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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.

In this paper, we study the regularity for viscosity solutions of locally uniformly elliptic equations and obtain a series of interior pointwise $C^{k,\alpha}$ ($k\geq 1$, $0<\alpha<1$) regularity with smallness assumptions on the solution and the right-hand term. As applications, we obtain various interior pointwise regularity for several classical elliptic equations, i.e., the prescribed mean curvature equation, the Monge-Amp\`{e}re equation, the $k$-Hessian equations, the $k$-Hessian quotient equations and the Lagrangian mean curvature equation. Moreover, the smallness assumptions are necessary in most cases (Remark 2.6, Remark 3.5, Remark 4.7, Remark 5.4 and Remark 6.5).

We investigate the interior pointwise $C^{\alpha}$ regularity for weak solutions of elliptic and parabolic equations with divergence-free drifts. For such equations, the integrability condition on the drift can be relaxed and the interior $C^{\alpha}$ regularity for some $0<\alpha<1$ has been obtained previously with the aid of Harnack inequality. In this paper, we prove the interior pointwise $C^{\alpha}$ regularity for any $0<\alpha<1$ provided that the drift is small. We obtain the regularity under three different types conditions on the drift. The proof is based on the energy inequality and the perturbation technique.