UNDERSTANDING SINGULARITIES IN FREE BOUNDARY …XAVIER ROS-OTON AND JOAQUIM SERRA Abstract. Free boundary problems are those described by PDEs that exhibit a priori unknown (free)

UNDERSTANDING SINGULARITIES INFREE BOUNDARY PROBLEMS

XAVIER ROS-OTON AND JOAQUIM SERRA

Abstract. Free boundary problems are those described by PDEs that exhibit a prioriunknown (free) interfaces or boundaries. The most classical example is the melting of iceto water (the Stefan problem). In this case, the free boundary is the liquid-solid interfacebetween ice and water.

A central mathematical challenge in this context is to understand the regularity andsingularities of free boundaries.

In this paper we provide a gentle introduction to this topic by presenting some classicalresults of Luis Caffarelli, as well as some important recent works due to Alessio Figalli andcollaborators.

Abstract. I problemi di frontiera libera sono quelli descritti da EDP che mostrano inter-facce o frontiere a priori sconosciuti (liberi). L’esempio più classico è lo scioglimento delghiaccio in acqua (problema di Stefan). In questo caso, la frontiera libera è l’interfacciasolido-liquido tra acqua e ghiaccio.

Una sfida matematica centrale in questo contesto è comprendere la regolarità e le singo-larità delle frontiere libere.

In questo articolo introduciamo questo argomento presentando alcuni risultati classici diLuis Caffarelli, oltre ad alcuni importanti lavori recenti dovuti ad Alessio Figalli e collabo-ratori.

2010 Mathematics Subject Classification. 35R35.Key words and phrases. Free boundaries; obstacle problem.This paper will appear in a special issue of the journal ‘Matematica, Cultura e Società’ (published by the

Unione Matematica Italiana) dedicated to the work of Alessio Figalli.1

2 XAVIER ROS-OTON AND JOAQUIM SERRA

1. Introduction

1.1. The Stefan problem.The Stefan problem, dating back to theXIXth century, is the most classical and im-portant free boundary problem. First consid-ered by Lamé and Clapeyron in 1831, aimsto describe the temperature distribution ina homogeneous medium undergoing a phasechange, typically a body of ice at zero de-grees centigrade submerged in water. It isnamed after Josef Stefan, a Slovenian physi-cist who introduced the general class of suchproblems around 1890; see [52, 53, 37].

The most classical formulation of the Ste-fan problem is as follows: Let Ω ⊂ R3 besome some bounded domain. For concrete-ness, we let us think that Ω is cylindric watertank as depicted in Figure 1. We denote

θ = θ(x, t)

the temperature of the water at the pointx ∈ Ω at time t ∈ R+ := [0,+∞). We as-sume that θ ≥ 0 in Ω × R+. Given are the(nonnegative) initial temperature and tem-perature at the boundary of the tank.

Water

Ice

Figure 1. The Stefan problem.

The set {(x, t) ∈ Ω×R+ : θ(x, t) > 0}, de-noted for brevity {θ > 0}, represents the wa-ter while its complement, denoted {θ = 0},represents the ice. The temperature θ satis-fies the heat equation

∂tθ −∆θ = 0 in the region {θ > 0},

while in the complement θ is just zero.Determining where is the interphase or

free boundary that separates the two re-gions (i.e., the surface ∂{θ > 0}) is partof the problem. For it, an extra equation(or boundary condition) on the interface isneeded. This is the so-called Stefan condi-tion:

∂tθ = |∇xθ|2 on ∂{θ > 0}. (1.1)This extra relation comes from two consid-erations. First, the normal velocity of theinterphase, V , is proportional to the amountof heat absorbed by it (which must be usedto melt the ice). In turn, this heat which“enters” the interphase is, by Fourier law,proportional to the gradient of temperature.Thus, we have |V | = C|∇θ| . Second, sinceθ = 0 on the moving interphase we ob-tain that, on it, V and ∇θ are parallel and(∂t + V · ∇)θ = 0. Combining the two pre-vious formations and choosing the physicalunits to make C = 1, we obtain the Stefancondition (1.1).

One can also see that, by the “maximumprinciple”, the ice {θ = 0} shrinks with time.In other words, if at some point of the tankthere is liquid water for some given time thenthat point will remain liquid at all futuretimes.

