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DCDS-S

We are interested in regularity results, up to the boundary, for the
second derivatives of the solutions of some nonlinear systems
of partial differential equations with
$p$-growth. We choose two representative cases: the ''full gradient
case'', corresponding to a $p$-Laplacian, and the ''symmetric
gradient case'', arising from
mathematical physics. The domain is either the ''cubic domain''
or a bounded open subset of $\mathbb{R}^3$ with a smooth boundary. Depending
on the model and on the range of $p$, $p<2$ or $p>2$, we prove
different regularity results. It is worth noting that in the full
gradient case with $p<2$ we cover the singular case and obtain
$W^{2,q}$-global regularity results, for arbitrarily large values of
$q$. In turn, the regularity achieved implies the Hölder
continuity of the gradient of the solution.

DCDS

Starting from the partial regularity results for suitable weak solutions to the Navier-Stokes Cauchy problem by Caffarelli, Kohn and Nirenberg [

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