freeboundary

Free boundary, in mathematics, refers to problems where part of the boundary of the domain where the solution is sought is not fixed in advance and must be determined as part of the solution. This occurs in partial differential equations and variational problems, where different equations or constraints hold in different subdomains separated by an interface whose location is unknown a priori.

Common examples include the Stefan problem, which models phase transitions such as melting or solidification. In the Stefan problem, the interface between phases moves with a velocity determined by heat flux and latent heat, and continuity conditions are imposed on the temperature across the moving boundary. The obstacle problem is another classic example, where a membrane is constrained to lie above an obstacle; the region where the constraint is active defines a free boundary that separates contacting and non-contacting regions, with the governing equation typically the Laplace equation in the non-contact region.

Mathematical approaches to free boundary problems often combine PDE theory with variational methods. Many problems are formulated as energy minimization or as variational inequalities, with the free boundary arising as the boundary of the active region. Regularity theory investigates when the free boundary is smooth and characterizes possible singularities. Techniques such as blow-up analysis, monotonicity formulas, and level set or phase-field methods are used to study both analytic properties and numerical approximations.

Applications span science and engineering, including materials science (solidification, melting), fluid dynamics (Hele-Shaw flow), combustion fronts, biological growth models (tumor boundaries), and in finance (pricing problems with early exercise boundaries in American options). The field sits at the intersection of analysis, geometry, and applied modeling.