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, (k 0, 0<<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.