A high regularity result of solutions to modified p-Stokes equations

Abstract

This paper is concerned with a special elliptic system, which can be seen as a perturbed $p$-Laplacean system, $p\in(1,2)$, and, for its "shape", it is close to the $p$-Stokes system. Since our "stress tensor" is given by means of $\nabla u $ and not by its symmetric part, then our system is not a $p$-Stokes system. Hence, the system is called {\it modified} $p$-Stokes system. We look for the high regularity of the solutions $(u,\pi)$, that is $D^2u,\nabla\pi \in L^q,q\in(1,\infty)$. In particular, we get $\nabla u,\pi\in C^{0,\alpha}$. As far as we know, such a result of high regularity is the first concerning the coupling of unknowns $(u,\pi)$. However, our result also holds for the $p$-Laplacean, and it is the first high regularity result in unbounded domains.