It can be shown that, after the transfor-mation

u(x, t) :=

∫ t0

θ(x, τ)dτ,

(see [2, 18]) the new function

u : Ω× R+ → R+

satisfies∂tu−∆u = −χ{u>0}

u ≥ 0∂tu ≥ 0,

(1.2)

where χA denotes the characteristic functionof the set A.

UNDERSTANDING SINGULARITIES IN FREE BOUNDARY PROBLEMS 3

Since we can easily recover θ form u bycomputing its time derivative, we see that(1.2) is an equivalent formulation of the Ste-fan problem. The new formulation is use-ful because it enjoys better mathematicalproperties (it has the structure of “varia-tional inequality”) than the original formu-lation with θ. For instance, while from theoriginal formulation with θ it is unclear howto show existence and uniqueness of solution(there was no rigurous proof for more than acentury!), it is much easier to do it with theequivalent formulation (1.2).

The stationary version of (1.2) is the well-known obstacle problem:

∆u = χ{u>0}

u ≥ 0. (1.3)

It is among the most famous problems in el-liptic PDE, as it arises in a variety of situa-tions.

1.2. Motivations and applications.Both the Stefan problem and the obstacleproblem appear in many different models inphysics, industry, biology, or finance. Wenext briefly comment on some of them, andrefer to the books [19, 34, 43, 27, 42] for moredetails and further applications of obstacle-type problems.

•Phase transitions. In the classical Stefanproblem, as explained in the previous sub-section, the solution u of (1.2) is the integralof the temperature of a solid undergoing aphase transition, such as ice melting to wa-ter.

• Fluid filtration. The so-called Damproblem aims to describe the filtration ofwater inside a porous dam. One considersa porous dam separating two reservoirs ofwater at two different heights; see Figure 2.Then, the interior of the dam has a wet part,where water flows, and a dry part. In thiscontext, an integral of the pressure solves the

obstacle problem (1.3), and the free bound-ary corresponds to the interface between thewet and dry parts of the dam.

Dam

Water

Figure 2. The Dam problem.

• Hele-Shaw flow. This model, datingback to 1898, describes a fluid flow betweentwo flat parallel plates separated by a verythin gap. Various problems in fluid mechan-ics can be approximated to Hele-Shaw flows,and that is why understanding these flows isimportant.

A Hele-Shaw cell (see Figure 3) is an ex-perimental device in which a viscous fluidis sandwiched in a narrow gap between twoparallel plates. In certain regions, the gapis filled with fluid while in others the gap isfilled with air. When liquid is injected insidethe device though some sinks (e.g. though asmall hole on the top plate) the region filledwith liquid grows. In this context, an in-tegral of the pressure solves, for each fixedtime t, the obstacle problem (1.3). Simi-larly as in the Dam problem, the free bound-ary corresponds to the interface between thefluid and the air regions.


Fluid

Air

Injection point

Figure 3. A Helle-Shaw cell.

• Optimal stopping, finance. In proba-bility and finance, both the Stefan problem(1.2) and the obstacle problem (1.3) appearwhen considering optimal stopping problemsfor stochastic processes.

A typical example is the Black-Scholesmodel for pricing of American options. AnAmerican option is a contract that entitlesits owner to buy some financial asset (typi-cally a share of some company) at some spec-ified price (the“strike price”) at any timebefore some specified date (the “maturitydate”). This option has some value, sincein case that the always fluctuating marketprice of the asset goes higher than the strikeprice then the option can be “exercised” tobuy the asset at the lower price.

The Black-Sholes model aims to calculatethe rational price u = u(x, t) of an option atany time t prior to the maturity date and de-pending on the price x ∈ R+ of the financialasset. Since the option can be exercised atany time before maturity, determining the“exercise region”, i.e. the pairs (x, t) forwhich it is better to exercise the option, is apart of the problem. Interestingly, this prob-lem leads to an Stefan problem of the type(1.2), and the free boundary corresponds tothe boundary of the exercise region.

• Interacting particle systems. Largesystems of interacting particles arise in phys-ical, biological, or material sciences.

