Obstacle problem

The obstacle problem is a classic motivating example in the mathematical study of variational inequalities and free boundary problems. It is deeply related to the study of minimal surfaces and the capacity of a set in potential theory as well. The problem consists of describing the properties of minimizers of an energy functional, such as Dirichlet's energy,

$J=\int _{D}(\nabla u)^{2}dx$

in some domain $D$ where the functions $u$ satisfy Dirichlet boundary conditions, and are in addition constrained to be greater than some given obstacle function $\phi (x)$.

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 2nd derivatives across the free boundary. The free boundary is characterized as a Holder continuous surface except at certain singular points, which reside on a smooth manifold.

The obstacle problem's applications include the study of fluid filtration in porous media, constrained heating, elasto-plasticity, optimal control, and financial math.

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 the membrane is constrained to lie above some obstacle $\phi (x)$in the interior of the domain as well. In this case, the energy functional to be minimized is the surface area integral, or

$J(u)=\int _{D}{\sqrt {1+|\nabla u|^{2}}}dx$

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

$J(u)=\int _{D}|\nabla u|^{2}dx$

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 $\phi (x)$.

In the simple case where the process is Brownian motion, and the process is forced to stop upon exiting the domain, the solution $u(x)$ of the obstacle problem can be characterized as the expected value of the payoff, starting the process at $x$, if the optimal stopping strategy is followed. The stopping criterion is simply that one should stop upon reaching the contact set.

Formal formulation

Given

1. a open bounded domain $D\subset \mathbb {R} ^{n}$ with smooth boundary
2. a smooth function $f(x)$ on $\partial D$
3. a smooth function $\phi (x)$ defined on all of $D$ such that $\phi |_{\partial D}<f$.

Consider the set

$K=\{u(x)\in H^{1}(D),u|_{\partial D}=f(x),u\geq \phi \}$,

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

$J(u)=\int _{D}|\nabla u|^{2}dx$

over all functions $u(x)$ belonging to $K$; the existence of such a minimizer is assured by considerations of Hilbert space theory.

Alternative formulations

Variational inequality

See also: Variational inequality

The obstacle problem can be reformulated as a standard problem in variational inequalities on Hilbert spaces. Seeking the energy minimizer in the set $K$ of suitable functions is equivalent to seeking

$u\in K$ such that $\int _{D}(\nabla u)\cdot (\nabla (v-u))dx\geq 0$ for every $v\in K$

This is a special case of the more general form for variational inequalities on Hilbert spaces, whose solutions are functions $u$ in some closed convex subset $K$ of the overall space, such that

$a(u,v-u)\geq f(v-u)$ for all $v$ in $K$.

for coercive, real-valued, bounded bilinear forms $a(u,v)$ and bounded linear functionals $f(v)$.

Least superharmonic function

See also: 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.

In fact, an application of the maximum principle then shows that the solution to the obstacle problem is the least superharmonic function in the set of admissible functions.

Regularity properties

Optimal regularity

The solution to the obstacle problem has $C^{1,1}$ regularity, or bounded second derivatives, when the obstacle itself has these properties. More precisely, the solution's modulus of continuity and the modulus of continuity for its derivative are related to those of the obstacle.

1. If the obstacle $\phi (x)$ has modulus of continuity $\sigma (r)$, that is to say that $|\phi (x)-\phi (y)|\leq \sigma (|x-y|)$, then the solution $u(x)$ has modulus of continuity given by $C\sigma (2r)$, where the constant depends only on the domain and not the obstacle.
2. If the obstacle's first derivative has modulus of continuity $\sigma (r)$, then the solution's first derivative has modulus of continuity given by $Cr\sigma (2r)$, where the constant again depends only on the domain.

Level surfaces and the free boundary

Subject to a degeneracy condition, level sets of the difference between the solution and the obstacle, $\{x:u(x)-\phi (x)=t\}$ for $t>0$ are $C^{1,\alpha }$ surfaces. The free boundary, which is the boundary of the set where the solution meets the obstacle, is also $C^{1,\alpha }$ except on a set of singular points, which are themselves either isolated or locally contained on a $C^{1}$ manifold.

Generalizations

The theory of the obstacle problem is extended to other divergence form uniformly elliptic operators, and their associated energy functionals. It can be generalized to degenerate elliptic operators as well.

The Signorini problem is a variant of the obstacle problem, where the energy functional is minimized subject to a constraint which only lives on a surface of one lesser dimension, which includes the boundary obstacle problem, where the constraint operates on the boundary of the domain.

The parabolic, time-dependent cases of the obstacle problem and its variants are also objects of study.

References

• Caffarelli, Luis (July 1998), "The obstacle problem revisited", The Journal of Fourier Analysis and Applications, vol. 4, no. 4–5, pp. 383–402, doi:10.1007/BF02498216
• Evans, Lawrence, An Introduction to Stochastic Differential Equations, Version 1.2 (PDF), pp. 110–114
• Frehse, Jens (1972), "On the regularity of the solution of a second order variational inequality", Bolletino della Unione Matematica Italiana, vol. 4, no. 6, pp. 312–215
• Friedman, Avner (1982), Variational principles and free boundary problems, New York: Wiley, ISBN 0471868493
• Kinderlehrer, David; Stampacchia, Guido (1980), An Introduction to Variational Inequalities and Their Applications, New York: Academic Press, ISBN 0-89871-466-4