The Bernstein problem for parametric elliptic functionals

A classical theorem of Bernstein says that entire minimal graphs over R^2 are planes. This result was extended to R^n for n < 8 in works of Fleming, De Giorgi, Almgren, and Simons. Finally, Bombieri - De Giorgi - Giusti constructed a non-flat entire minimal graph over R^8, settling the Bernstein problem for the area functional. We will discuss the Bernstein problem for more general parametric elliptic functionals, whose minimizers model crystal surfaces. In particular, we will show that the graph of a certain quadratic polynomial over R^6 minimizes such a functional.