Abstract
In this paper, we obtain the interior gradient estimate of some nonlinear equations which arise naturally from prescribed curvature problem of graphs in hyperbolic space. The method depends on the maximum principle.
Highlights
In this paper, we study the interior gradient estimate of the equation σk (λ{Aij }) = H (x, u, ν)(1 + |∇u|2) k 2 (1.1) whereAij := δij + u uij −n uluj uli − w(w + 1)n uluiulj + w(w + 1)
The interior gradient estimate for high order prescribed curvature equation in Euclidean space has been obtained by Korevaar [12] for Weingarten equations, Li
The interior gradient estimate has been obtained in [1] for quasilinear parabolic equations having coefficients depending on the gradient, which include both isotropic and anisotropic flows, and [16] for some modified mean curvature flow (MMCF) in hyperbolic space
Summary
The interior gradient estimate for high order prescribed curvature equation in Euclidean space has been obtained by Korevaar [12] for Weingarten equations, Li We adapt the method used in [5] and [20] to obtain the interior gradient estimate for equation (1.1), but here more terms are involved and should be handled with care.
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