An Overview of the

Donatella Danielli

1. Introduction air, or water and ice. In addition to the usually prescribed Free Boundary Problems naturally occur in physics and en- initial and boundary values, other conditions, arising from gineering when a conserved quantity or relation changes the physical laws governing the model, are imposed at the discontinuously across some value of the variables under free boundary. Since in general, with few exceptions, it is consideration. In mathematical terms, they consist in solv- impossible to explicitly determine the solution and the un- ing partial differential equations (PDEs) in a domain, a derlying domain, the ultimate goal consists in establishing part of whose boundary is a priori unknown, and which analytic and geometric properties for both. In the past few has to be determined as part of the problem. Said portion decades this area of research has seen deep and broad ad- of the boundary is called the free boundary, and in models vancements, due to the assimilation of problems coming it appears, for instance, as the interface between a fluid and from other applied sciences, such as finance, mathematical biology, and population dynamics, just to name a few. Its Donatella Danielli is a professor of at Purdue University. Her development has been intrinsically intertwined with the email address is [email protected]. theory of Variational Inequalities, born in Italy in the early Communicated by Notices Associate Editor Daniela De Silva. 1960s. Its founding fathers were , who was motivated by questions in , and Gae- For permission to reprint this article, please contact: [email protected]. tano Fichera, who instead was interested in problems in with unilateral constraints (more on this later...). DOI: https://doi.org/10.1090/noti2165

NOVEMBER 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1487 퐽(푣) is continuous and strictly convex on the convex 풦, the existence and uniqueness of minimizers are guaranteed. The obstacle problem can be reformulated as a Varia- tional on 푊 1,2(Ω). In fact, solving the obstacle problem is equivalent to determining a function 푢 ∈ 풦 such that

∫ ∇푢 ⋅ ∇(푣 − 푢) 푑푥 ≥ 0 for all 푣 ∈ 풦. Ω Figure 1. Example of an obstacle problem. Standard variational arguments show that the solution is harmonic away from the contact set, or Over the years, Variational Inequalities have become an essential tool in various sectors of applied mathematics. Δ푢 = 0 in {푢 > 휙}, From a more theoretical standpoint, they appear in the Cal- culus of Variations when a function is minimized over a set and superharmonic on the contact set, i.e., of constraints, giving rise to a set of differential inequal- Δ푢 ≤ 0 in {푢 = 휙}. ities which replace the classical Euler-Lagrange equation. Free Boundary Problems and Variational Inequalities have Hence, the solution is a superharmonic function in Ω, in generated new and exciting ideas based on the interplay of the sense of distributions. Based on these considerations, methods from PDEs, the , Geomet- we can rewrite the obstacle problem in yet another way, ric Measure Theory, and Mathematical Modeling. One of namely as the main driving behind the development of their min{−Δ푢, 푢 − 휑} = 0, (1) theory has been the study of the obstacle problem. subject to the boundary condition 푢|휕Ω = 푓(푥). 