Partial boundary regularity of non-linear parabolic systems in low dimensions

Abstract

Abstract In this paper we establish partial boundary regularity for non-linear parabolic systems ∂ t u - div a ( x , t , u , D u ) = 0 $\partial _t u - \operatorname{div}a(x,t,u,Du) =0$ with quadratic growth in dimensions n ≤ 2 ${n\le 2}$ . In particular, we prove that almost every lateral boundary point is a Hölder continuity point for the spatial gradient of the solution. We are also able to treat particular vector fields in the higher dimensional case. In the case of vector fields a ( x , t , D u ) ${a(x,t,Du)}$ not depending on u, the partial boundary regularity has been established in [Ann. Inst. H. Poincaré, Anal. Non Linéaire 27 (2010), 145–200].