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

Abstract

Listen

Translate article

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.