Pointwise regularity for locally uniformly elliptic equations and applications

TL;DR

This paper establishes interior pointwise regularity results for viscosity solutions of locally uniformly elliptic equations, with applications to classical elliptic equations like Monge-Ampère and mean curvature equations, under smallness conditions.

Contribution

It provides new interior pointwise $C^{k,eta}$ regularity results for viscosity solutions of elliptic equations, extending classical regularity theory with smallness assumptions.

Findings

Interior pointwise regularity for viscosity solutions

Applications to Monge-Ampère and mean curvature equations

Smallness assumptions are necessary in most cases

Abstract

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).