Asymptotic values of minimal graphs in a disc (Q717798)

Let \(D\) be the open unit disk \(\{ (r,\theta) : 0\leq r<1,\;0\leq \theta\leq 2\pi\}\). For certain functions solving variational equations on \(D\), the authors investigate several questions regarding the existence of radial limits (in the extended reals) at the boundary of \(D\). In the first instance, the authors use the Poincaré disk model of hyperbolic space. They construct an unbounded minimal graph over the disk which fails to have a radial limit (in the extended reals) for almost every direction \(\theta\). In the second instance, the authors prove that a solution \(u\) of the prescribed mean curvature equation (over the Euclidean unit disk) \[ \text{div} \left( \frac{\nabla u}{\sqrt{1 + |Du|^2}}\right) = 2H \,, \] with \(H\) bounded and continuous, does have a radial limit (in the extended reals) for almost every direction \(\theta\). The proof is succinct and self-contained. In the third instance, the authors construct an example of a minimal graph over the Euclidean unit disk for which there is but a measure zero set of directions \(\theta\) for which there exists a finite radial limit (in the extended reals). This construction is based on extending Scherk surfaces and is inspired by the example in the authors' paper [Ann. Math. (2) 172, No. 3, 1879--1906 (2010; Zbl 1209.53010)].