Parabolic p-systems: Hölder continuity of D u

The space gradient Du of local weak solutions of the quasilinear system (1.10) of Chap. VIII are locally Holder continuous in Ω_T provided the structure conditions (Si)-(S6) are in force. We will show this first for the homogeneous system (1.1) and then will indicate how to extend it to the general systems (1.10). The estimates of this chapter hold in the interior of 12T and deteriorate near its parabolic boundary T. If K is a compact subset ofΩ_T we let dist (К; Γ) denote the parabolic distance from К to the parabolic boundaryΓof Ω_T,i.e., dist$$ \left( {\mathcal{K};\Gamma } \right) \equiv \mathop {\mathop {\inf }\limits_{\left( {x,t} \right) \in \mathcal{K}} }\limits_{\left( {y,s} \right) \in \Gamma } \left( {\left| {x - y} \right| + \sqrt {\left| {t - s} \right|} } \right) $$