x FRANK DUZAAR, GIUSEPPE MINGIONE AND KLAUS STEFFEN
of solutions, and the first Calder´ on-Zygmund type results on the higher integrabil-
ity of solutions to general non-linear parabolic systems, also treating systems with
possibly discontinuous coefficients.
More precisely we introduce new methods allowing to give the best possible
forms of the rather preliminary versions of the desired regularity results already
present in the literature, on one hand. On the other hand we hereby face several
untouched issues: we give the first answers to a few non-clarified and difficult
aspects of regularity theory such as the singular set reduction and general sharp
gradient integrability estimates. Specifically: the partial regularity results in the
interior are achieved via the method of A-caloric approximation, which extends the
classical DeGiorgi’s harmonic approximations lemma [21, 72]. Here such a method,
originally introduced in [33] for quadratic growth parabolic systems i.e. p = 2, is
extended to cover general systems with polynomial super-quadratic growth p 2
and allows to deduce optimal partial regularity results for solutions in a direct
way, without employing tools as reverse older inequalities. In turn, the singular
set estimates are obtained via a novel comparison method to deduce space-time
fractional differentiability.
Finally, the gradient higher integrability results are new even for scalar para-
bolic equations that is when considering scalar valued solutions N = 1, or when
applied to the basic model given by the classical evolutionary p-Laplacean system
with (possibly discontinuous) coefficients
ut div (e(x,
t)|Du|p−2Du)
= H .
Moreover we find the optimal form, in the case of general non-linear parabolic
equations and of p-Laplacean type systems, of results available up to now only for
linear parabolic problems. Such results are obtained with Harmonic Analysis-free
proofs, since classical tools as singular integrals and maximal operators cannot be
used for evolutionary problems with polynomial growth. Nevertheless some intrinsic
principles of Harmonic Analysis such as local representation formulas and stopping
time arguments will be employed directly at the suitable pde level. A main point
of the paper is to show a central bulk of techniques that simultaneously apply to
the three main issues treated here.
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