In this paper, we use the maximum principle to get the gradient estimate for the solutions of the prescribed mean curvature equation with Neumann boundary value problem, which gives a positive answer for the question raised by Lieberman [16] in page 360. As a consequence, we obtain the corresponding existence theorem for a class of mean curvature equations.
In this note, we study the mean curvature flow and the prescribed mean curvature type equation with general capillary-type boundary condition, which is uν = −φ(x)(1 + |Du| 2) 1−q 2 for any parameter q > 0. Using the maximum principle, we prove the gradient estimates for the solutions of such a class of boundary value problems. As a consequence, we obtain the corresponding existence theorem for a class of mean curvature equations. In addition, we study the related additive eigenvalue problem for general boundary value problems and describe the asymptotic behavior of the solution at infinity time. The originality of the paper lies in the range 0 < q < 1, since there are no any related results before. For parabolic case, we generalize the result of Ma-Wang-Wei [25] to any q > 0. And in elliptic case, we generalize the results in [32] to any q ≥ 0 and to any bounded smooth domain.
In this paper, we will use the maximum principle to give a new proof of the gradient estimates for mean curvature equations with some oblique derivative problems. Specially, we shall give a new proof for the capillary problem with zero gravity in any dimension n ≥ 2 and Neumann problem in n = 2, 3 dimensions.
This paper concerns the gradient estimates for Neumann problem of a certain Monge-Ampère type equation with a lower order symmetric matrix function in the determinant. Under a one-sided quadratic structure condition on the matrix function, we present two alternative full discussions of the global gradient bound for the elliptic solutions.
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