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
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 ,
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