Regularity for subquadratic parabolic systems: higher integrability and dimension estimates

Abstract

We consider weak solutions of parabolic systems of the type

where the structure function b is differentiable with respect to x and satisfies standard ellipticity and growth properties with polynomial growth rate p ∊ (2n/(n + 2), 2). We investigate regularity properties of the solution, including the existence of second-order spatial derivatives, the existence of the time derivative and the higher integrability of the spatial gradient. As an application, we derive dimension estimates for the singular set of solutions of homogeneous parabolic systems. More precisely, we establish the bound

provided the structure function depends Höolder continuously on the space variable with Höolder exponent β ∊(0, 1].