2 FRANK DUZAAR, GIUSEPPE MINGIONE AND KLAUS STEFFEN
describing in a few lines, very few results are available for parabolic systems, and
a natural extension to the parabolic case of the elliptic partial regularity results is
still missing. As a matter of fact, the first result we are going to present gives the
precise extension to the case of parabolic systems with general polynomial growth
of the optimal partial regularity results available for elliptic systems. To emphasize
the main points, for the sake of simplicity we shall restrict our attention to the case
of homogeneous systems of the type
(1.3) ut div a(x, t, u, Du) = 0
the non-homogeneous case being treatable with a slight generalization of the tech-
niques hereby introduced.
Theorem 1.1. Let u C0(−T, 0; L2(Ω, RN )) Lp(−T, 0; W 1,p(Ω, RN )) be a
weak solution to the system (1.3) under the assumptions (2.2)–(2.5) in Chapter 2
below. Then there exists an open subset Ωu ΩT such that
Du
Cβ,β/2(Ωu, RNn)
and |ΩT \ Ωu| = 0 .
The previous result means that the spatial derivative Du is older continuous
in Ωu with exponent β with respect to the parabolic metric given by
(1.4) dpar((x, t), (y, s)) := max |x y|, |t s| , x, y
Rn,
s, t R .
The set Ωu is called the regular set of u
(1.5) Ωu := z ΩT : Du is
Cβ,β/2-regular
in a neighborhood of z ,
and is open by its very definition, while its complement is called the singular set of
u
(1.6) Σu := ΩT \ Ωu .
Main points of Theorem 1.1 are: first, the older continuity exponent of the spatial
derivative is the optimal one, that is the same as the exponent appearing in (2.5).
This is in clear accordance with the classical Schauder estimates for linear parabolic
equations. Second, and strictly related to the previous point, a full treatment
of coefficients depending on the zero order quantities (x, t, u) under the natural
assumptions (2.5) is given here, something which was not possible with previous
techniques; is fact, assumptions (2.2)–(2.5) considered in the previous theorem are
minimal in order to obtain the claimed result. From a different viewpoint a third
feature of the result actually concerns the technique used. In the basic elliptic case
the proofs of partial regularity of solutions which seems to be most suitable for
parabolic extensions typically involve the use of higher integrability of solutions,
and specifically of reverse older inequalities in the style of Gehring’s lemma; see for
instance [39] while a reverse older inequality free approach valid for the elliptic
case is in [42]. On the other hand no reverse older inequality is available for
solutions to (1.3) unless p = 2; see for instance the recent interesting papers of
Kinnunen & Lewis [48, 49] and ogelein [7, 9]. The reason is that the presence in
(1.3) of both an evolutionary - that is ut - and a diffusion term with p-growth - that
is div a(z, u, Du) - makes the system exhibit a lack of scaling that in turn reflects
in the fact that if u is a solution, then cu, with c R, is not any longer a solution
of a similar system. This phenomenon clearly rules out the possibility of getting
reverse-type inequalities, since they are homogeneous in nature. Indeed, the higher
integrability estimates available for solutions to (1.3) exhibit an inhomogeneity
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