Partial regularity for subquadratic parabolic systems by $\mathcal{A}$-caloric approximation

Abstract

We establish a partial regularity result for weak solutions of nonsingular parabolic systems with subquadratic growth of the type $$ \partial_t u - \mathrm{div} a(x,t,u,Du) = B(x,t,u,Du), $$ where the structure function $a$ satisfies ellipticity and growth conditions with growth rate $\frac{2n}{n+2} < p < 2$. We prove Hölder continuity of the spatial gradient of solutions away from a negligible set. The proof is based on a variant of a harmonic type approximation lemma adapted to parabolic systems with subquadratic growth.