On Singular Points of Solutions of the Minimal Surface Equation on Sets of Positive Measure

It is shown that, for any compact set $K\subset\mathbb{R}^n$ ($n\ge 2$) of positive Lebesgue measure and any bounded domain $G\supset K$, there exists a function in the Hölder class $C^{1, 1}(G)$ that is a solution of the minimal surface equation in $G\setminus K$ and cannot be extended from $G\setminus K$ to $G$ as a solution of this equation.