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{{Short description|Motivating example in mathematical study}}
The '''obstacle problem''' is a classic motivating example in the [[mathematical]] study of [[variational inequalities]] and [[free boundary problem]]s. The problem is to find the [[Mechanical equilibrium|equilibrium]] position of an [[Solid mechanics|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.<ref name="Caf384">See {{harvnb|Caffarelli|1998|p=384}}.</ref>
 
The mathematical formulation of the problem is to seek minimizers of the [[Dirichlet energy]] functional,
 
:{{bi|left=1.6|<math>\displaystyle J = \int_D |\nabla u|^2 \mathrm{d}x</math>}}
in some domain ''<math>D</math>'' where the functions ''<math>u</math>'' represent the vertical displacement of the membrane. In addition to satisfying [[Dirichlet boundary conditions]] corresponding to the fixed boundary of the membrane, the functions ''<math>u</math>'' are in addition constrained to be greater than some given ''obstacle'' function ''<math>\phi</math>''<math>(x)</math>. 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 some domaindomains ''<math>D</math>'' where the functions ''<math>u</math>'' represent the vertical displacement of the membrane. In addition to satisfying [[Dirichlet boundary conditions]] corresponding to the fixed boundary of the membrane, the functions ''<math>u</math>'' are in addition constrained to be greater than some given ''obstacle'' function ''<math>\phi</math>''<math>(x)</math>. 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 continuity|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 continuity|Hölder continuous]] surface except at certain singular points, which reside on a smooth manifold.
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==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 ''<math>\phi</math>''<math>(x)</math> in the interior of the domain as well.<ref name="Caf383">See {{harvnb|Caffarelli|1998|p=383}}.</ref> In this case, the energy functional to be minimized is the surface area integral, or
 
:{{bi|left=1.6|<math>\displaystyle J(u) = \int_D \sqrt{1 + |\nabla u|^2}\,\mathrm{d}x.</math>}}
 
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]]
 
:{{bi|left=1.6|<math>\displaystyle J(u) = \int_D |\nabla u|^2 \mathrm{d}x.</math>}}
 
===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 ''<math>\phi</math>''<math>(x)</math>.
 
In the simple case wherewherein the process is [[Brownian motion]], and the process is forced to stop upon exiting the domain, the solution <math>u(x)</math> of the obstacle problem can be characterized as the expected value of the payoff, starting the process at <math>x</math>, if the optimal stopping strategy is followed. The stopping criterion is simply that one should stop upon reaching the ''contact set''.<ref>See the lecture notes by {{harvtxt|Evans|Version 1.2| pp=110–114}}.</ref>
 
==Formal statement==
Suppose the following data is given:
#an [[open set|open]] [[bounded set|bounded]] [[Domain_Domain (mathematicsmathematical analysis)#Real and complex analysis|domain]] <math>D\subseteq\mathbb{R}^n</math> ⊂ ℝ''<sup>n</sup>'' with [[smooth function|smooth]] [[boundary (topology)|boundary]]
#a [[smooth function]] <math>f (x)</math> on '''∂'''<math>\partial D</math> (the [[boundary (topology)|boundary]] of <math>D</math>)
#a smooth function ''<math>\varphi</math>''<math>(x)</math> defined on all of <math>D</math> such that <math>\scriptstyle\varphi|_{\partial D}</math> < <math>f</math>, i.e., the restriction of ''<math>\varphi</math>''<math>(x)</math> to the boundary of <math>D</math> (its [[Trace operator|trace]]) is less than <math>f</math>.
 
Then consider the set
 
:{{bi|left=1.6|<math>\displaystyle K = \left\{ u(x) v\in H^1(D): uv|_{\partial D} = f(x)\text{ and } uv \geq \varphi \right\},</math>}}
which is a [[closed set|closed]] [[convex set|convex]] [[subset]] of the [[Sobolev space]] of square [[integrable function]]s with square integrable [[weak derivative|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]]▼
 
