---
tag: YR42E2W19Q
kind: problem
name: Free boundary problem
msc: [35R35, 35R37]
uses: [GC5BDZ6QBC, 0DEB0JMN0W, 3B0HB3DHW1]
about: [GC5BDZ6QBC]
sources:
  - url: https://en.wikipedia.org/wiki/Free_boundary_problem
same_as:
  wikidata: Q5500196
  wikipedia: Free boundary problem
history:
  posed_by: "Gabriel Lamé and Émile Clapeyron"
  first_stated: "1831"
---

## Problem

A free boundary problem is a problem for a [[partial differential equation|GC5BDZ6QBC]] 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.

## Remarks

Free boundary problems are nonlinear even when the equation in clause 1 is linear, since the domain depends on the solution.
The [[Stefan problem|3B0HB3DHW1]] for melting ice is the classical evolution example.
The heat equation holds in the liquid, and the melting front moves with a speed fixed by the heat flux.
The obstacle problem is the classical stationary example.
There the free boundary separates the region where a membrane touches an obstacle from the region where it is [[harmonic|0DEB0JMN0W]].
A common method is to recast the problem as a minimization or a variational inequality on a fixed domain, solve it there in a weak sense, and recover $\Gamma$ afterwards.
The main questions are then existence and uniqueness of a solution, the regularity of $u$, and the regularity of the free boundary $\Gamma$.
Lamé and Clapeyron studied the solidification of a liquid sphere in 1831, a problem of this type.

## Explanations

### L1 human

In most problems for differential equations the region is given, and only the function is unknown.
In a free boundary problem the edge of the region is also unknown and must be found together with the function.
Think of ice melting in water: the temperature in the water is unknown, and so is the position of the ice front.
Because the region has an extra unknown, the problem needs one extra condition on its edge.