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Source: Obstacle problem

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.
The mathematical formulation of the problem is to seek minimizers of the Dirichlet energy functional,

in some domains

D

{\displaystyle D}

where the functions

u

{\displaystyle u}

represent the vertical displacement of the membrane. In addition to satisfying Dirichlet boundary conditions corresponding to the fixed boundary of the membrane, the functions

u

{\displaystyle u}

are in addition constrained to be greater than some given obstacle function

ϕ
(
x
)

{\displaystyle \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 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

Motivating problems

= Shape of a membrane above an obstacle

= Optimal stopping

Formal statement

, i.e., the restriction of

φ

{\displaystyle \varphi }

to the boundary of

D

{\displaystyle D}

(its trace) is less than

f

{\displaystyle f}

.
Then consider the set

which is a closed convex subset of the Sobolev space

H

1

(
D
)

{\displaystyle H^{1}(D)}

of square integrable functions with domain

D

{\displaystyle D}

whose weak first derivatives is square integrable, containing those functions with the desired boundary conditions and whose values above the obstacle's. A solution to the obstacle problem is a function

u
∈
K

{\displaystyle u\in K}

which minimizes the energy integral

over all functions

v

{\displaystyle v}

belonging to

K

{\displaystyle K}

; in symbols

J
(
u
)
=

min

v
∈
K

⁡
J
(
v
)

or

u
∈

Argmin

K

⁡
J
.

{\displaystyle J(u)=\operatorname {min} _{v\in K}J(v){\text{ or }}u\in \operatorname {Argmin} _{K}J.}

The existence and uniqueness 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 on Hilbert spaces. Seeking the energy minimizer in the set

K

{\displaystyle K}

of suitable functions is equivalent to seeking

where

⋅
:

R

n

×

R

n

→

R

{\displaystyle \cdot :\mathbb {R} ^{n}\times \mathbb {R} ^{n}\to \mathbb {R} }

is the ordinary scalar product in the finite-dimensional real vector space

R

n

{\displaystyle \mathbb {R} ^{n}}

. This is a special case of the more general form for variational inequalities on Hilbert spaces, whose solutions are functions

u

{\displaystyle u}

in some closed convex subset

K

{\displaystyle K}

of the overall space, such that

for coercive, real-valued, bounded bilinear forms

(
v
,
w
)
↦
a
(
v
,
w
)

{\displaystyle (v,w)\mapsto a(v,w)}

and bounded linear functionals

v
↦
l
(
v
)

{\displaystyle v\mapsto l(v)}

on

H

1

(
D
)

{\displaystyle H^{1}(D)}

.

= 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 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

C

1
,
1

{\displaystyle C^{1,1}}

regularity, or bounded second derivatives, when the obstacle itself has these properties. 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

ϕ
(
x
)

{\displaystyle \phi (x)}

has modulus of continuity

σ
(
r
)

{\displaystyle \sigma (r)}

, that is to say that

|

ϕ
(
x
)
−
ϕ
(
y
)

|

≤
σ
(

|

x
−
y

|

)

{\displaystyle |\phi (x)-\phi (y)|\leq \sigma (|x-y|)}

, then the solution

u
(
x
)

{\displaystyle u(x)}

has modulus of continuity given by

C
σ
(
2
r
)

{\displaystyle 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

σ
(
r
)

{\displaystyle \sigma (r)}

, then the solution's first derivative has modulus of continuity given by

C
r
σ
(
2
r
)

{\displaystyle Cr\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,

{
x
:
u
(
x
)
−
ϕ
(
x
)
=
t
}

{\displaystyle \{x:u(x)-\phi (x)=t\}}

for

t
>
0

{\displaystyle t>0}

are

C

1
,
α

{\displaystyle C^{1,\alpha }}

surfaces. The free boundary, which is the boundary of the set where the solution meets the obstacle, is also

C

1
,
α

{\displaystyle C^{1,\alpha }}

except on a set of singular points, which are themselves either isolated or locally contained on a

C

1

{\displaystyle C^{1}}

manifold.

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.

Notes

Historical references

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.

References

Caffarelli, Luis (July 1998), "The obstacle problem revisited", The Journal of Fourier Analysis and Applications, 4 (4–5): 383–402, doi:10.1007/BF02498216, MR 1658612, S2CID 123431389, Zbl 0928.49030
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, MR 0318650, Zbl 0261.49021.
Friedman, Avner (1982), Variational principles and free boundary problems, Pure and Applied Mathematics, New York: John Wiley & Sons, pp. ix+710, ISBN 0-471-86849-3, MR 0679313, Zbl 0564.49002.
Kinderlehrer, David; Stampacchia, Guido (1980), An Introduction to Variational Inequalities and Their Applications, Pure and Applied Mathematics, vol. 88, New York: Academic Press, pp. xiv+313, ISBN 0-12-407350-6, MR 0567696, Zbl 0457.35001
Petrosyan, Arshak; Shahgholian, Henrik; Uraltseva, Nina (2012), Regularity of Free Boundaries in Obstacle-Type Problems. Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, ISBN 978-0-8218-8794-3

External links

Caffarelli, Luis (August 1998), The Obstacle Problem (PDF), draft from the Fermi Lectures, p. 45, retrieved July 11, 2011, delivered by the author at the Scuola Normale Superiore in 1998.