Interior pointwise regularity for elliptic and parabolic equations in divergence form and applications to nodal sets

In this paper, we obtain the interior pointwise $C^{k,\alpha }$ ($k\ge 0$, $0<\alpha <1$) regularity for weak solutions of elliptic and parabolic equations in divergence form. The compactness method and perturbation technique are employed. The pointwise regularity is proved in a very simple way and the results are optimal. In addition, these pointwise regularity can be used to characterize the structure of the nodal sets of solutions.