Remarks on a Class of Solutions to the Minimal Surface System

is responsible for the progress of nonlinear elliptic PDE theory in the last century. Indeed, most early works on nonlinear elliptic problems focused on this particular equation. There are many beautiful existence, uniqueness, regularity theorems for the minimal surface equation, see for example [13]. The graph of a solution to (1.1) is naturally a minimal hypersurface in R. In general, we can consider a vector-valued function whose graph is a minimal submanifolds of the Euclidean space; the function then satisfies a nonlinear elliptic system. Indeed, a C vector-valued function f = (f, · · · f) : D → R is said to be a solution to the minimal surface system (see Osserman [16] or Lawson-Osserman [13]) if