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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