Introduction
In this memoir we aim at presenting, in a unified way, various and intercon-
nected basic aspects of the regularity theory of solutions to general, non-linear
second order parabolic systems with polynomial p-growth
p 2 .
Specifically, we are dealing with systems of the type
(0.1) ut div a(x, t, u, Du) = 0
and related non-homogeneous ones
ut div a(x, t, u, Du) + H = 0 , H D (ΩT ) ,
under natural p-growth and ellipticity assumptions on the vector field
a: ΩT ×
RN
×
RNn

RNn
that is
(0.2) |a(x, t, u, w)| L(1 +
|w|2)
p−1
2
holds for every (x, t) ΩT , u
RN
and w
RNn.
The system in question is
considered in the cylindrical domain
ΩT = Ω × (−T, 0) ,
where Ω denotes a bounded domain in
Rn,
n 2, and T 0; in the following Du
will always denote the gradient with respect tot he space variables x, i.e. Du(x, t)
Dxu(x, t). The precise assumptions will be every time specified when presenting
the various theorems obtained in the paper; such assumptions will be in most of
the cases natural and optimal with respect to the implied conclusions. The notion
of (weak) solution adopted here, and in the rest of the paper, prescribes of course
that a map
u
C0(−T,
0;
L2(Ω, RN
))
Lp(−T,
0; W
1,p(Ω, RN
))
is a (weak) solution to (0.1), under the assumption (0.2) for p 2, iff
ΩT
u · ϕt a(x, t, u, Du),Dϕ dx dt = H, ϕ
holds for everyϕ C0
∞(ΩT
,
RN
).
We shall provide three basic types of interconnected regularity results for so-
lutions to (0.1), and in fact one of our aims is also to give a unifying approach to
different regularity issues, emphasizing the interactions between the various techni-
cal aspects of the problems treated. We shall give optimal partial regularity results
for solutions, the first results on the Hausdorff dimension size of the singular sets
ix
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