In some some models the particles attracteach other when they are far, but experiencea repulsive force when they are close [14]. Inother related models in statistical mechan-ics, the particles (e.g. electrons) repel with aCoulomb force and one wants to understandtheir behaviour in presence of some externalfield that confines them [48].

In the previous models, a natural and in-teresting question is to determine the “equi-librium configurations”. For instance, inCoulomb systems the charges accumulate insome region with a well definite boundary;see Figure 4. Interestingly, this problemsare equivalent to obstacle problems of thetype (1.3) — for instance the electric poten-tial u = u(x) generated by the charges solvesa problem like (1.3) and free boundary cor-responds to the boundary of the region inwhich the particles concentrate.

Charges

External field

Figure 4. The equilibriumconfiguration for a Coulombsystem.

• Elasticity. Let us consider the equilib-rium position v(x) of an elastic membranewhose boundary is held fixed, and whichis constrained to lie above a given obstacleϕ(x).


Figure 5. An elastic mem-brane above an obstacle ϕ.

In the region where the membrane is abovethe obstacle ϕ, the solution v solves a PDE(i.e., ∆v = 0 in {v > ϕ}), while in the otherregion the membrane coincides with the ob-stacle (i.e., v = ϕ). By considering the func-tion u := v−ϕ ≥ 0, we are led to an obstacleproblem of the type (1.3).

2. Regularity of free boundaries

From the mathematical point of view, acentral question in the Stefan problem (1.2)and the obstacle problem (1.3) is to un-derstand the regularity of free boundaries[12, 42].

For example, in the Stefan problem:is there a regularization mechanism thatsmoothes out the free boundary, indepen-dently of the initial data? (Notice that apriori the free boundary could be a very ir-regular set, even a fractal set!) Such type ofquestions are usually very hard, and even inthe simplest cases almost nothing was knownbefore the 1970s. The development of theregularity theory for free boundaries startedin the late seventies, and since then it hasbeen a very active area of research.

2.1. Some examples of Schaeffer.The first thing one might try is to construct

some explicit solutions to the obstacle prob-lem, and see how their free boundaries be-have. What one finds is that, in most sim-ple cases, free boundaries seem to be verysmooth.

It was Schaeffer who first realized thatwith a bit more effort one can actually con-struct several free boundaries with some sin-gularities. Namely, he constructed differentsolutions to the obstacle problem in R2 inwhich the free boundary has a cusp.

These examples are actually constructedby using complex analysis, and the cusps arerepresented by the curves

x2 = ±x2k+ 1

21 , 0 ≤ x1 ≤ 1.

Figure 6. A one-sided cusp.

The set {u > 0} is actually the image of{|z| ≤ 1, Im z > 0} under the conformalmapping f(z) = z2 + i z4k+1, and u satisfiesnear the origin

u(z) ≈ x22

2+ ck Im

(z2k+

32

)+ ...,

where z = x1 + ix2.Another type of singularities (a two-sided

cusp) was also constructed by Schaeffer. Inthis case, these two-sided cusps are repre-sented by the curves

x2 = ± |x1|2k, −1 ≤ x1 ≤ 1.


Figure 7. A two-sided cusp.

By considering slightly more general ob-stacle problems of the type (1.3) in R2, Scha-effer noticed that it is even possible to con-struct examples in which the free boundaryhas infinitely many cusps.

Figure 8. The free bound-ary could even create an infi-nite number of cusps.

2.2. The breakthrough of Caffarelli.Despite all these examples, before the late1970s there was no general regularity resultfor free boundaries. A natural guess, giventhese examples, could be that free bound-aries are smooth outside a certain set of “sin-gular points”. However, no result of thistype was known, and this seemed to be anextremely challenging open problem.

This changed in 1977 with the ground-breaking paper of Luis Caffarelli [6]. Suchwork developed for the first time the regu-larity theory for free boundaries in both theobstacle problem (1.3) and the Stefan prob-lem (1.2).

The main results from [6] (see also [33])may be summarized as follows:

• The free boundary splits into regularpoints and singular points.

• The set of regular points is an open subsetof the free boundary and it is C∞.

• Singular points x0 can be characterized asthose at which the set “contact set” {u = 0}has density zero (as in a cusp), i.e.

limr→0

|{u = 0} ∩Br(x0)||Br(x0)|

= 0. (2.1)

In other words:

The free boundary is smooth, outside ofa certain set of cusp-like singularities.

