Regularity estimates for equations and systems of mean curvature type.

摘要

e first consider a general class of non-uniformly elliptic equations in divergence form. For several special examples of this type (i.e. the minimal surface equation), one is able to exploit the geometrical significance of the equation to prove a priori estimates. On the other hand, general regularity theorems do not seem to be known. By extending a classic technique of De Giorgi we are able to prove a partial regularity theorem.;When one steps up to codimension greater than one very little appears to be known, even in the simplest case of the minimal surface system. We establish a priori estimates for the gradient of solutions of the minimal surface system in two independent variables and for the curvature of their graphs. These estimates are fairly straightforward consequences of some old results of Heinz. With the intent of extending these results to graphs with non-zero mean curvature vectors, we analyze the compactness properties of smooth