Interior gradient estimates for mean curvature equations (Q1127653)

The author considers the mean curvature equation \[ (\delta_{ij}-u_iu_j/(1+| \nabla u| ^2))u_{ij}=H(x)(1+| \nabla u| ^2)^{1/2} \] in a ball \(B_r(0)\subset \mathbb{R}^n\). It is proved that if \(u\) is a nonnegative solution of the above equation, of class \(C^3\), if \(| H(x)| \leq C_0\), \(| \nabla H(x)| \leq C_0\), then \(| \nabla u(0)| \leq \exp(C_1+C_2M^2/r^2)\), where \(M=\text{ sup}_{\Omega}u(x)\), \(C_1\) depends on \(n\), \(M\), and \(C_0\); and \(C_2\) depends on \(n\) and \(C_0\). The proof is based on a suitable choice of an auxiliary function. The result is then extended to higher order mean curvature equations. The same method is used to prove Liouville type result.