Free boundary problem
Encyclopedia
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 location of the ice/liquid interface is controlled dynamically by the solution of the PDE.
for the temperature field T, which is formulated as follows. Consider a medium occupying a region Ω consisting of two phases, phase 1 which is present when T > 0 and phase 2 which is present when T < 0. Let the two phases have thermal diffusivities
α1 and α2. For example, the thermal diffusivity of water is 1.4×10−7 m2/s, while the diffusivity of ice is 1.335×10−6 m2/s.
In the regions consisting solely of one phase, the temperature is determined by the heat equation: in the region T > 0,
while in the region T < 0,
This is subject to appropriate conditions on the (known) boundary of Ω; Q represents sources or sinks of heat.
Let Γt be the surface where T = 0 at time t; this surface is the interface between the two phases. Let ν denote the unit outward normal vector to the second (solid) phase. The Stefan condition determines the evolution of the surface Γ by giving an equation governing the velocity V of the free surface in the direction ν, specifically
where L is the latent heat of melting. By T1 we mean the limit of the gradient as x approaches Γt from the region T > 0, and for T2 we mean the limit of the gradient as x approaches Γt from the region T < 0.
In this problem, we know beforehand the whole region Ω but we only know the ice-liquid interface Γ at time t = 0. 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 Γ.
The one-phase Stefan problem corresponds to taking either α1 or α2 to be zero; it is a special case of the two-phase problem. In the direction of greater complexity we could also consider problems with an arbitrary number of phases.
, which bears close connections to the classical Poisson equation. The solutions of the differential equation
satisfy a variational principle, that is to say they minimize the functional
over all functions u taking the value g on the boundary. In the obstacle problem, we impose an additional constraint: we minimize the functional E subject to the condition
in Ω, for some given function φ.
Define the coincidence set C as the region where u = φ. Furthermore, define the non-coincidence set N = Ω\C as the region where u is not equal to φ, and the free boundary Γ as the interface between the two. Then u satisfies the free boundary problem
on the boundary of Ω, and
Note that the set of all functions v such that v ≤ φ is convex. 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.
for the sake of analysis. To illustrate this point, we first turn to the minimization of a function F of n real variables over a convex set C; the minimizer x is characterized by the condition
If x is in the interior of C, then the gradient of F must be zero; if x is on the boundary of C, the gradient of F at x must be perpendicular to the boundary.
The same idea applies to the minimization of a differentiable functional F 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 u have one derivative, in much the same way as the weak formulation of elliptic boundary value problems.
, 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 H1
, but it may not have second derivatives. 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. Secondly, one must also demonstrate the regularity of the free boundary itself. For example, for the Stefan problem, the free boundary is a C1/2 surface.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a free boundary problem is a partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
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 location of the ice/liquid interface is controlled dynamically by the solution of the PDE.
Two-phase Stefan problems
The melting of ice is a Stefan problemStefan problem
In mathematics and its applications, particularly to phase transitions in matter, a Stefan problem is a particular kind of boundary value problem for a partial differential equation , adapted to the case in which a phase boundary can move with time...
for the temperature field T, which is formulated as follows. Consider a medium occupying a region Ω consisting of two phases, phase 1 which is present when T > 0 and phase 2 which is present when T < 0. Let the two phases have thermal diffusivities
Thermal diffusivity
In heat transfer analysis, thermal diffusivity is the thermal conductivity divided by density and specific heat capacity at constant pressure. It has the SI unit of m²/s...
α1 and α2. For example, the thermal diffusivity of water is 1.4×10−7 m2/s, while the diffusivity of ice is 1.335×10−6 m2/s.
In the regions consisting solely of one phase, the temperature is determined by the heat equation: in the region T > 0,
while in the region T < 0,
This is subject to appropriate conditions on the (known) boundary of Ω; Q represents sources or sinks of heat.
Let Γt be the surface where T = 0 at time t; this surface is the interface between the two phases. Let ν denote the unit outward normal vector to the second (solid) phase. The Stefan condition determines the evolution of the surface Γ by giving an equation governing the velocity V of the free surface in the direction ν, specifically
where L is the latent heat of melting. By T1 we mean the limit of the gradient as x approaches Γt from the region T > 0, and for T2 we mean the limit of the gradient as x approaches Γt from the region T < 0.
In this problem, we know beforehand the whole region Ω but we only know the ice-liquid interface Γ at time t = 0. 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 Γ.
The one-phase Stefan problem corresponds to taking either α1 or α2 to be zero; it is a special case of the two-phase problem. In the direction of greater complexity we could also consider problems with an arbitrary number of phases.
Obstacle problems
Another famous free-boundary problem is the obstacle problemObstacle problem
The obstacle problem is a classic motivating example in the mathematical study of variational inequalities and free boundary problems. The problem is to find the equilibrium position of an elastic membrane whose boundary is held fixed, and which is constrained to lie above a given obstacle...
, which bears close connections to the classical Poisson equation. The solutions of the differential equation
satisfy a variational principle, that is to say they minimize the functional
over all functions u taking the value g on the boundary. In the obstacle problem, we impose an additional constraint: we minimize the functional E subject to the condition
in Ω, for some given function φ.
Define the coincidence set C as the region where u = φ. Furthermore, define the non-coincidence set N = Ω\C as the region where u is not equal to φ, and the free boundary Γ as the interface between the two. Then u satisfies the free boundary problem
on the boundary of Ω, and
Note that the set of all functions v such that v ≤ φ is convex. 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.
Connection with variational inequalities
Many free boundary problems can profitably be viewed as variational inequalitiesVariational inequality
In mathematics, a variational inequality is an inequality involving a functional, which has to be solved for all the value of a given variable, belonging usually to a convex set...
for the sake of analysis. To illustrate this point, we first turn to the minimization of a function F of n real variables over a convex set C; the minimizer x is characterized by the condition
If x is in the interior of C, then the gradient of F must be zero; if x is on the boundary of C, the gradient of F at x must be perpendicular to the boundary.
The same idea applies to the minimization of a differentiable functional F on a convex subset of a Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
, 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 u have one derivative, in much the same way as the weak formulation of elliptic boundary value problems.
Regularity of free boundaries
In the theory of elliptic partial differential equationsElliptic operator
In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which implies the key property that the principal symbol is...
, one demonstrates the existence of a weak solution
Weak solution
In mathematics, a weak solution to an ordinary or partial differential equation is a function for which the derivatives may not all exist but which is nonetheless deemed to satisfy the equation in some precisely defined sense. There are many different definitions of weak solution, appropriate for...
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 H1
Sobolev space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself as well as its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space...
, but it may not have second derivatives. 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. Secondly, one must also demonstrate the regularity of the free boundary itself. For example, for the Stefan problem, the free boundary is a C1/2 surface.