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. It is deeply related to the study of minimal surfaces and the capacity of a set in potential theory as well. Applications include the study of fluid filtration in porous media, constrained heating, elasto-plasticity, optimal control, and financial mathematics.
The mathematical formulation of the problem is to seek minimizers of the Dirichlet energy functional,
in some domains where the functions represent the vertical displacement of the membrane. In addition to satisfying Dirichlet boundary conditions corresponding to the fixed boundary of the membrane, the functions are in addition constrained to be greater than some given obstacle function . The solution breaks down into a region where the solution is equal to the obstacle function, known as the contact set, and a region where the solution is above the obstacle. The interface between the two regions is the free boundary.
In general, the solution is continuous and possesses Lipschitz continuous first derivatives, but that the solution is generally discontinuous in the second derivatives across the free boundary. The free boundary is characterized as a Hölder continuous surface except at certain singular points, which reside on a smooth manifold.
Historical note
Motivating problems
Shape of a membrane above an obstacle
The obstacle problem arises when one considers the shape taken by a soap film in a domain whose boundary position is fixed (see Plateau's problem), with the added constraint that the membrane is constrained to lie above some obstacle in the interior of the domain as well. In this case, the energy functional to be minimized is the surface area integral, or
This problem can be linearized in the case of small perturbations by expanding the energy functional in terms of its Taylor series and taking the first term only, in which case the energy to be minimized is the standard Dirichlet energy
Optimal stopping
The obstacle problem also arises in control theory, specifically the question of finding the optimal stopping time for a stochastic process with payoff function .
In the simple case wherein the process is Brownian motion, and the process is forced to stop upon exiting the domain, the solution of the obstacle problem can be characterized as the expected value of the payoff, starting the process at , if the optimal stopping strategy is followed. The stopping criterion is simply that one should stop upon reaching the contact set.
Formal statement
Suppose the following data is given: an open bounded domain ⊂ ℝn with smooth boundary a smooth function on ∂ (the boundary of ) a smooth function defined on all of such that < , i.e. the restriction of to the boundary of (its trace) is less than . Then consider the set
which is a closed convex subset of the Sobolev space of square integrable functions with square integrable weak first derivatives, containing precisely those functions with the desired boundary conditions which are also above the obstacle. The solution to the obstacle problem is the function which minimizes the energy integral
over all functions belonging to ; the existence of such a minimizer is assured by considerations of Hilbert space theory.
Alternative formulations
Variational inequality
The obstacle problem can be reformulated as a standard problem in the theory of variational inequalities on Hilbert spaces. Seeking the energy minimizer in the set of suitable functions is equivalent to seeking
such that
where ⟨ . , . ⟩ : ℝn × ℝn → ℝ is the ordinary scalar product in the finite-dimensional real vector space ℝn. This is a special case of the more general form for variational inequalities on Hilbert spaces, whose solutions are functions in some closed convex subset of the overall space, such that
for coercive, real-valued, bounded bilinear forms and bounded linear functionals .
Least superharmonic function
A variational argument shows that, away from the contact set, the solution to the obstacle problem is harmonic. A similar argument which restricts itself to variations that are positive shows that the solution is superharmonic on the contact set. Together, the two arguments imply that the solution is a superharmonic function.