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 D {\displaystyle D} where the functions u {\displaystyle u} represent the vertical displacement of the membrane. In addition to satisfying Dirichlet boundary conditions corresponding to the fixed boundary of the membrane, the functions u {\displaystyle u} are in addition constrained to be greater than some given obstacle function ϕ ( x ) {\displaystyle \phi (x)} . 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 Qualche tempo dopo Stampacchia, partendo sempre dalla sua disequazione variazionale, aperse un nuovo campo di ricerche che si rivelò importante e fecondo. Si tratta di quello che oggi è chiamato il problema dell'ostacolo.
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 ϕ ( x ) {\displaystyle \phi (x)} 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 ϕ ( x ) {\displaystyle \phi (x)} . In the simple case wherein the process is Brownian motion, and the process is forced to stop upon exiting the domain, the solution u ( x ) {\displaystyle u(x)} of the obstacle problem can be characterized as the expected value of the payoff, starting the process at x {\displaystyle x} , 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 D ⊆ R n {\displaystyle D\subseteq \mathbb {R} ^{n}} with smooth boundary a smooth function f {\displaystyle f} on ∂ D {\displaystyle \partial D} (the boundary of D {\displaystyle D} ) a smooth function φ {\displaystyle \varphi } defined on all of D {\displaystyle D} such that φ | ∂ D < f {\displaystyle \varphi |_{\partial D}<f} , i.e., the restriction of φ {\displaystyle \varphi } to the boundary of D {\displaystyle D} (its trace) is less than f {\displaystyle f} . Then consider the set
which is a closed convex subset of the Sobolev space H 1 ( D ) {\displaystyle H^{1}(D)} of square integrable functions with domain D {\displaystyle D} whose weak first derivatives is square integrable, containing those functions with the desired boundary conditions and whose values above the obstacle's. A solution to the obstacle problem is a function u ∈ K {\displaystyle u\in K} which minimizes the energy integral
over all functions v {\displaystyle v} belonging to K {\displaystyle K} ; in symbols
J ( u ) = min v ∈ K J ( v ) or u ∈ Argmin K J . {\displaystyle J(u)=\operatorname {min} _{v\in K}J(v){\text{ or }}u\in \operatorname {Argmin} _{K}J.}
The existence and uniqueness of such a minimizer is assured by considerations of Hilbert space theory.
Alternative formulations
Variational inequality
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