Free boundary problem

In mathematics, a free boundary problem is a partial differential equation to be solved for both an unknown function u and an unknown domain Ω. The segment Γ of the boundary of Ω which is not known at the outset of the problem is the free boundary.

The classic example is the melting of ice. Given a block of ice, one can solve the heat equation given appropriate initial and boundary conditions to determine its temperature. But, if in any region the temperature is greater than the melting point of ice, this domain will be occupied by liquid water instead. The boundary formed from the ice/liquid interface is controlled dynamically by the solution of the PDE.