This is one of the main results for whichCaffarelli received the Wolf Prize in 2012 andthe Shaw Prize in 2018.

2.3. Blow-ups.To prove such regularity result, one considersblow-ups. This is a key idea that is commonin many problems in PDEs and GeometricAnalysis.

Given a free boundary point x0, one firstconsiders the rescaled functions

ur(x) :=u(x0 + rx)

r2,

for r ∈ (0, 1). Notice that, when taking r > 0smaller and smaller, we are zooming in thesolution u around the point x0. Caffarellishowed that for any free boundary point x0and r > 0 one has

cr2 ≤ supBr(x0)

u ≤ Cr2

where c and C are positive constants. Thus,the rescaling factor r−2 is just taken so thatthe functions ur satisfy c ≤ maxB1 ur ≤ C.


Figure 9. The rescaled func-tion ur at a singular point.

Then, the idea is to take the limit r ↓ 0,and prove that

ur(x) :=u(x0 + rx)

r2r↓0−−−→ u0(x)

for some function u0 which is a global so-lution of the obstacle problem in the entirespace. The function u0 is called a blow-up ofu at x0.

Roughly speaking, the function u0 shouldgive us information about how the solutionu looks like at the point x0.

The main difficulty is actually to classifyblow-ups, i.e., to show that either

(a) u0 is 1D solution of the form

u0(x) = (x · e)2+,where e ∈ Sn−1 is some unit vector;

Figure 10. A 1D solution u0.

or

(b) u0 is a quadratic polynomial of theform

u0(x) =12xTAx,

where A ≥ 0 is any positive semidef-inite matrix satisfying tr(A) = 1.

Figure 11. A quadraticpolynomial u0.

Notice that, after the blow-up, the contactset {u0 = 0} becomes either a half-space —in the case (a)— or it is a linear proper sub-space of Rn and hence it has zero measure—in the case (b). The first case is what weexpect at regular points, while the secondis what we expect at singular points. Thisintuition is not only correct but actually istranslated into a rigorous mathematical def-inition: we say that a free boundary pointx0 is regular if the blow-up u0 at x0 is of thetype (a) and we say that it is singular if theblow-up u0 at x0 is of the type (b).

To complete the prove of Caffarelli’s theo-rem, one has to “transfer information” fromu0 to u, and show that if x0 was a regu-lar point, then the free boundary is C1 ina neighborhood of x0. This is delicate andis another of the main achievements in [6].On the other hand, the fact that fact that{u = 0} satisfies (2.1) at singular points isa more immediate consequence of the defini-tion of singular point and follows easily usingthe fact that the convergence of ur to u0 isuniform and the that u0 > 0 outside of aproper linear sub-space of Rn. We refer to[7, 8] or [42] for more details.


3. Understanding singularities

After the results of Caffarelli [6], a natu-ral question is to understand better the setof singular points.

The fine understanding of singularities isactually a central research topic in a numberof areas related to nonlinear PDE and Geo-metric Analysis. In the present context, it isnatural to ask:

• How large may the singular set be?• In case it is a large set, does it enjoy somenice regularity properties?

3.1. The 2D case.The first results in this direction were ob-tained by L. Caffarelli and N. Riviere in twodimensions [11]. They proved that, at ev-ery singular point x0, any solution u to theobstacle problem in R2 satisfies

u(x) = p2(x) + o(|x− x0|2

), (3.1)

where p2 is a quadratic polynomial satisfy-ing p2 ≥ 0 and ∆p2 = 1. Moreover, thisimplies that the singular set is contained ina C1 curve.

It took a long time before these resultscould be improved. The next important re-sult in this direction was obtained by Sakaiin 1991 [44, 45], who proved that the isolatedcusps of Schaeffer (described above) are es-sentially the only ones that may appear forthe obstacle problem (1.3) in R2. This niceand sharp result gives a complete picture forthe obstacle problem in the plane, and itsproof uses crucially tools from complex anal-ysis.

