On the global regularity for nonlinear systems of the $p$-Laplacian type

Abstract

We are interested in regularity results, up to the boundary, for thesecond derivatives of the solutions of some nonlinear systems of partial differential equations with$p$-growth. We choose two representative cases: the ''full gradientcase'', corresponding to a $p$-Laplacian, and the ''symmetricgradient 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. Dependingon the model and on the range of $p$, $p2$, we provedifferent regularity results. It is worth noting that in the fullgradient 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 Höldercontinuity of the gradient of the solution.