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

TL;DR

This paper establishes optimal interior pointwise regularity results for solutions of divergence form elliptic and parabolic equations, using compactness and perturbation methods, with applications to understanding nodal set structures.

Contribution

It provides a simple, optimal proof of interior pointwise $C^{k,eta}$ regularity for divergence form equations, enhancing understanding of solution structures.

Findings

Proved optimal interior pointwise regularity for elliptic and parabolic equations.

Applied regularity results to analyze the structure of nodal sets.

Demonstrated the effectiveness of compactness and perturbation techniques in regularity theory.

Abstract

In this paper, we obtain the interior pointwise $C^{k,\alpha}$ ($k\geq 0$, $0<\alpha<1$) 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.