3.2. Higher dimensions.In dimensions n ≥ 3, where complex analysisis of no use, the first results on the singularset were established by Caffarelli in 1998 [8](see also [40]). He proved that, if u is anysolution to the obstacle problem (1.3) in Rn,then (3.1) holds at every singular point x0.

Moreover, this implies that the singular setis contained in a C1 manifold of dimension(n− 1).

For almost two decades, this was the bestknown result for the singular set in dimen-sions n ≥ 3. Still, it was an open question tounderstand whether (3.1) can be improvedor not.

In two dimensions, because of the resultsof Sakai, at every singular free boundarypoint we have that

u(x) = p2(x) +O(|x− x0|3

). (3.2)

It is however important to notice that Sakai’smethods, based on complex anaysis, cannotwork in higher dimensions (nor for the Ste-fan problem). Thus, improving (3.1) in di-mensions n ≥ 3 would require completelydifferent ideas.

The first new result in this direction for di-mensions n ≥ 3 was established by Colombo,Spolaor, and Velichkov in 2017 [15], by im-proving and refining the methods of Weiss[54]. They proved that at every singularpoint the expansion (3.1) holds with an addi-tional logarithmic modulus of continuity onthe term o

(|x− x0|2

).

Independently and with different meth-ods, Figalli and the second author provedin [23] the following result for the obstacleproblem in Rn:Theorem 1 ([23]). Outside a small set ofHausdorff dimension n− 3, (3.2) holds.

In other words, in higher dimensions, (3.2)holds at most singular points x0.

It is important to remark that the fine de-scription of u (3.2) actually gives a fine de-scription of the cusp at x0. Namely, it followsfrom the results of [23] that, at most singularpoints x0, (possibly after a rotation) near x0we have

{u = 0} ⊆ {|xn| ≤ C|x′|2},where x = (x′, xn), x

′ ∈ Rn−1.


Furthermore, it was shown in [23] that theresult is sharp, in the sense that there ex-ist (isolated) singular points in R3 at which(3.2) fails. In fact, at these points, the loga-rithmic modulus of continuity of [15] cannotbe improved.

This gives a very complete picture of sin-gularities for the obstacle problem in all di-mensions.

3.3. The Stefan problem.As explained above, a major question inPDE problems where singularities appear isto establish estimates for the size of the sin-gular set.

Some famous results in this direction in-clude those on minimal surfaces [50, 1], onthe Navier-Stokes equations [10], and onmean curvature flow [56].

In the Stefan problem (1.2), the sametechniques used in the study of the (elliptic)obstacle problem yield that for problem (1.2)the singular set Σt is (n− 1)-dimensional forevery t; see [55, 5, 38]. This result is optimalin space, in the sense that one can constructexamples in which the singular set containssome (n − 1)-dimensional hypersurface (forinstance a sphere) for some fixed time t = t0.In this respect it would seem that the set ofsingular points can be as large as the regularpoints, which is always (n− 1)-dimensional.However, one could still hope that, jointlyin space and time, the size of singular setshould be in some sense much smaller thanthe set of regular points!

For instance, there are no examples inwhich singular points appear for every timet, but it is (or was) not clear a priori if thiscould happen or not. In case this cannothappen, the next question is to establish esti-mates on the size of the set of singular times.Namely, if we let Σt be the set of singularpoints at time t, then we may denote

S = {t : Σt 6= ∅}

the set of all singular times (i.e., the set ofall times at which singularities appear).

The following result was announced atthe ICM lecture of Alessio Figalli [22], andproved in the paper [24]:

Theorem 2 ([24]). Let u(x, t) be any solu-tion to the Stefan problem (1.2) in R3. As-sume that

ut > 0 in {u > 0}.

Then,

dimH(S) ≤1

2.

Here, dimH(S) denotes the Hausdorff di-mension of the set S.

In particular, the previous result impliesthat, for the Stefan problem in R3:

The free boundary is C∞ foralmost every time t.

This is the first result on the size of theset of singular times for the Stefan problem.

The result is stated for simplicity in thephysical dimension n = 3, but [24] actuallyestablishes new results for the Stefan prob-lem in Rn for any dimension n ≥ 2.

As said above, prior to this result it wasnot even known if solutions to the Stefanproblem (1.2) in R3 could have singularitiesfor all times t.

