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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 dimensionsn  ≤  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 fieldsa  (  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].

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