Interior gradient estimates for mean curvature equations

Abstract

The interior gradient estimate for the prescribed mean curvature equation has been extensively studied, see [9] and the references therein. For high order mean curvature equations it has also been obtained in [11, 18]. In most articles such estimates were obtained by carrying out analysis on the graphs of solutions and so the arguments depend on the invariance of the equations under rigid motion. From the view point of partial differential equations such estimate should hold for equations with similar structural conditions. In [13, 6] the gradient estimate was obtained for certain fully nonlinear elliptic equations under various conditions. Different proofs for the gradient estimate for mean curvature equations have been given in [1, 3]. In this note we show that for curvature and Hessian equations the interior gradient estimate can be obtained very easily. Our proof, which is based on suitable choice of auxiliary functions, is elementary and avoids geometric computations on the graph of solutions. The technique in this note has actually been widely used in literature, see, e.g., [9].