Non-linear Calderón–Zygmund theory for parabolic systems with subquadratic growth

Abstract

For vector-valued solutions of parabolic systems $$\partial_tu-{\rm div}\, a(x,t,Du)={\rm div}\left(|F|^{p-2}F\right)$$ with polynomial growth of rate \({p\in\Big(\frac{2n}{n+2},2\Big)}\), we prove Calderon–Zygmund type estimates for the spatial gradient. In order to deal with the anisotropic scaling behaviour of the above system, we employ the concept of intrinsic geometry by DiBenedetto. Following ideas of Mingione, we avoid tools from harmonic analysis such as singular integrals and maximal functions. Our methods apply to systems that are merely continuous with respect to the space variable as well as to certain systems with a VMO-type regularity. With respect to the time variable, we do not impose any regularity except measurability.