Year: 2026
Type: preprint
Abstract: In this paper, we study interior gradient estimates for solutions of \[ \operatorname{div}_{M}\left( \frac{\nabla u}{\sqrt{1+|\nabla u|_g^2}} \right) =f(x,\nabla u) \] on a complete Riemannian manifold of dimension $n\ge2$. Under a Ricci curvature lower bound and a Hessian comparison for the distance function, we derive polynomial and exponential gradient bounds under structural conditions on $f$ and its horizontal and vertical derivatives. The estimates quantify the dependence on the radius, the oscillation of the solution, and the geometric and analytic parameters. In the Euclidean case, we also improve the dependence on the radius in the gradient estimate of Xu~\cite[Theorem~1.1(d)]{Xu2026}. As applications, we obtain Liouville-type results for entire solutions under suitable curvature and growth assumptions.
Source: arXiv (Cornell University)
Authors: Li, Weiyang, Xu, Fanheng
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G4758409503
OpenAlex ID: W7223017291