As
far as we know, mainly due to a few severe technical obstructions we are shortly
1
Previous Page Next Page