:<math>J(u) = \int_D |\nabla u|^2\mathrm{d}x</math>▼
▲which is a [[closed set|closed]] [[convex set|convex]] [[subset]] of the [[Sobolev space]] <math>H^1(D)</math> of square [[integrable function]]s with squaredomain integrable<math>D</math> whose [[weak derivative|weak first derivatives]] is square integrable, containing precisely those functions with the desired boundary conditions whichand arewhose alsovalues above the obstacle's. TheA solution to the obstacle problem is thea function <math>u\in K</math> which minimizes the energy [[integral]]
over all functions <math>u(x)</math> belonging to <math>K</math>; the existence of such a minimizer is assured by considerations of [[Hilbert space]] theory.<ref name="Caf383"/><ref>See {{harvnb|Kinderlehrer|Stampacchia|1980|pp=40–41}}.</ref>▼
 
▲:{{bi|left=1.6|<math>\displaystyle J(uv) = \int_D |\nabla uv|^2\mathrm{d}x</math>}}
 
over all functions <math>v</math> belonging to <math>K</math>; in symbols
:<math>J(u)=\operatorname{min}_{v\in K} J(v)\text{ or }u\in\operatorname{Argmin}_K J.</math>
 
▲overThe allexistence functionsand <math>u(x)</math> belonging to <math>K</math>; the existenceuniqueness of such a minimizer is assured by considerations of [[Hilbert space]] theory.<ref name="Caf383"/><ref>See {{harvnb|Kinderlehrer|Stampacchia|1980|pp=40–41}}.</ref>
 
==Alternative formulations==
===Variational inequality===
{{See also|Variational inequality}}
The obstacle problem can be reformulated as a standard problem in the theory of [[variational inequality|variational inequalities]] on [[Hilbert space]]s. Seeking the energy minimizer in the set ''<math>K</math>'' of suitable functions is equivalent to seeking
 
:{{bi|left=1.6|<math>\displaystyle u \in K</math> '''such that''' <math>\int_D\langle {\nabla u} , \cdot{\nabla (v - u)}\rangle \mathrmoperatorname{d} x \geq 0\qquad\forall v \in K, </math>}}
 
where ⟨ . , . ⟩ : ℝ''<supmath>\cdot:\mathbb{R}^n</sup>''\times × ℝ''<sup>\mathbb{R}^n\to\mathbb{R}</supmath>'' → ℝ is the ordinary [[scalar product]] in the [[Dimension (mathematics)|finite-dimensional]] [[Real number|real]] [[vector space]] ℝ''<supmath>\mathbb{R}^n</supmath>''. This is a special case of the more general form for variational inequalities on Hilbert spaces, whose solutions are functions ''<math>u</math>'' in some closed convex subset ''<math>K</math>'' of the overall space, such that
 
:{{bi|left=1.6|<math>\displaystyle a(u,v-u) \geq fl(v-u)\qquad\forall v \in K.\,</math>}}
 
for [[Coercive function|coercive]], [[Real numbers|real-valued]], [[bounded operator|bounded]] [[bilinear form]]s <math>a(uv,w)\mapsto a(v,w)</math> and bounded [[linear functional]]s <math>fv\mapsto l(v)</math>
on <math>H^1(D)</math>.<ref name="KS-chapter2">See {{harvnb|Kinderlehrer|Stampacchia|1980|pp=23–49}}.</ref>
 
===Least superharmonic function===
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===Optimal regularity===
The solution to the obstacle problem has <math>\scriptstyle C^{1,1}</math> regularity, or [[bounded function|bounded]] [[Derivative#Higher derivatives|second derivative]]s, when the obstacle itself has these properties.<ref>See {{harvnb|Frehse|1972}}.</ref> 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 <math>\scriptstyle\phi(x)</math> has modulus of continuity <math>\scriptstyle\sigma(r)</math>, that is to say that <math>\scriptstyle|\phi(x) - \phi(y)|\leq \sigma(|x-y|)</math>, then the solution <math>\scriptstyle u(x)</math> has modulus of continuity given by <math>\scriptstyle C\sigma(2r)</math>, where the constant depends only on the domain and not the obstacle.
#If the obstacle's first derivative has modulus of continuity <math>\scriptstyle\sigma(r)</math>, then the solution's first derivative has modulus of continuity given by <math>\scriptstyle C r \sigma(2r)</math>, where the constant again depends only on the domain.<ref>See {{harvnb|Caffarelli|1998|p=386}}.</ref>
 
