Interior Gradient Estimates for General Prescribed Curvature Equations

Abstract

In this paper, we derive the interior gradient estimate for solutions to general prescribed curvature equations. The proof is based on a fundamental observation of Gårding's cone and some delicate inequalities under a suitably chosen coordinate chart. As an application, we obtain a Liouville type theorem.

The Dirichlet problem for fully nonlinear elliptic equations on Riemannian manifolds

“…In this paper, we shall focus on the interior gradient estimate. In fact, this is the first of a series of papers on interior gradient estimates [6] [7]. However, the technique does not work well on complex equations.…”

“…We say that C 2 function u is admissible if λ(χ u ) ∈ Γ. Similar to [6], we assume that f satisfies the following conditions:…”

“…For more details and discussions on the structure conditions, we refer the readers to [6]. However, we do not need the condition that ∂f ∂λ j > δ i ∂f ∂λ i when λ j < 0, for some δ > 0.…”