Pointwise regularity for locally uniformly elliptic equations and applications

Yuanyuan Lian; Kai Zhang

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arXiv · 2405.07199

Pointwise regularity for locally uniformly elliptic equations and applications

Yuanyuan Lian·Kai Zhang

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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).

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Yuanyuan Lian, Kai Zhang. 2024-05-12. Pointwise regularity for locally uniformly elliptic equations and applications. https://arxiv.org/abs/2405.07199

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