  Global solutions to the obstacle problem were first completely classified in
two dimensions by Sakai using complex analysis techniques. Although the complex
analysis approach produced a very succinct proof in two dimensions, it left the
higher dimensional cases, and even closely related problems in two dimensions,
unresolved. A complete classification in dimensions $n\geq 3$ was recently
given by Eberle, Figalli and Weiss, forty years after Sakai published his
proof. In this paper we give a proof of Sakai's classification result for
unbounded coincidence sets in the spirit of the recent proof by Eberle, Figalli
and Weiss. Our approach, in particular, avoids the need for complex analysis
techniques and offers new perspectives on two-dimensional problems that complex
analysis cannot address.