It is not known whether the dimensionbound 1

2is optimal or not. What is clear

from the proofs is that such exponent is criti-cal in several ways. This is somewhat similar(even though the results and proofs are verydifferent) to what happens in the Navier-Stokes equations; see the classical result ofCaffarelli, Kohn, and Nirenberg [10].

4. Generic regularity

A second major question in the under-standing of singularities is the development


of methods to prove generic regularity re-sults.1 This is one of the big challenges incontemporary PDE theory.

Indeed, in PDE problems in which singu-larities may appear, it is important to un-derstand whether these singularities appear“often”, or if instead “most” solutions haveno singularities.

In the context of the obstacle problem(1.3), the key question is to understand thegeneric regularity of free boundaries. Aswe saw earlier, explicit examples show thatsingular points in the obstacle problem canform a very large set, of dimension n − 1(as large as the regular set). Still, singularpoints are expected to be rare ([46]):

Conjecture (Schaeffer, 1974):Generically, the weak solution of the obsta-cle problem is also a strong solution, in thesense that the free boundary is a C∞ mani-fold.

In other words, the conjecture states that,generically, the free boundary has no singu-lar points.

The conjecture was only known to holdin the plane R2 [40], and until very recentlynothing was known in R3 or in higher dimen-sions.

Notice that in the obstacle problem thequestion of generic regularity is particularlyrelevant, since the singular set can be aslarge as the regular set —while in other prob-lems the singular set has lower Hausdorff di-mension [31]. Also, from the point of view ofapplications, it is particularly relevant to un-derstand the problem in the physical spaceR3.

A key result established by Figalli and theauthors in [24] is the following:

Theorem 3 ([24]).Schaeffer’s conjecture holds in R3.

We remark that very few results are knownin this direction in elliptic PDE, and mostof them deal only with the cases in whichthe singular set is known to be very small(e.g. the obstacle problem in R2 [40], or area-minimizing hypersurfaces in R8 [51]).

Due to the general character of the proofsin [24], one can even apply these results tothe Hele-Shaw flow (explained above).

Corollary 4 ([24]). Let u(x, t) be any solu-tion to the Hele-Shaw flow in R2 or R3.

Then, the free boundary is C∞ for almostevery time t.

Such result is completely new even inthe 2D case (the most relevant one for thismodel). Moreover, as in the Stefan problem,[24] establishes a new bound on the size ofthe singular set: for the Hele-Shaw flow inR2, the set of singular times has Hausdorffdimension at most 1

4.

5. The fractional obstacle problem

5.1. Optimal stopping and Finance.As explained in subsection 1.2, a nice mo-tivation of the study of obstacle problemscomes from Probability and Finance. In par-ticular they arise in pricing of American op-tions. Let us explain it in more detail next.

We recall that an American option entitlesits owner with the possibility to buy a givenasset at a fixed price at any time before thematurity date. In the Black-Scholes model,the (logarithmic) price of the underlying as-set Xt is modelled as a Wiener process (orBrownian motion) with a drift parameterthat recreates long-term growth and a vari-ance parameter that matches the volatilityof the asset. Under this assumption, as said

1Here, by generic regularity we mean that there is an open and dense set of boundary datums for whichthe corresponding solution has no singularities.


in subsection 1.2, the rational price of theoption solves a parabolic obstacle problemlike (1.2). However, in finance price fluctua-tions are often better modelled by more gen-eral stochastic processes Xt: Lévy processes(see for instance [16]). These are stochasticprocesses with jumps, and were introducedin option pricing models by the Nobel Prizewinner R. Merton [39] in the 1970s.

Under this more general assumption, therational price of an American option stillsolves an obstacle problem similar to (1.2)but where the Lapacian is replaced by someother operator (actually the so-called infin-itesimal generator of the process Xt, whichis an elliptic operator of integro-differentialtype). The most important and canonicalexample of a Lévy process (other than theBrownian motion) corresponds to the casein which the law of Xt is rotationally invari-ant and satisfies a scaling property. In suchcase, what we get is the obstacle problem forthe fractional Laplacian

(−∆)su(x) := cn,s∫Rn

u(x)− u(x+ y)|y|n+2s

dy,

with s ∈ (0, 1).The obstacle problem for the fractional

Laplacian is often called the fractional ob-stacle problem. Because of its connec-tion to Probability and Finance, in the lastdecade there have been considerable effortsto understand the fractional obstacle prob-lem (both the stationary and the evolution-ary problem).

