4 1. INTRODUCTION AND RESULTS
or

Q(λ)
Du dz λ .
The problematic aspect in (1.12) clearly relies in the fact that the value of the
integral average must be comparable to a constant which is in turn involved in the
construction of its support
Q(λ)

Q(λ)(z0),
exactly according to (1.9).
Let us now briefly turn to the case of standard non-degenerate systems with
linear growth i.e. p = 2, to recall the known approaches of partial regularity
and local linearization. On the side of classical partial regularity proofs, the
main technique usually employed is basically a linearization one. The basic idea
can be now summarized as follows: A point z0 ΩT is by definition regular iff
the oscillations of the gradient of the solution are small in a quantifiable way in
a neighborhood of it. Vice versa, the viewpoint of partial regularity is that this
situation is achieved provided the oscillations of the gradient are a priori small in
a neighborhood of the point in question, this smallness being measured, as usual,
in an averaged integral way. Indeed, functionals as the mean square deviation of
the gradient with respect to its average are useful at this stage to express a small
oscillation property. The basic assertion of partial regularity is now that a point z0
is regular iff a smallness condition of the type
(1.13)
Q
|Du
(Du)Q|2
dz ε , with (Du)Q
Q
Du dz ,
is satisfied for a standard cylinder Q = Q (z0) centered at z0. Here the number ε
implying the regularity of the gradient in general depends on the structure condi-
tions imposed on the system, and in most of the cases also on the point z0 where Q
is centered. Condition (1.13) is in turn used to implement a comparison argument
aimed at comparing the original solution u to the solution v of a linear parabolic
system with constant coefficients of the type
(1.14) ∂tv = div
(
Da
(
(Du)Q
)
Dv
)
,
with v agreeing with u on the parabolic boundary of Q, or at least with v close in an
integral sense to u. The role of a smallness assumption as (1.12) is then to quantify
the closeness of the original solution to an affine map whose coefficient is given
by (Du)Q, so that the system (1.14) can be considered as a Taylor approximation
of the original system. On the other hand, since (1.14) is a linear system, good
regularity estimates are available for the solution v and in series such estimates can
be conveyed to u, thereby proving that Du is older continuous in a neighborhood
of the center z0 of Q. It is of course at the core of partial regularity to prove
that a smallness condition formulated in an integral way as in (1.13) is sufficient to
make the whole machinery work and to prove the older continuity of Du. This
is the standard approach in elliptic and parabolic regularity theory: to commute
integral bounds on the oscillation of Du in
L2
or something near it in pointwise
bounds, that is in
L∞.
Needless to say, since a condition of the type (1.13) is
only satisfied almost everywhere, the above techniques ultimately lead to almost
everywhere regularity in the sense of (1.1).
The first of the methods outlined in the preceding lines, i.e. the intrinsic scaling
method, is, as already mentioned above, at the core of DiBenedetto’s viewpoint on
parabolic regularity and allows for the proof of interior regularity of solutions to
systems as (1.5), while the linearization method used to prove partial regularity
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