Free Boundary Problems

Core Idea

In a free boundary problem, the domain on which the PDE holds is not given in advance but is itself part of the unknown, determined simultaneously with the solution. The classic examples are the obstacle problem (find u ≥ ψ minimizing ∫|∇u|²dx, where the contact set {u = ψ} has an unknown boundary) and the Stefan problem (modeling ice-water phase transitions, where the interface between phases moves according to the heat flux jump). The mathematical challenge is that the PDE holds only on an unknown region, and the free boundary must satisfy additional conditions coupling it to the solution.

Explainer

Free boundary problems are among the most challenging and physically important problems in PDE theory. Unlike standard boundary value problems where the equation is posed on a fixed domain, in a free boundary problem the domain itself must be determined as part of the solution. This introduces a fundamental nonlinearity even when the underlying PDE is linear: the Laplacian is a linear operator, but the obstacle problem (where we solve Δu = 0 on an unknown region determined by u ≥ ψ) is nonlinear because the domain depends on the solution.

The obstacle problem is the canonical free boundary problem. A membrane (modeled by its displacement u) is pushed against an obstacle ψ from below. Where the membrane lifts off, it is free and satisfies Δu = 0; where it touches, u = ψ. The variational formulation seeks u minimizing ∫|∇u|²dx subject to u ≥ ψ, which is a constrained optimization problem in H¹. The Euler-Lagrange condition becomes a variational inequality: ∫∇u·∇(v-u)dx ≥ 0 for all v ≥ ψ. Existence and uniqueness follow from the theory of variational inequalities (Lions-Stampacchia).

The regularity of the free boundary is the central mathematical question. Caffarelli's groundbreaking work established that the free boundary ∂{u > ψ} is a C^1,α surface near regular points—points where the solution detaches from the obstacle in a non-degenerate way (quadratic growth). At degenerate points, cusps and singularities can form, but Caffarelli showed the singular set has codimension at least 1. This regularity theory uses a blow-up analysis: one rescales the solution near a free boundary point and studies the limiting profiles, which are classified as either half-plane solutions (regular points) or polynomial solutions (singular points).

The Stefan problem for phase transitions is another fundamental free boundary problem. The temperature satisfies the heat equation in each phase (solid and liquid), and the interface Γ(t) between phases moves according to the Stefan condition: V_n = [∂T/∂n], where V_n is the normal velocity and [∂T/∂n] is the jump in heat flux. The weak (enthalpy) formulation avoids tracking the interface explicitly and is the basis for existence theory, while the classical formulation provides regularity results for the interface. Modern free boundary problems arise in fluid mechanics (water waves, Hele-Shaw flow), finance (American option pricing), and biology (tumor growth models).