In this paper, we study some regularity issues concerning the gradient of weak solutions of $u_t - { \rm div} \mathcal{ A}(x,t,\nabla u) = g$, where $\mathcal{ A}(x,t,\nabla u)$ is modeled after the $p$-Laplace operator. The main results we are interested in is to obtain optimal conditions on the datum $g$ (independent of time) such that borderline higher integrability of the gradient and Lipschitz estimates for the weak solution holds. Moreover, we develop a theory where we can obtain elliptic type estimates using parabolic theory, which gives improved potential estimates for the elliptic systems.
Categories: math.AP