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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 wherein 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| pp=110–114}}.</ref>
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==Formal statement==
Suppose the following data is given:
#an [[open set|open]] [[bounded set|bounded]] [[Domain (mathematical 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|pp=394 and 397}}.</ref>
 
==Generalizations==
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== See also ==
*[[Barrier option]]
*[[Minimal surface]]
*[[Variational inequality]]
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==Notes==
{{reflist|30em}}
 
==Historical references==
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| mr = 1658612
| zbl =0928.49030
| s2cid=123431389
}}
*{{Citation
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}}.
*{{Citation
| lastlast1=Kinderlehrer
| firstfirst1=David
| author-link=David Kinderlehrer
| last2=Stampacchia
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| zbl=0457.35001
}}
*{{citation|lastlast1=Petrosyan|firstfirst1=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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