A Free Boundary Problem: Contributions from modern analysis

We exemplify the role of Free Boundary Problems as an important source of ideas in modern analysis. With the help of a model problem we illustrate the use of analytical, algebraic and geometrical techniques obtaining uniqueness of weak solutions via the use of entropy inequalities, existence through nonlinear semigroup theory, and regularity using a method, called intrinsic scaling, based on interpreting a partial differential equation in a geometry dictated by its own structure.