2. The Classical Theory Two natural questions arise at this point: In its classical formulation, the obstacle problem consists 1. How regular is the function 푢? in finding the equilibrium configuration of an elastic mem- 2. What are the geometric properties of the contact brane whose boundary is held fixed, and which is con- set? Is the free boundary a smooth surface? strained to lie above a given obstacle. Mathematically, we Concerning the first question, it was shown in [Fre72] that seek to minimize the Dirichlet energy the solution is in 퐶1,1(Ω) (assuming that 휑 is at least in the same class), i.e., it has bounded second . It 2 퐽(푣) = ∫ |∇푣| is readily verified that such regularity is optimal. Heuristi- Ω cally, Δ푢 jumps from 0 on the set where 푢 is detached from in a domain Ω ⊂ ℝ푛, among all configurations 푣(푥) (repre- 휑, to Δ휑 on the contact set, and therefore it is unreasonable senting the vertical of the membrane) with to expect continuity of the second derivatives. prescribed boundary values 푣| = 푓(푥), and constrained 휕Ω The second question is significantly more delicate and to remain above the obstacle 휑(푥), that is, in the class complex. Before we begin to outline the key steps in its an- 1,2 풦 = {푣 ∈ 푊 (Ω) | 푣|휕Ω = 푓(푥), 푣 ≥ 휑}, swer, let us mention that from the point of view of appli- given the compatibility condition 휑 ≤ 푓 on 휕Ω. We will cations, smoothness properties of the interface are crucial denote by 푢 the solution of this minimization problem. to develop, for instance, robust numerical methods. The The domain Ω then breaks down into a region where 푢 first fundamental accomplishment in this direction isdue coincides with the obstacle function, known as the contact to D. Kinderlehrer and L. Nirenberg [KN77], who showed 1 set, and a region where the solution is above the obstacle. that 퐶 free boundaries are, in fact, analytic. Applicabil- The free boundary is defined as the topological boundary of ity of this result, however, rests on the initial knowledge the contact set, that is, of a certain degree of smoothness, whereas a priori the in- terface could be a very irregular object. The breakthrough ℱ(푢) = 휕{푥 ∈ Ω | 푢(푥) = 휑(푥)}. came from the seminal paper [Caf77], where L. Caffarelli, We explicitly observe that, as it often happens in the Calcu- inspired by De Giorgi’s approach for the regularity of area- lus of Variations, searching for solutions in an apparently minimizing surfaces, introduced one of the most trans- natural but too narrow family of admissible functions may formative ideas in the theory of free boundary problems, produce no results. This justifies the choice of the class of namely the use of blow-up arguments. In layman’s terms, competitors as the 푊 1,2(Ω), endowed with they consist in zooming in on a fixed free boundary point, the inner product ⟨푢, 푣⟩ = ∫Ω (푢푣 + ∇푢 ⋅ ∇푣) 푑푥. Since the as if using a magnifying glass, and observing the properties

