A Brief Overview on The Obstacle Problem

Regis Monneau

Abstract. We present a short survey on the obstacle problem including thetheory developed by L. A. Caffarelli and the theory developed independentlyby G. S. Weiss. We also present some other recent results on the regularity ofthe free boundary.

1. Introduction

In this note we will present a short survey on the obstacle problem. This presen­tation is obviously far from being exhaustive. The author has decided to presentsome new and old important results chosen from a wide literature. In particularthe references of this note are deliberately limited.

A number of books deal with the obstacle problem, e.g., the 1980 book ofD. Kinderlehrer and G. Stampacchia [17] on variational inequalities, and the bookof A. Friedman [12], which covers some results on the obstacle problem through1982, as well as many other free boundary problems. For examples of applicationsof the obstacle problem to Mathematical Physics and many references until 1987look at the book of J. F. Rodrigues [20]. Its Chapter 6 gives in particular a goodoverview and detailed results on the obstacle problem.

1.1. A physical example

Let us consider a horizontal wire and a membrane hanging on this wire. We assumethat this membrane is above a plate. We push up the plate on the membrane. Weget a contact area between the membrane and the obstacle which is the plate.This contact area is called the coincidence set (see Figure 1). The boundary of thecoincidence set is called the free boundary for the obstacle problem. The goal ofthis note is to throw some light on the properties of this free boundary.

1.2. Mathematical formulationWe now give a simple mathematical model for this problem. We assume that themembrane is given by the graph of a function

u: f2cRn~R

1991 Mathematics Subject Classification. 35R35.Key words and phrases. Obstacle problem, Coincidence set, Free boundary, Blowup, Liouvilleresult, Monotonicity formula, Stability, Convexity, Generic regularity.

C. Casacuberta et al. (ed.), European Congress of Mathematics© Springer Basel AG 2001

304 R. Monneau

where 0 is a smooth bounded domain.

u(X)

wire

FIGURE 1. Membrane over a plate obstacle

We assume that the wire is given by the value of u on the fixed boundary 80which is assumed to be a constant

u = constant = g on 80 . (1)

We assume that the plate is at the O-level, and we assume that the membranestays above the plate, i.e., the obstacle

u 2: 0 on O.

We assume that u minimizes an energy on 0 which is given by

In lV'ul2 + 2u (2)

where the minimization is mathematically taken on the Sobolev space of functions

K={UEH1(0), u=gon80, u2:0onO}. (3)

The first term in the energy (2) is a kind of elastic energy and the second term isa kind of gravity energy for a heavy membrane.

The well-posedness and good properties of the obstacle problem are guaran­teed by the following result for g > 0:

Theorem 1.1. (J. Frehse [11], el,l regularity of the minimizer) There exists aunique solution u minimizing energy (2) on the convex set K defined in (3). More­over this solution satisfies

{~u= 1

u>Ou E C1,1(O)

on {u>O}nOon 0 (4)

A Brief Overview on The Obstacle Problem

The coincidence set is

305

{u = O}

and the free boundary is 8{u = O}. Moreover it is proved that the free boundaryhas a finite (hyper)area:

Theorem 1.2. (H. Brezis, D. Kinderlehrer [3], L. A. Caffarelli [6], Finite Hausdorffmeasure)

Tln -1(8{u = O}) 5, C.

1.3. Examples of singularities

FIGURE 2. Contact point

FIGURE 3. Cusp

Let us now give some examples of singularities presented by D. G. Schaefferin (23]. In two dimensions, the free boundary can have a contact point (see Figure 2)or a cusp point (see Figure 3). Following Schaeffer [23], the intuitive origin of thesesingularities can be obtained by "continuous deformation" with varying obstacleswith different components, the case of Figure 2 being the "product" of the joining oftwo components, and the case of Figure 3 a further consequence of the vanishing ofthe diameter in the right part of Figure 2. Notice that the double point of 8{u = O}consists of two tangent curves and not two curves intersecting at a non-zero angle.

2. The Blowup Method

To prove regularity results on the free boundary, the main tool (first introducedfor the obstacle problem by L. A. Caffarelli in [4]) is the notion of blowup.

Let us consider a solution u of (4) and assume that Xo is a point on the freeboundary 8{u = O}. Let us consider the sequence of functions

u€(X) = u(Xo~ EX) .E

306 R. Monneau

oU (x ,x )= 1/2 (max(x 0)) 2

I 2 2

By Theorem 1.1, u€(O) = V'u€(O) = 0 and the second derivatives ID2u€1 arebounded by a constant independent of E > O. Up to extraction of a subsequencewe get a limit as E --+ 0,

u=O

oU (x I'X

2)= 1/2

FIGURE 4. Blowup limits

In Figure 4 we have represented some possible limits in two dimensions. Inany dimensions, the main result for the regularity of the free boundary is thefollowing

Theorem 2.1. (L. A. Caffarelli [5, 7], Characterization of the blowup limit) Theblowup limit is unique and depends only on the point Xo on the free boundary.Moreover either there exists a unit vector vXo E sn-l such that

1uO(X) = 2(max( (X, VXo), 0)2

and the point Xo is called regular,or UO is a quadratic form, i. e.,

uO(X) = ~ tX .Qxo . X ~ 0

where Qxo is a symmetric matrix n x n such that tr rv Qxo = 1. Such a point Xois called singular.

