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.[1]
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
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.[2]
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.[3] 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.[4]
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
closedconvexsubset 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.[3][5]
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 is the ordinary
scalar product in the
finite-dimensionalrealvector space. 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
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.[1]
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.[6]
Regularity properties
Solution of a one-dimensional obstacle problem. Notice how the solution stays superharmonic (concave down in 1-D), and matches derivatives with the obstacle (which is the condition)
Optimal regularity
The solution to the obstacle problem has regularity, or
boundedsecond derivatives, when the obstacle itself has these properties.[7] More precisely, the solution's
modulus of continuity and the modulus of continuity for its
derivative are related to those of the obstacle.
If the obstacle has modulus of continuity , that is to say that , then the solution has modulus of continuity given by , where the constant depends only on the domain and not the obstacle.
If the obstacle's first derivative has modulus of continuity , then the solution's first derivative has modulus of continuity given by , where the constant again depends only on the domain.[8]
Level surfaces and the free boundary
Subject to a degeneracy condition, level sets of the difference between the solution and the obstacle, for are surfaces. The free boundary, which is the boundary of the set where the solution meets the obstacle, is also except on a set of singular points, which are themselves either isolated or locally contained on a manifold.[9]
Generalizations
The theory of the obstacle problem is extended to other divergence form uniformly
elliptic operators,[6] and their associated energy functionals. It can be generalized to degenerate elliptic operators as well.
The double obstacle problem, where the function is constrained to lie above one obstacle function and below another, is also of interest.
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.
^"Some time after Stampacchia, starting again from his variational inequality, opened a new field of research, which revealed itself as important and fruitful. It is the now called obstacle problem" (English translation). The
Italic type emphasis is due to the author himself.
Faedo, Sandro (1986), "Leonida Tonelli e la scuola matematica pisana", in Montalenti, G.;
Amerio, L.; Acquaro, G.; Baiada, E.; et al. (eds.),
Convegno celebrativo del centenario della nascita di Mauro Picone e Leonida Tonelli (6–9 maggio 1985), Atti dei Convegni Lincei (in Italian), vol. 77, Roma:
Accademia Nazionale dei Lincei, pp. 89–109, archived from
the original on 2011-02-23, retrieved 2013-02-12. "Leonida Tonelli and the Pisa mathematical school" is a survey of the work of Tonelli in
Pisa and his influence on the development of the school, presented at the International congress in occasion of the celebration of the centenary of birth of Mauro Picone and Leonida Tonelli (held in
Rome on May 6–9, 1985). The Author was one of his pupils and, after his death, held his chair of mathematical analysis at the
University of Pisa, becoming dean of the faculty of sciences and then rector: he exerted a strong positive influence on the development of the university.
Evans, Lawrence,
An Introduction to Stochastic Differential Equations(PDF), p. 130, retrieved July 11, 2011. A set of lecture notes surveying "without too many precise details, the basic theory of probability, random differential equations and some applications", as the author himself states.
Frehse, Jens (1972), "On the regularity of the solution of a second order variational inequality", Bolletino della Unione Matematica Italiana, Serie IV, vol. 6, pp. 312–315,
MR0318650,
Zbl0261.49021.