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

TL;DR

This paper derives sharp interior gradient estimates for solutions to the mean curvature equation under nonlinear conditions, enabling regularity results and Liouville-type theorems for global solutions.

Contribution

It introduces new gradient bounds under weaker assumptions, applicable to various nonlinear forms, and derives consequences like regularity and rigidity results.

Findings

Established sharp gradient bounds depending on solution oscillation.

Proved uniform ellipticity away from critical points.

Derived Liouville-type theorems for global solutions.

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.