vi ABSTRACT
not necessarily satisfying a structure assumption as (0.3). The partial regular-
ity rather than the everywhere one, is natural since even in the non-degenerate
case, when considering systems with general structure, singularities may occur.
The proof of the almost everywhere regularity of solutions is then achieved via
an extremely delicate combination of local linearization methods, together with a
proper use of DiBenedetto’s intrinsic geometry: the general approach that consists
in performing the local analysis by considering parabolic cylinders whose space-
time scaling depend on the local behavior of the solution itself. The combination
of these approaches was exactly the missing link to prove partial regularity for gen-
eral parabolic systems considered in (0.1). In turn, the implementation realizing
such a matching between the two existing theories is made possible by the p-caloric
approximation lemma. More precisely, the proof involves two different kinds of
linearization techniques: a more traditional one in those zones where the system
is non-degenerate and the original solution is locally compared to solutions of a
suitable linear system, and a degenerate one in the zones where the system is truly
degenerate and the solution can be compared with solutions of systems as (0.2) via
the p-caloric approximation lemma.
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