Equations of Mean Curvature Type

Abstract

In this chapter we focus attention on both the prescribed mean curvature equation,

$$\mathfrak{M}\mathfrak{u} = \left( {1 + {{{\left| {Du} \right|}}^{2}}} \right)\Delta u - {{D}_{i}}u{{D}_{j}}u{{D}_{{ij}}}u = nH{{\left( {1 + {{{\left| {Du} \right|}}^{2}}} \right)}^{{{{3} \left/ {2} \right.}}}}$$

(16.1)

and a related family of equations in two variables. Our main concern is with interior derivative estimates for solutions. We shall see that not only can interior gradient bounds be established for solutions of these equations but that also their non-linearity leads to strong second derivative estimates which distinguish them from uniformly elliptic equations such as Laplace’s equation. In particular we shall derive an extension of the classical result of Bernstein that a C ²(ℝ²) solution of the minimal surface equation in ℝ² must be a linear function (Theorem 16.12).