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 C^k, (k 1, 0<<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\`ere 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).