A Hölder estimate for entire solutions to the two-valued minimal surface equation (Q2790198)

Bernstein's problem is a natural question: The graphs of which functions of \(\mathbb R^{n-1}\) have minimal surface area in \(R^n\)? These surfaces, known to result from linear functions for \(n\leq 8\), are called minimal surfaces and satisfy the minimal surface equation. In this paper, the author studies the two-valued Minimal Surface Equation (2MSE), resulting from the \textit{two-valued minimal surface operator} \(\mathcal{M}_0\). The two-valued function \(u\) corresponding to a solution \(u_0\) of \(\mathcal{M}_0(u_0) = 0\) on \(\mathbb R^2\setminus \{0\}\) (called an \textit{entire solution}) is defined in polar coordinates by \(u(r,\theta) = u_0(r^{1/2} ,\theta/2)\). The author first proves a Bernstein-type theorem for the 2MSE: If \(u_0\) is an entire solution to the 2MSE which can be extended continuously across the origin, then \(u_0(r,\theta) = ar^2 \cos 2\theta + br^2\sin 2\theta + c\) for some \(a,b,c\in \mathbb R\). This result relies on a strong maximum principle proven in one of the author's previous works. The author then proves a strong decay condition on \(Du\) for \(u\) corresponding to an entire solution \(u_0\): There exists a vector \(\vec{a}\in \mathbb R^2\) such that NEWLINE\[NEWLINE \sup_{r\geq R} |Du-\vec{a}| \leq c \left( \frac{R_{u_0}}{R}\right)^{1/64\pi} NEWLINE\]NEWLINE for all \(R\geq 2R_{u_0}\), where \(R_{u_0} = \max\{ |\liminf_{r\to 0} u_0|, |\limsup_{r\to 0} u_0|\}\). He concludes with a few corollaries of the last result. The author proves his results by modifying methods used to study the minimal surface equation.