In mathematics, a free boundary problem (FB problem) is a partial differential equation to be solved for both an unknown function
which is not known at the outset of the problem is the free boundary.
FBs arise in various mathematical models encompassing applications that ranges from physical to economical, financial and biological phenomena, where there is an extra effect of the medium.
This effect is in general a qualitative change of the medium and hence an appearance of a phase transition: ice to water, liquid to crystal, buying to selling (assets), active to inactive (biology), blue to red (coloring games), disorganized to organized (self-organizing criticality).
An interesting aspect of such a criticality is the so-called sandpile dynamic (or Internal DLA).
The most classical 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.
The melting of ice is a Stefan problem for the temperature field
denote the unit outward normal vector to the second (solid) phase.
The Stefan condition determines the evolution of the surface
To solve the Stefan problem we not only have to solve the heat equation in each region, but we must also track the free boundary
The one-phase Stefan problem corresponds to taking either
In the direction of greater complexity we could also consider problems with an arbitrary number of phases.
Another famous free-boundary problem is the obstacle problem, which bears close connections to the classical Poisson equation.
In the obstacle problem, we impose an additional constraint: we minimize the functional
Where the Poisson problem corresponds to minimization of a quadratic functional over a linear subspace of functions, the free boundary problem corresponds to minimization over a convex set.
Many free boundary problems can profitably be viewed as variational inequalities for the sake of analysis.
To illustrate this point, we first turn to the minimization of a function
The same idea applies to the minimization of a differentiable functional
on a convex subset of a Hilbert space, where the gradient is now interpreted as a variational derivative.
To concretize this idea, we apply it to the obstacle problem, which can be written as This formulation permits the definition of a weak solution: using integration by parts on the last equation gives that This definition only requires that
have one derivative, in much the same way as the weak formulation of elliptic boundary value problems.
In the theory of elliptic partial differential equations, one demonstrates the existence of a weak solution of a differential equation with reasonable ease using some functional analysis arguments.
However, the weak solution exhibited lies in a space of functions with fewer derivatives than one would desire; for example, for the Poisson problem, we can easily assert that there is a weak solution which is in
One then applies some calculus estimates to demonstrate that the weak solution is in fact sufficiently regular.
For free boundary problems, this task is more formidable for two reasons.
For one, the solutions often exhibit discontinuous derivatives across the free boundary, while they may be analytic in any neighborhood away from it.
From a purely academic point of view free boundaries belong to a larger class of problems usually referred to as overdetermined problems, or as David Kinderlehrer and Guido Stampacchia addressed it in their book: The problem of matching Cauchy data.
Other related FBP that can be mentioned are Pompeiu problem, Schiffer’s conjectures.