A free boundary problem is a boundary value problem in which part of the boundary of the domain is unknown and must be determined together with the solution of the governing partial differential equation. Conditions on the moving boundary close the problem. Classical examples include the melting interface in the Stefan problem and the boundary separating the exercise and continuation regions in an American option.
Free boundary problems are often reformulated as variational inequalities. In optimal stopping, the unknown boundary is the locus at which the value first meets the reward or obstacle.
