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 H¨ 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 H¨ 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