Obstacle problem: Difference between revisions

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(x)</math>'' in the interior of the domain as well.<ref name="Caf383">Caffarelli, ppSee {{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}\, dx\mathrm{d}x.</math>}}

:{{bi|left=1.6|<math>\displaystyle J(u) = \int_D |\nabla u|^2 \,dxmathrm{d}x.</math>}}

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(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,| pp =110–114}}.</ref>

==Formal formulationstatement==

#aan [[open set|open]] [[bounded set|bounded]] [[Domain_Domain (mathematicsmathematical analysis)#Real and complex analysis|domain]] <math>\scriptstyle D \subset subseteq\mathbb{R}^n</math> with [[smooth function|smooth]] [[boundary (topology)|boundary]]

#a [[smooth function]] <math>f(x)</math> on <math>\scriptstyle \partial D</math> (the [[boundary (topology)|boundary]] of <math>D</math>)

#a smooth function ''<math>\phi(x)varphi</math>'' defined on all of <math>D</math> such that <math>\scriptstyle\varphi|_{\partial D} < f</math>, i.e., the restriction of ''<math>\phi(x)varphi</math>'' to the boundary of <math>D</math> (its [[Trace operator|trace]]) is less than <math>f.</math>.

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\text{</math> '''such that''' } <math>\int_D ({\nabla u) }\cdot ({\nabla (v - u))} \,operatorname{d} dxx \geq 0\qquad\forall v \in K., </math>}}

where <math>\cdot:\mathbb{R}^n\times \mathbb{R}^n\to\mathbb{R}</math> is the ordinary [[scalar product]] in the [[Dimension (mathematics)|finite-dimensional]] [[Real number|real]] [[vector space]] <math>\mathbb{R}^n</math>. 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 ''a''<math>(''u''v,&nbsp;''w)\mapsto a(v'',w)</math> and bounded [[linear functional]]s ''ƒ''<math>v\mapsto l(''v'')</math>
on <math>H^1(D)</math>.<ref name="KS-chapter2">See {{harvnb|Kinderlehrer and |Stampacchia, |1980|pp =23–49}}.</ref>

{{See also|Superharmonic function|Viscosity solution}}

The solution to the obstacle problem has <math>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>\phi(x)</math> has modulus of continuity <math>\sigma(r)</math>, that is to say that <math>\scriptstyle|\phi(x) - \phi(y)|\leq \sigma(|x-y|)</math>, then the solution <math>u(x)</math> has modulus of continuity given by <math>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>\sigma(r)</math>, then the solution's first derivative has modulus of continuity given by <math>C r \sigma(2r)</math>, where the constant again depends only on the domain.<ref>See {{harvnb|Caffarelli, pp |1998|p=386}}.</ref>

Subject to a degeneracy condition, level sets of the difference between the solution and the obstacle, <math>\{x: u(x) -\phi(x) = t\}</math> for <math>t > 0</math> are <math>C^{1,\alpha}</math> surfaces. The free boundary, which is the boundary of the set where the solution meets the obstacle, is also <math>C^{1,\alpha}</math> except on a set of ''singular points,'' which are themselves either isolated or locally contained on a <math>C^1</math> manifold.<ref>Caffarelli,See {{harvnb|Caffarelli|1998|pp=394, and 397}}.</ref>

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.