Improved Interior Gradient Estimates for the Mean Curvature Equation under Nonlinear Assumptions

TL;DR

This work analyzes interior gradient estimates for the mean curvature equation $\mathrm{div}\left( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}} \right)=f(\nabla u)$ under diverse nonlinear growth conditions on $f=f(p)$. Using a Bernstein-type approach, it rewrites the equation as $a^{ij}(p)u_{ij}=g$ with $a^{ij}=\delta^{ij}-\frac{u_i u_j}{1+|\nabla u|^2}$ and $g=(1+|\nabla u|^2)^{1/2}f(\nabla u)$, and constructs an auxiliary function $\varphi(hF)$ to obtain pointwise differential inequalities at a maximum point. The authors establish sharp interior gradient bounds that depend on the oscillation $L$ and a growth parameter $\theta$ across four regimes (A1)–(A4), covering cases such as $f(p)=|p|^{\theta}$, $f(p)=\log^{\theta}(1+|p|^2)$, as well as the elliptic regularization of the inverse mean curvature flow and the minimal surface equation. Consequently, the gradient bounds imply uniform ellipticity away from the critical set, enabling standard elliptic regularity and leading to Liouville-type rigidity for global solutions under sublinear or linear growth, with explicit decay rates in $R$.

Abstract

In this paper, we investigate interior gradient estimates for solutions to the mean curvature equation $$ \dive \left( \frac{\nabla u}{\sqrt{1 + |\nabla u|^2}} \right) = f(\nabla u)$$ under various nonlinear assumptions on the right-hand side. Under the weakened initial assumption $u\in C^1(B_R) \cap C^3(\{|\nabla u|>0\})$, we establish sharp gradient bounds that depend on the oscillation of the solution. These estimates are applicable to a wide class of nonlinear terms, including the specific forms arising from the elliptic regularization of the inverse mean curvature flow ($f=\varepsilon\sqrt{1+|\nabla u|^2}$ ), minimal surface equation ($f=0$) and several polynomial and logarithmic growth regimes. As applications, the gradient bounds imply uniform ellipticity of the equation away from the critical set,which allows one to apply classical elliptic regularity theory and obtain higher regularity of solutions in the noncritical region. Moreover, when the solution grows at most linearly, all cases of our results can be applied in Moser's theory to establish the affine linear rigidity of global solutions. This directly leads to the Liouville-type theorems for global solutions without requiring additional proofs.

Theorems & Definitions (1)

• Theorem 1.1