Pogorelov Estimates and Liouville Theorem for Hessian Quotient Equations in Half Space

Abstract

In this paper, we consider the Pogorelov estimates up to the flat boundary of convex solutions for Hessian quotient equations ${{{S_{k}}(U[u])} \over {S_{l}(U[u])}}= {{{C}_{n}^{k}} \over {{C}_{n}^{l}}}{(n - 1)}^{k-l}$, where $U[u] = (\Delta {u})I - D^{2}u, \, 0 \leq l < k \leq n$. Furthermore, we obtain the Liouville theorem for such equation.

Keywords

Hessian quotient equation / Pogorelov estimate / Liouville theorem / half space / 35B45 / 35J60

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