1488 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 10 of the free boundary at a very large scale. The mathemati- As one may surmise from the nomenclature, the objec- cal tool corresponding to the magnifying glass is the rescal- tive is now to show that the free boundary can be expressed ing (see Figure 2). To illustrate its workings, we follow here as a smooth hypersurface in a neighborhood of a regular point, whereas smoothness is not expected around a sin- gular one. Cusp-like or pinched bottleneck singularities, in fact, may occur even when the obstacle is smooth. The main difficulty in the study of the free boundary atthis point lies in transferring the information acquired on the shape of the blow-ups back to the original configuration. In principle, in fact, a point 푥0 ∈ ℱ(푢) could be regular and singular at the same time, depending on the choice of the subsequence 푟푗 taken to pass to the limit. Geometric considerations, however, prove that this is not possible. Figure 2. The effect of rescaling. ∗ 1 + 2 In case (i), to fix ideas assume 푢 (푥) = (푥푛 ) . The new 2 ∗ contact set {푢 = 0} is the half-space {푥푛 ≤ 0} and there- the approach in [Caf98]. First of all, we reduce the prob- fore, undoing the rescaling, one can see that the original lem to a zero-obstacle one by replacing 푢 with 푤 = 푢 − 휑. contact set {푢 = 0} has positive Lebesgue density at 푥0, that Then, in the region {푤 > 0}, one has is, |퐵 (푥 ) ∩ {푢 = 0}| Δ푤 = Δ(푢 − 휑) = −Δ휑 ∶= 푔(푥). 푟 0 lim sup 푛 > 0. 푟→0+ 푟 For the sake of simplicity we will assume 푔(푥) ≡ 1, and con- This is a very stable situation, in the sense that the flatness tinue to denote the solution to this normalized problem of the free boundary for 푢∗ (namely the hyperplane {푥 = by 푢. The starting point is the observation that 푢 exhibits 푛 0}) translates into “almost-flatness” of the free boundary quadratic growth near a free boundary point 푥 ∈ ℱ(푢). 0 of the rescaled functions 푢 for 푟 sufficiently small, and More precisely, there exist two constants 푐, 퐶 > 0 such that 푟 thus for 푢 at a small scale. More precisely, for 휎 > 0 and 2 2 푐푟 ≤ sup 푢 ≤ 퐶푟 , (2) 0 < 푟 < 푟휍 it holds that 퐵푟(푥0) ℱ(푢 ) ∩ 퐵 (푥 ) ⊂ {|푥 | < 휎}. 푛 푟 1/2 0 푛 (5) where 퐵푟(푥0) = {푥 ∈ ℝ | |푥 − 푥0| < 푟}. This estimate suggests the following family of rescaled functions: The first step in establishing the smoothness of the free boundary is to prove its Lipschitz regularity. This easily 1 푢 (푥) = 푢(푥 + 푟푥) (3) follows from the fact that the directional derivatives 휕 푢 푟 푟2 0 푒 are nonnegative near a regular point for 푒 in a cone of di- for 푟 > 0. The boundedness of the second derivatives rections of 푢 allows one to apply the classical compactness result 퐶 = {푥 ∈ ℝ푛 | 푥 > 훿|푥′|}, of Ascoli-Arzelà. We thus infer that, possibly passing to 훿 푛 + with 푥′ = (푥 , … , 푥 ) and 0 < 훿 < 1; see Fig- a subsequence 푟푗 → 0 , 푢푟(푥) converges to a function 1 푛−1 푢∗(푥), called a blow-up limit (or simply blow-up) relative to ure 4. Without going into the technical details, the guiding principle here is that this monotonicity property, which 푥0 ∈ ℱ(푢). The following classification of blow-ups holds: clearly holds true for 푢∗, can be transferred to 푢 (for 푟 (i) either 푢∗ is a half-space solution to the obstacle 푟 problem of the form 1 푢∗(푥) = max{푥 ⋅ 푒, 0}2 2 for some 푒 unit vector in ℝ푛, (ii) or 푢∗ is a parabola solution of the form 1 푢∗(푥) = 푥 ⋅ 퐴푥 (4) 2 for some matrix 퐴, with 퐴 nonnegative definite and trace(퐴) = 1. This alternative leads to a corresponding characterization of the free boundary point 푥 . If (i) holds, we say that 푥 0 0 Figure 3. Blow-up limits at regular (left) and singular (right) is a regular free boundary point, and otherwise we say that free boundary points. 푥0 is singular; see Figure 3.

