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

Abstract

In this paper, we obtain the interior pointwise $C^{k,\hat{I}\pm }$ ($k\ge 0$, $0<\hat{I}\pm <1$) 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.