1.2. SINGULAR SETS ESTIMATES 3

exponent, that is p/2, which is exactly the “scaling deficit” of the system; see

also Paragraph 1.3 below. This circumstance also affects other methods of proof

involving blow-up procedures. To overcome the lack of such a basic technical tool we

shall use an upgraded version of the A-caloric approximation technique, originally

introduced in [33] to treat the case of quadratic growth parabolic systems, that is

when p = 2, and that is now seen to crucially work in the case p 2 in a new ad-

hoc version. A delicate combination of energy estimates and caloric approximations

will finally give the proof of Theorem 1.1. We shall anyway go back later to the

lack of scaling described above, this is indeed of the main leading to consider, when

studying certain regularity issues in non-linear parabolic problems, to consider the

so called DiBenedetto’s intrinsic geometry approach [25].

Remark 1.2. In (1.5) we defined the regular set Ωu as actually the regular

set of Du. Now, while in the elliptic case the regularity of u immediately follows

from that of Du for obvious general reasons, in the parabolic case this is not the

case, since no regularity is achieved in Ωu for the time derivative ut, and it actually

cannot be since solutions of parabolic problems as those considered in this paper

exhibit very low regularity with respect to the time variable. Nevertheless, the

H¨ older continuity of u, with every possible exponent, still follows in the regular set

Ωu. For such issues and related statements we refer to Chapter 7 below, and in

particular to Theorem 7.1.

1.2. Singular sets estimates

After establishing partial regularity in the sense of Theorem 1.1, the first nat-

ural problem is the estimation of the Hausdorff dimension of the singular set Σu

defined in (1.6). Here we are of course using the Hausdorff dimension related to

the Hausdorff measure generated by the parabolic distance in (1.4), see Paragraph

(2.6) below for the precise definition. When using such a measure the “ambient”

dimension is n + 2, which is larger than the topological dimension n + 1 due to

the faster time-direction shrinking of the parabolic metric in (1.4); therefore the

natural question is whether or not

dimpar(Σu) n + 2

that is whether or not the dimension of the singular is strictly less than the ambient

dimension. In the case p = 2 this problem has been settled in [33], a paper whose

techniques in this respect fail to apply to the case p 2; see [61, 62] for elliptic

estimates. The problem of giving an estimate for the singular set in the case p 2

is, as far as we know, unsettled up to now, even in the simpler cases covered in

the literature. In this paper we give a partial answer to such a question, which

applies to a family of parabolic systems with a structure simpler than the general

one in (1.3), and provided suﬃcient regularity on the coeﬃcients is assumed; the

result, we think, is nevertheless significant being actually the first one in this

direction. More precisely we shall consider systems of the type

(1.7) ut − div a(x, t, Du) = 0 in ΩT ,