  By a classical result, solutions of analytic elliptic PDEs, like the Laplace
equation, are analytic. In many instances, the properties that come from being
analytic are more important than analyticity itself. Many important equations
are degenerate elliptic and solutions have much lower regularity. Still, one
may hope that solutions share properties of analytic functions. These
properties are closely connected to important open problems.
  In this survey, we will explain why solutions of an important degenerate
elliptic equation have analytic properties even though the solutions are not
even $C^3$.

Differential Geometry;Analysis of PDEs