COMPACTNESS AND GRADIENT BOUNDS FOR SOLUTIONS OF THE MEAN-CURVATURE SYSTEM IN 2 INDEPENDENT VARIABLES

We establish here a priori estimates for the gradient of solutions of the minimal surface system in two independent variables and for the curvature of their graphs. With the intent of extending these results to graphs with nonzero mean curvature vectors, we then analyze the compactness properties of smooth (C2) solutions of the mean curvature system. Using a geometric measure theory approach we are able to classify the possible behaviors of a sequence {u(l)(x)} of such solutions, under the assumption that a uniform bound on the area of the graphs holds and suitable hypotheses on the length of the mean curvature vector H(x). In particular, this implies the existence of an a priori gradient bound depending on the oscillation of the solution u(x).