Variants of Bernstein's theorem for variational integrals with linear and nearly linear growth (Q6635347)

The graph of a smooth function \(u:\mathbb R^N\rightarrow \mathbb R\) is a minimal surface in \(\mathbb R^{N+1}\) if and only if \(u\) satisfies the minimal surface equation:\N\[\N-\mbox{div}\biggl[\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\biggr]=0.\tag{1}\N\]\NIn his celebrated work, S.N. Bernstein proved that any smooth solution \(u\)\Nof the minimal surface equation in \(\mathbb R^2\) must be an affine function.\N\NMore generally, the question arises as to which classes of second-order equations the Bernstein property extends. In particulare, one could replace (1) with the more general form\N\[\N-\mbox{div}\biggl[Df(\nabla u)\biggr]=0.\N\]\NFor this particular class of energy\Ndensities, Bildhauer and Fuchs proved the Bernstein property assuming that the function \( f : \mathbb{R}^2 \to \mathbb{R} \) is of class \( C^2 \) and exibit nearly linear growth namely satisfies the following conditions\N\[\ND^2 f(p)(q, q) > 0 \quad \text{for all } p, q \in \mathbb{R}^2, \; q \neq 0,\N\]\Nand that, for a constant \(\lambda > 0\), it holds that\N\[\ND^2 f(p)(q, q) \leq \lambda \frac{\ln(2 + |p|)}{1 + |p|} |q|^2, \quad p, q \in \mathbb{R}^2.\N\]\NFinally they assume that for some numbers \( 0 \leq m < 1 \) and \( K > 0 \), the solution \(u\) satisfies\N\[\N|\partial_1 u(x)| \leq K \frac{|\partial_2 u(x)|^{m + 1}}{m + 1}, \quad x \in \mathbb{R}^2.\N\]