This theorem says two things:

i) A Liouville result which classifies all possible blowup limits on Rn.ii) The uniqueness of the blowup limit for each point X o.The Liouville result was obtained by L. A. Caffarelli in [5] and essentially uses themaximum principle. We can also cite the work of G. S. Weiss [25] which helps toclassify all solutions on Rn invariant by some dilatations.

A Brief Overview on The Obstacle Problem 307

The uniqueness of the blowup limit is difficult to prove in the case of regularpoints and will be discussed in Section 3. On the contrary, in the case of singularpoints, this uniqueness is quite easy to prove. In [7] L. A. Caffarelli uses themonotonicity formula of H. W. Alt, L. A. Caffarelli and A. Friedman [1] appliedto the first derivatives of the solution to prove it. A more elementary fact whichavoids deriving the solution u is the following:

Theorem 2.2. (R. Monneau [19], Monotonicity formula for singular points) Letv(X) = ~ tX . Q . X 2: 0 where Q is a symmetric matrix n x n such thattr rv Q = 1. If u is a solution to (4) and the origin 0 is a singular point, then

_1_ { (u _ V)2 is nondecreasing in r.rn+3 J8Br (O)

This result is a simple corollary of the following monotonicity formula dis­covered by G. S. Weiss:

Theorem 2.3. (G. S. Weiss [25], Monotonicity formula for the obstacle problem)For each point X o E 0 such that the ball Br(Xo) cO we have

eJ>xo,u(r) := ( n~2 { IV'ul 2 + 2u - n~3 ( 2U2

)r JBr(Xo) r J8Br(Xo)

is nondecreasing in r .

In particular with these monotonicity formulas it is possible to prove

Theorem 2.4. (L. A. Caffarelli [7], Continuity on the set of singular points) Themap

Xo~Qxo

is continuous on the set of singular points of the free boundary. Moreover the setof singular points is a closed set included in a 0 1 (n - 1) -dimensional submanifold.

More precise results can be found in [7].

Remark 2.5. We have

and taking the limit we get

eJ>o,uo(r) = constant = eJ>xo,u(O+).

Using Theorem 2.1, it is possible to compute for a point X o of the free boundary

eJ> (0+) = {an for a regular point X o (5)Xo,u 2 f . l . Xan J or a smgu ar pomt °

where an is a constant which depends only on the dimension n.

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3. Regular Points

R. Monneau

The expected property for regular points is the following

Theorem 3.1. (L. A. Caffarelli [4, 5], G. S. Weiss [25], Smoothness of the freeboundary near a regular point) If Xo is a regular point, then in some appropri­ate coordinates, the coincidence set is a subgraph {xn ::::; h(Xl,'" ,Xn -l)} in aneighborhood of X O, where the function h is of class C 1 •

This result is completed by

Theorem 3.2. (D. Kinderlehrer, L. Nirenberg [16], V. Isakov [14], Higher regular­ity of the free boundary) If for a solution of (4), the free boundary is locally C1 ,

this free boundary is analytic.

3.1. Criteria for regular points

We will present an energetic criterion first and then a geometric criterion.Using the monotonicity formula (Theorem 2.3) and the property (5), we get

Theorem 3.3. (G. S. Weiss [25], Energetic criterion) If

<I>xo,u(r) < 2an

for some r, then the point X o is a regular point.

To introduce the geometric criterion found by L. A. Caffarelli, we need thefollowing

Definition 3.4. (Thickness of the coincidence set) We define the thickness of thecoincidence set {u = o} in a ball Br(Xo) by

1Or(XO) = -m.d. ({u = o} n Br(Xo))

rwhere the minimum diameter (m.d.) of {u = o} n Br(Xo) is the infimum of thedistances between pairs of parallel hyperplanes whose strip determined by themcontains it.

Theorem 3.5. (L. A. Caffarelli [7], Geometric criterion) For each r > 0, thereexists a critical thickness a(r) with a(r) ~°as r ~ 0, such that if

or(XO) > a(r)

for some point Xo of the free boundary and for one radius r > 0, then the point Xois regular.

In fact the uniqueness of the blowup limit for regular points is a consequenceof Theorem 3.1. The main difficulty is to avoid the free boundary rotating (slowlyand slowly) around X o. In the approach of G. S. Weiss, the proof of Theorem 3.1is based on an energy power decay estimate on <I>xo,u(r) - <I>xo,u(O+) ~ °whichis enough to control the axis of the blowup limit.