===Level surfaces and the free boundary===
Subject to a degeneracy condition, level sets of the difference between the solution and the obstacle, <math>\scriptstyle\{x: u(x) -\phi(x) = t\}</math> for <math>\scriptstyle t > 0</math> are <math>\scriptstyle C^{1,\alpha}</math> surfaces. The free boundary, which is the boundary of the set where the solution meets the obstacle, is also <math>\scriptstyle C^{1,\alpha}</math> except on a set of ''singular points,'' which are themselves either isolated or locally contained on a <math>\scriptstyle C^1</math> manifold.<ref>See {{harvnb|Caffarelli|1998|ppp=394 and 397}}.</ref>
 
==Generalizations==
The theory of the obstacle problem is extended to other divergence form uniformly [[elliptic operator]]s,<ref name="KS-chapter2"/> 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 partial differential equation|parabolic]], time-dependent cases of the obstacle problem and its variants are also objects of study.
 
== See also ==
*[[Barrier option]]
*[[Minimal surface]]
*[[Variational inequality]]
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==Notes==
{{reflist|30em}}
 
==Historical references==
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| editor-first = G.
| editor2-last = Amerio
| editor2-first = L.
| editor2-link = Luigi Amerio
| editor3-last = Acquaro
| editor3-first = G.
| editor4-last = Baiada
| editor4-first = E.
| editor5-last = Cesari
| editor5-first = L.
| editor5-link = Lamberto Cesari
| editor6-last = Ciliberto
| editor6-first = C.
| editor7-last = Cimmino
| editor7-first = G.
| editor7-link = Gianfranco Cimmino
| editor8-last = Cinquini
| editor8-first = S.
| editor9-last = De Giorgi
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| editor11-last = Fichera
| editor11-first = G.
| editor11-link = Gaetano Fichera
| editor12-last = Galligani
| editor12-first = I.
| editor13-last = Ghizzetti
| editor13-first = A.
| editor13-link = Aldo Ghizzetti
| editor14-last = Graffi
| editor14-first = D.
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| editor17-last = Magenes
| editor17-first = E.
| editor17-link = Enrico Magenes
| editor18-last = Martinelli
| editor18-first = E.
| editor18-link = Enzo Martinelli
| editor19-last = Pettineo
| editor19-first = B.
| editor20-last = Scorza
| editor20-first = G.
| editor20-link = Giuseppe Scorza Dragoni
| editor21-last = Vesentini
| editor21-first = E.
| editor21-link = Edoardo Vesentini
| display-editors = 4
| contribution = Leonida Tonelli e la scuola matematica pisana
| title = Convegno celebrativo del centenario della nascita di Mauro Picone e Leonida Tonelli (6–9 maggio 1985)
| language = Italian
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| publisher = [[Accademia Nazionale dei Lincei]]
| doi =
| access-date = 2013-02-12
| archive-url = https://web.archive.org/web/20110223030014/http://www.lincei.it/pubblicazioni/catalogo/volume.php?lg=e&rid=32847
| archive-date = 2011-02-23
| url-status = dead
}}. "''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.
 
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| journal=The Journal of Fourier Analysis and Applications
| volume=4
| issue=4-54–5
| pages=383–402
| url=http://www.springerlink.com/content/u2105854886n6282/
| doi=10.1007/BF02498216
| mr = 1658612
| zbl =0928.49030
| s2cid=123431389
}}
*{{Citation
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| year=1972
| periodical=Bolletino della Unione Matematica Italiana
| series = Serie IV,
| volume=6
| issue=
| pages=312–215312–315
| id=
| mr=318650
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}}.
*{{Citation
| lastlast1=Kinderlehrer
| firstfirst1=David
| author-link=David Kinderlehrer
| last2=Stampacchia
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| zbl=0457.35001
}}
*{{citation|last1=Petrosyan|first1=Arshak|last2=Shahgholian|first2=Henrik|last3=Uraltseva|first3=Nina|title=Regularity of Free Boundaries in Obstacle-Type Problems. Graduate Studies in Mathematics|publisher=American Mathematical Society, Providence, RI|year=2012|isbn=978-0-8218-8794-3 }}
 
== External links ==
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| series = draft from the [[Fermi Lecture]]s
|date=August 1998
| pagespage = 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.
 
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