5.2. Known results.The main questions in this context are:

• What is the optimal regularity of solu-tions?

• Can one prove the regularity of free bound-aries?

The first results in this direction were ob-tained by Silvestre in [49] in the stationary

setting, who established the almost-optimalregularity of solutions, u ∈ C1,s−ε for allε > 0. The optimal regularity of solutionswas established later by Caffarelli, Salsa, andSilvestre [13]:

Solutions u are C1,s.

Furthermore, they also established the fol-lowing:

The free boundary is smooth, possiblyoutside a certain set of degenerate points.

More precisely, they proved that if usolves the obstacle problem for the fractionalLaplacian (−∆)s in Rn, then u ∈ C1,s, andfor every free boundary point x0 one has adichotomy: either the solution u is degen-erate at x0, or the free boundary is smootharound such point.

After the results of [13], many more re-sults have been obtained for the (stationary)fractional obstacle problem: the higher reg-ularity of free boundaries [35, 17, 36, 32], thestudy of singular points [28, 3, 25, 30, 20, 26],or the case of operators with drift [41, 29, 21].

Still, there was one question that wasmuch less understood: what happens in theevolutionary setting?

5.3. The parabolic case.Despite many known results for the frac-tional obstacle problem in the stationary set-ting, much less was known in the paraboliccase, in which the solution evolves with time.

The first result in this direction was estab-lished by L. Caffarelli and A. Figalli in [9],where they established the optimal regular-ity of solutions:

Theorem 5 ([9]). Let u(x, t) be any solutionto the parabolic fractional obstacle problem.

Then, u is C1,s in x.


The question of free boundary regularitywas still open for some years. The key dif-ficulty here was that in the stationary set-ting the proofs rely very strongly on certainmonotonicity formulas which do not seem toexist in the parabolic context. Thus, a quitedifferent argument must be developed.

This was accomplished in [4], where B.Barrios, A. Figalli, and the first author es-tablished the following:

Theorem 6 ([4]). Let s > 12, and u(x, t) be

any solution to the parabolic fractional ob-stacle problem.

Then, the free boundary is smooth, possi-bly outside a certain set of degenerate points.

This extended for the first time the resultsof [13] to the parabolic setting, with a com-pletely different proof.

References

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[2] C. Baiocchi, Free boundary problems in the the-ory of fluid flow through porous media, in Pro-ceedings of the ICM 1974.

[3] B. Barrios, A. Figalli, X. Ros-Oton, Global regu-larity for the free boundary in the obstacle prob-lem for the fractional Laplacian, Amer. J. Math.140 (2018), 415-447.

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Xavier Ros-Oton.Born in Barcelona in 1988. PhD in Mathematics (Barcelona, 2014), his research is mainlyfocused on Partial Differential Equations. He is currently Assistant Professor of Mathematicsat Universität Zürich.

Joaquim Serra.Born in Barcelona in 1986. PhD in Mathematics (Barcelona, 2014), his research is mainlyfocused on Partial Differential Equations. He is currently SNF Ambizione Fellow at ETHZürich.

Universität Zürich, Institut für Mathematik, Winterthurerstrasse, 8057 Zürich, Switzer-land

Email address: [email protected]

ETH Zürich, Department of Mathematics, Rämistrasse 101, 8092 Zürich, SwitzerlandEmail address: [email protected]

1. Introduction1.1. The Stefan problem1.2. Motivations and applications

2. Regularity of free boundaries2.1. Some examples of Schaeffer2.2. The breakthrough of Caffarelli2.3. Blow-ups

3. Understanding singularities3.1. The 2D case3.2. Higher dimensions3.3. The Stefan problem

4. Generic regularity5. The fractional obstacle problem5.1. Optimal stopping and Finance5.2. Known results5.3. The parabolic case

References

Documents

UNDERSTANDING SINGULARITIES IN FREE BOUNDARY …XAVIER ROS-OTON AND JOAQUIM SERRA Abstract. Free boundary problems are those described by PDEs that exhibit a priori unknown (free)