UC Irvine

In this thesis we discuss the Bernstein problem for parametric elliptic functionals. In the first part, we give a proof by foliation that the cones over $\mathbb{S}^k \times \mathbb{S}^l$ minimize parametric elliptic functionals for each $k, l \geq 1$. We also analyze the behavior at infinity of the leaves in the foliations. This analysis motivates conjectures related to the existence and growth rates of nonlinear entire solutions to equations of minimal surface type that arise in the study of such functionals. In the second part we construct nonlinear entire anisotropic minimal graphs over $\mathbb{R}^4$, which completes the solution to the anisotropic Bernstein problem. The examples we construct have a variety of growth rates, and our approach both generalizes to higher dimensions and recovers and elucidates known examples of nonlinear entire minimal graphs over $\mathbb{R}^n, n \geq 8$. The first two parts are joint works with C. Mooney. In the final part, we discuss the work of Ecker-Huisken about a Bernstein theorem for the minimal surface equation with controlled growth condition, and also its possible generalization to the anisotropic case.