The interior gradient estimate of prescribed Hessian quotient curvature equations
If \(u \in C^2({\mathbb R}^{n})\), let \(W(u)\) be the Weingarten curvature matrix, i.e. \[ \begin{aligned} W(u)_{ij}&={1 \over {\sqrt{1+|\nabla u|^2}}}\Bigg(\partial_i \partial_j u-{{\sum_{k=1}^n\partial_i u\;\partial_k u\;\partial_k \partial_j u} \over {\sqrt{1+|\nabla u|^2}(1+\sqrt{1+|\nabla u|^2})}}\\&\quad-{{\sum_{k=1}^n \partial_j u\;\partial_k u\;\partial_i \partial_k u} \over {\sqrt{1+|\nabla u|^2}(1+\sqrt{1+|\nabla u|^2})}}+{{\sum_{k,l=1}^{n}\partial_{i}u\;\partial_{j}u\;\partial_{k}u\;\partial_{l}u\;\partial_{k}\partial_{l}u} \over {(1+|\nabla u|^2) (1+\sqrt{1+|\nabla u|^2)^2}}}\Bigg), \end{aligned} \] for \(1\leq i,\;j\leq n\). If \(a\) is a matrix and \(\lambda(a)\) is the vector of eigenvalues of \(a\), let \(\sigma_k(a)={\sum_{1\leq i_1 < i_2 < \cdots < i_k\leq n}} \lambda_{i_1}(a)\lambda_{i_2}(a)\cdots\lambda_{i_k}(a)\) for \(1 \leq k \leq n\), \(\sigma_0 (a)=1\). Theorem 1. If \(r, \; M>0, \; u \in C^{3}(B(0,r))\) and \(f \in C^{1}(B(0,r)), \;f \geq 0 ,\; |\nabla f^{1 \over {k-l}}|\leq M\) and such that \({{\sigma_{k}(W(u))} \over {\sigma_{l}(W(u))}}=f\) in \(B(0,r)\) with \(0 \leq l < k \leq n\), then there exists \(C>0\) depending only on \(n,\; k, \; l, \; M\) and \(|f|_{l^{\infty}}\) such that \[ |\nabla u(0)|\leq e^{C\big(1+({osc(u)}_{B(0,r)})^2/r^2+{osc(u)}_{B(0,r)}+({osc(u)}_{B(0,r)})^2/r\big)}. \] Theorem 2. If \(u\in C^{3}({\mathbb R}^{n})\) is such that \({{\sigma_{k}(W(u))} \over {\sigma_{l}(W(u))}}=0\) in \({\mathbb R}^{n}\) with \(0 \leq l < k \leq n\) and \({\lim}_{{x \rightarrow \infty}}{{u(x)} \over{|x|}}=0\), then \(u\) is constant.
scientific article; zbMATH DE number 6714549