The approach of L. A. Caffarelli to prove Theorem 3.1 is based on classicaltools like the Maximum Principle and Harnack Inequality, used very cleverly tocontrol the rotation of the axis.

A Brief Overview on The Obstacle Problem 309

(6)

3.2. A counterexample

Let us consider the slightly more general obstacle problem

{~U=f~1 on {u>O}nOu ~ 0 on 0u E Cl,l(O)

where f E CQ:(O); then Theorems 2.1 and 3.1 are still true. But we cannot go upto the regularity f E CO as shown by the following counterexample

Theorem 3.6. (I. Blank [2], Example of non-uniqueness of the blowup limit) InR 2 , there exists a continuous function f, a solution u to (6) and a point X o of thefree boundary such that the blowup limit in X o is not unique. More precisely foreach II E 81, there exists a sequence Ek ---+ 0 such that

1U€k(X) ~ 2(max((X,II),0))2.

4. More Results in Two Dimensions

Let us recall

Theorem 4.1. (L. A. Caffarelli, N. M. Riviere [8], Regularity of the free boundaryin two dimensions) If C is a connected component of the interior of {u = O} inR 2, then 8C is analytic except in a finite number of points.

Moreover these authors give a precise behaviour of the free boundary neara singularity in R 2 in [9]. See also [24]. We now give more information on thenumber of singularities:

Theorem 4.2. (R. Monneau [18], Small components have at most two singularpoints) If C is the connected component of the interior of {u = O} in R 2 , thenthere exists

such thati) If diam(C) < p, then 8C has at most two singular points.ii) If diam(C) ~ p and Xl, X 2 E 8C are singular points, then IX2 - XII > ~.

5. Stability and Genericity

In this section we consider solutions u to the obstacle problem (4) such that u =constant = g on 80. We will make perturbations varying the parameter g > O. Tothis end we denote u by ug to make explicit the dependence of the solution on g.

Theorem 5.1. (Stability of the free boundary) If the free boundary 8{ugo =O} isanalytic for a particular value go, then 8{ug = O} is analytic for g in a neighbor­hood of go·

310 R. Monneau

This result was first proved by D. G. Schaeffer in [22] when an E Coo, usingthe Nash-Moser inverse function theorem. Using the geometric criterion (Theo­rem 3.5) of L. A. Caffarelli, it is possible to see that Theorem 5.1 is a consequenceof

Theorem 5.2. (L. A. Caffarelli [6], Measure stability) If Ul, U2 satisfy the sameobstacle problem (4), then we can estimate the measure of the symmetric differenceof the coincidence sets by

1

I{U2 = O}~{UI = O}I ~ C1u2 - ullioo(fl)

where C = C(ID2UIILOO(fl)' ID2U2ILOO(fl)' n, n).

See also the book of J. F. Rodrigues [20] for detailed results on more generaloperators.

Let us recall

Conjecture 5.3. (D. G. Schaeffer [21] (1974)) We conjecture that generically theweak solution of the obstacle problem that one obtains variationally is also a strongsolution, by which we mean that the free boundary is a Coo manifold.

The following result is a first answer in a simple case:

Theorem 5.4. (R. Monneau [18], Generic regularity ofthe free boundary) In R 2,

for almost every constant g > 0, the free boundary a{ug = O} is analytic.

The question remains open in higher dimensions.

6. Convexity Property of The Coincidence Set

Theorem 6.1. (B. Kawobl [15], Convexity of the coincidence set) Let n c R n

be a smooth bounded domain. If n is convex, then for each constant g > 0, thecoincidence set {ug = O} is convex (and analytic).

This result was first proved in the particular case n = 2 by A. Friedman andD. Phillips in [13]. Let us give an example of a generalization of this result in twodimensions

Theorem 6.2. (J. Dolbeault, R. Monneau [10], Convexity for nonlinear obstacleproblems) Let n c R 2 a smooth bounded domain. Let ug E H1(n) minimizing

In F(I'VuI 2) +G(u)

under the constraints u 2 0 on nand u = constant = g on an.We assume that F, G E C2 are nondecreasing convex functions, with F'(O) >

0, G' (0) > O. Ifn is convex, then the coincidence set {ug = O} is convex (and C1).

A Brief Overview on The Obstacle Problem 311

Acknowledgements

I would like to thank Jose Francisco Rodrigues who invited me to present this noteat the third European Congress of Mathematics (2000), and for helpful commentson the redaction of this note. Part of this work was done with the help of a NATOgrant for a postdoctoral position at MIT (Cambridge, USA) and I would like tothank these two institutions. In particular I would like to thank David Jerison forstimulating discussions and his help for the references.

Most of all I would like to express my deepest gratitude to my advisors AlexisBonnet and Henri Berestycki.

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Ecole Nationale des Ponts et ChausseesCERMICS6 et 8 avenue Blaise PascalCite Descartes Champs-sur-Marne77455 Marne-la-Vallee Cedex 2, FranceE-mail address:monneau@cermics.enpc.fr