Pointwise regularity for locally uniformly elliptic equations and applications

In this paper, we study the regularity for viscosity solutions of locally uniformly elliptic equations and obtain a series of interior pointwise Ck,α (k≥ 1, 0<α<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).