Obstacle problem
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The obstacle problem is a classic motivating example in the
mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
study of
variational inequalities In mathematics, a variational inequality is an inequality involving a functional, which has to be solved for all possible values of a given variable, belonging usually to a convex set. The mathematical theory of variational inequalities was initi ...
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 In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces tha ...
and the
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in
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as well. Applications include the study of fluid filtration in porous media, constrained heating, elasto-plasticity, optimal control, and financial mathematics.See . The mathematical formulation of the problem is to seek minimizers of the
Dirichlet energy In mathematics, the Dirichlet energy is a measure of how ''variable'' a function is. More abstractly, it is a quadratic functional on the Sobolev space . The Dirichlet energy is intimately connected to Laplace's equation and is named after the ...
functional, :J = \int_D , \nabla u, ^2 \mathrmx in some domains ''D'' where the functions ''u'' represent the vertical displacement of the membrane. In addition to satisfying
Dirichlet boundary conditions In the mathematical study of differential equations, the Dirichlet (or first-type) boundary condition is a type of boundary condition, named after Peter Gustav Lejeune Dirichlet (1805–1859). When imposed on an ordinary or a partial differentia ...
corresponding to the fixed boundary of the membrane, the functions ''u'' 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 In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there e ...
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 Hölder: * ''Hölder, Hoelder'' as surname * Hölder condition * Hölder's inequality * Hölder mean In mathematics, generalized means (or power mean or Hölder mean from Otto Hölder) are a family of functions for aggregating sets of number ...
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 In mathematics, Plateau's problem is to show the existence of a minimal surface with a given boundary, a problem raised by Joseph-Louis Lagrange in 1760. However, it is named after Joseph Plateau who experimented with soap films. The problem ...
), with the added constraint that the membrane is constrained to lie above some obstacle ''\phi''(x) in the interior of the domain as well.See . In this case, the energy functional to be minimized is the surface area integral, or :J(u) = \int_D \sqrt\,\mathrmx. This problem can be ''linearized'' in the case of small perturbations by expanding the energy functional in terms of its
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
and taking the first term only, in which case the energy to be minimized is the standard
Dirichlet energy In mathematics, the Dirichlet energy is a measure of how ''variable'' a function is. More abstractly, it is a quadratic functional on the Sobolev space . The Dirichlet energy is intimately connected to Laplace's equation and is named after the ...
:J(u) = \int_D , \nabla u, ^2 \mathrmx.


Optimal stopping

The obstacle problem also arises in
control theory Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
, specifically the question of finding the optimal stopping time for a stochastic process with payoff function ''\phi''(x). In the simple case wherein the process is
Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...
, 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 statement

Suppose the following data is given: #an
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bounded domain D ⊂ ℝ''n'' with
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boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment * ''Boundaries'' (2016 film), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film *Boundary (cricket), the edge of the pla ...
#a smooth function f (x) on ∂D (the
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of D) #a smooth function ''\varphi''(x) defined on all of D such that \scriptstyle\varphi, _ < f, i.e. the restriction of ''\varphi''(x) to the boundary of D (its
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) is less than f. Then consider the set :K = \left\, which is a closed
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subset of the
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 together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
of square
integrable function In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with d ...
s 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 In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...
:J(u) = \int_D , \nabla u, ^2\mathrmx 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

The obstacle problem can be reformulated as a standard problem in the theory of
variational inequalities In mathematics, a variational inequality is an inequality involving a functional, which has to be solved for all possible values of a given variable, belonging usually to a convex set. The mathematical theory of variational inequalities was initi ...
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\langle , \rangle \mathrmx \geq 0\qquad\forall v \in K, where ⟨ . , . ⟩ : ℝ''n'' × ℝ''n'' → ℝ is the ordinary
scalar product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
in the finite-dimensional
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ℝ''n''. 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)\qquad\forall v \in K.\, for
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,
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, bounded bilinear forms a(u,v) and bounded
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s f(v).See .


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. In fact, an application of the
maximum principle In the mathematical fields of partial differential equations and geometric analysis, the maximum principle is any of a collection of results and techniques of fundamental importance in the study of elliptic and parabolic differential equations. ...
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 \scriptstyle C^ regularity, or bounded
second derivative In calculus, the second derivative, or the second order derivative, of a function is the derivative of the derivative of . Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, ...
s, when the obstacle itself has these properties. More precisely, the solution's
modulus of continuity In mathematical analysis, a modulus of continuity is a function ω : , ∞→ , ∞used to measure quantitatively the uniform continuity of functions. So, a function ''f'' : ''I'' → R admits ω as a modulus of continuity if and only if :, f(x)-f ...
and the modulus of continuity for its
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
are related to those of the obstacle. #If the obstacle \scriptstyle\phi(x) has modulus of continuity \scriptstyle\sigma(r), that is to say that \scriptstyle, \phi(x) - \phi(y), \leq \sigma(, x-y, ), then the solution \scriptstyle u(x) has modulus of continuity given by \scriptstyle C\sigma(2r), where the constant depends only on the domain and not the obstacle. #If the obstacle's first derivative has modulus of continuity \scriptstyle\sigma(r), then the solution's first derivative has modulus of continuity given by \scriptstyle C r \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, \scriptstyle\ for \scriptstyle t > 0 are \scriptstyle C^ surfaces. The free boundary, which is the boundary of the set where the solution meets the obstacle, is also \scriptstyle C^ except on a set of ''singular points,'' which are themselves either isolated or locally contained on a \scriptstyle C^1 manifold.See .


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 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.


See also

*
Minimal surface In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces tha ...
* Variational inequality * Signorini problem


Notes


Historical references

*. "''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 , established_title = Founded , established_date = 753 BC , founder = King Romulus (legendary) , image_map = Map of comune of Rome (metropolitan city of Capital Rome, region Lazio, Italy).svg , map_caption ...
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 The University of Pisa ( it, Università di Pisa, UniPi), officially founded in 1343, is one of the oldest universities in Europe. History The Origins The University of Pisa was officially founded in 1343, although various scholars place ...
, becoming dean of the faculty of sciences and then rector: he exerted a strong positive influence on the development of the university.


References

* *. 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. *. *. * *


External links

*{{Citation , last = Caffarelli , first = Luis , author-link = Luis Caffarelli , title = The Obstacle Problem , place = , publisher = , series = draft from the Fermi Lectures , date=August 1998 , page = 45 , language = , url = http://www.ma.utexas.edu/users/combs/obstacle-long.pdf , accessdate = July 11, 2011 , ref=none , delivered by the author at the Scuola Normale Superiore in 1998. Partial differential equations Calculus of variations