Free boundary problem

Problem

A free boundary problem is a problem for a partial differential equation in which part of the domain is unknown. A typical stationary form is the following. Let $D \subseteq \R^n$ be a bounded open set and let $g$ be given boundary data on $\partial D$. Find an open set $\Omega \subseteq D$ and a function $u$ on $\overline{\Omega}$ such that:

1. $u$ solves a given partial differential equation in $\Omega$;
2. $u = g$ on the known part $\partial \Omega \cap \partial D$ of the boundary;
3. $u$ satisfies two conditions on the unknown part $\Gamma = \partial \Omega \cap D$, the free boundary.

One condition on $\Gamma$ alone would fix $u$ for each choice of $\Omega$. The second condition is what determines $\Gamma$. A standard example is the one-phase Bernoulli problem: find $u \ge 0$ with $\Delta u = 0$ in $\Omega = \set{u > 0}$, $u = 0$ on $\Gamma$ and $\abs{\nabla u} = 1$ on $\Gamma$. When the free boundary moves in time, one also speaks of a moving boundary problem.