The interior gradient estimate of prescribed Hessian quotient curvature equations

Abstract

In this paper, we introduce a new auxiliary function, and establish the interior C 2 estimate for prescribed Gauss curvature equation in dimension two.

Boundary pointwise C1, and C2, regularity for fully nonlinear elliptic equations

“…• For the k-Hessian equations, see [19,59,65,66,68,77,113,131,134]. • For the k-Hessian quotient equations, see [8,32,33,34,37,44,49,89,91,113] • For the Lagrangian mean curvature equation, see [10,11,19,36,90,127,137,139,143,147] In addition, there exist Pogorelov's type interior C 1,1 estimates for some equations. The smoothness requirements of the boundary can be relaxed.…”

Interior estimates for Translating solitons of the $Q_k$-flows in $\mathbb{R}^{n+1}$

“…It is well know that the eigenvalues of this matrix correspond to the principal curvatures of the graph the function u. This matrix was first used in [CNS88] for deriving gradient estimates for Dirichlet curvature equations, it was also used in [Hol14] for parabolic curvature equations and a result analogous to Proposition 3.3 was announced in [CXZ17], but their proof is incomplete.…”

Erratum to: The interior gradient estimate of prescribed Hessian quotient curvature equations

“…Since σ 1 (B + tC) = σ 1 (B) and 1] and σ m (B + tC) = σ m (B − tC), which can also be obtained from…”

“…We can give a detailed proof as follows. Following the proof in [1], we assume A = {a i j } n×n with a i j = a ji , and at x 0 , we have…”