4 FRANK DUZAAR, GIUSEPPE MINGIONE AND KLAUS STEFFEN
where the main simplifying point is the absence of the dependence on the variable
u in the vector field a(·); therefore we assume
(1.8)





|a(z, w)| +
(
1 +
|w|2
)
1
2
|∂wa(z, w)| L
(
1 +
|w|2
)
p−1
2
∂wa(z, w) ˜ w, ˜ w ν
(
1 +
|w|2
)
p−2
2
|
˜|2
w
for any z ΩT and w, ˜ w RNn, where p 2, 0 ν 1 L. The reason for
this restriction will be soon clarified. The main assumption coming into the play
at this stage is the older continuity of the vector field a(·) with respect to the
“coefficients” (x, t), i.e.
(1.9) |a(x, t, w) a(x0,t0,w)| L |x x0| + |t t0|
β
(1 +
|w|2)
p−1
2
,
whose rate β directly determines the possibility to estimate the Hausdorff dimension
of the singular set. Specifically, our main result in this respect is
Theorem 1.3. Let u C0(−T, 0; L2(Ω, RN )) Lp(−T, 0; W 1,p(Ω, RN )) be a
weak solution to the system (1.7) under the assumptions (1.8)–(1.9), with
(1.10) max 1
2
p
,
1
2
β 1 .
Then, denoting by Σu = ΩT \ Ωu the singular set of u in the sense of Theorem 1.1,
we have that
(1.11) dimpar(Σu) n + 2
(2β 1)p
p 1
n + 1 .
Note that the assumptions considered in the previous result are slightly stronger
than those considered in Theorem 1.1. We do not known whether the bound in
(1.11) is natural or not, although its shape could depend on the technique we are
using. Indeed, under the assumptions (1.8)–(1.9), the bounds available in the case
p = 2 turns out to be better than (1.11) when this last one is particularized for
p = 2, see [33], moreover, no lower bound is required on β; anyway, look at Remark
1.4 below. On the other hand we point out that in the case p 2 the techniques
available in the case p = 2 do not any longer apply, since, as already mentioned in
the previous paragraph, the systems in question scale differently in the space and
time variables: while the evolutionary part scales quadratically - when considered
with respect to the standard parabolic metric (1.4) - the diffusive part does not, and
solutions follow a sort of local intrinsic geometry as explained in the basic work of
DiBenedetto [25, 23, 24]. As a matter of fact, the worsening of (1.10)-(1.11) when
p increases reflects the following circumstance: we are trying to prove singular sets
estimates with respect to a fixed standard parabolic metric, that is the one (1.4),
while, on the other hand, the natural metric followed by the system is adaptive
with respect to the solution in the sense that if |Du| λ 1 in a suitably defined
small cylinder - see Paragraph 5.2 and Lemma 5.6 below - then in such cylinder the
natural metric associated to the system according to [25] is
(1.12) dpar,λ((x, t), (y, s)) := max |x y|, λp−2|t s| .
Clearly, the more p differs from 2, the more dpar,λ differs from dpar, hence the
worsening of (1.10)-(1.11) when p gets large. In some sense one should perhaps
compute the singular sets estimate using this metric, something which is hardly
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