NOVEMBER 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1489 small enough) combining the convergence in 퐶1 with (5), and then, “undoing” the rescaling, back to 푢 itself.

Figure 5. The different nature of singular points.

Figure 4. The cone of monotonicity. functional 1 풲(푟; 푢) = ∫ (|∇푢|2 + 2푢) Another relatively simple application of the directional 푟푛+2 monotonicity property yields that ℱ(푢) is locally a 퐶1- 퐵푟(푥0) 1 graph and this, combined with the result in [KN77], al- − ∫ 2푢2, 푟푛+3 lows one to conclude that the free boundary is an analytic 휕퐵푟(푥0) hypersurface in a neighborhood of a regular point. We now turn our attention to the case discovered by G. Weiss in [Wei99]. From Monneau’s result one infers, in fact, not only uniqueness of the blow-up, but |퐵푟(푥0) ∩ {푢 = 0}| also the continuous dependence from the singular point: lim sup 푛 = 0. 푟→0+ 푟 if we denote by Σ(푢) the collection of singular points, the ∗ Σ(푢) ∋ 푥0 ↦ 푢푥0 is locally uniformly continuous. Then it can be shown that 푥0 is singular, and it holds that With this information at our disposal, it is finally possible for any 푟 > 0 there exists a unit vector 푒 such that to describe the set Σ(푢). We begin by observing that the nature of the free boundary can vary at different singular 푛 ∗ ℱ(푢) ∩ 퐵푟(푥0) ⊂ {푥 ∈ ℝ | |푒 ⋅ (푥 − 푥0)| ≤ 표(푟)}. points, depending on the 푑푥0 of the set {푢푥0 = ∗ 1 2 0}. For instance, in the case 푛 = 3, if 푢푥 = 푥 , then the What we have just described is the essence of the dichotomy 0 2 1 theorem in [Caf77]. While the first part, relative to case contact set is asymptotically close to a , whereas it is ∗ 1 2 2 asymptotically close to a line if 푢푥 = (푥 +푥 ); see Figure (i), is definitive, the second one is not. In principle, in 0 2 1 2 fact, the vector 푒 could depend, once more, on the subse- 5. It is thus natural to consider a classification of points in quence chosen in the limiting process. In order to describe Σ(푢) based on 푑푥0 . More precisely, for 푑 = 1, … , 푛 − 1, we the structure of the free boundary near singular points, it introduce the set is then pivotal to establish the uniqueness of the blow- Σ = {푥 ∈ Σ(푢) | 푑 = 푑}. ups. This is where monotonicity formulas enter the scene. In 푑 0 푥0 [Caf98], the uniqueness in the limiting process is accom- Caffarelli established in [Caf98] the first decisive result plished by applying the celebrated Alt-Caffarelli-Friedman concerning the structure of the singular set. Namely, he monotonicity formula to the first derivatives of the solu- proved that Σ푑 is locally contained in a 푑-dimensional tion. A few years later, R. Monneau devised in [Mon03] an manifold of class 퐶1. We illustrate here the main ideas of alternative approach (which does not require to differen- the proof. Note that, since 푢 ≥ 0 in Ω and 푢 = 0 on ℱ(푢), tiate the solution), based on a monotonicity formula that necessarily ∇푢 = 0 on ℱ(푢). In particular, 푢 = ∇푢 = 0 in nowadays bears his name: if 푥0 is a singular point, and ∗ Σ푑 and thus, keeping in mind (3) and (4), we infer that 푢푥 푢∗ = 푢∗ is as in (4), then the functional 0 푥0 corresponds, at least in principle, to the second-order term in the Taylor expansion of 푢 at 푥0. Thanks to the continuity ∗ ∗ 2 ∗ ℳ(푟; 푢, 푢 ) = ∫ (푢 − 푢 ) of the map Σ(푢) ∋ 푥0 ↦ 푢푥0 , it is possible to apply Whit- 푛 푛 휕퐵푟(푥0) ney’s extension theorem to find a function 퐹 ∶ ℝ → ℝ of class 퐶1 such that is nondecreasing in 푟 > 0. This important property is de- ∗ 2 ∗ duced, in turn, from the nondecreasing character of the 퐹(푥0) = ∇푢푥0 = 0 and ∇퐹(푥0) = 퐷 푢푥0

1490 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 10 for all 푥0 ∈ Σ푑. Invoking the definition of Σ푑, we have control, and in financial mathematics. Since the first in- stance is the simplest to describe, we will derive the math- dim ker ∇퐹(푥 ) = dim ker 퐷2푢∗ = 푑 0 푥0 ematical formulation of the problem from it. Semiperme- in Σ푑. Applying the Implicit Function Theorem we can able membranes allow the passage of fluid freely in one di- finally conclude that rection, but prevent all outflow of fluid in the other. They are used, for instance, in the process of hyperfiltration, or Σ = {퐹 = 0} ∩ Σ 푑 푑 reverse osmosis. This is a pressure-driven mechanism to is locally contained in a 퐶1 푑-dimensional manifold. remove low molecular solutes, e.g., microorganic When 푛 = 2, Weiss showed in [Wei99] that 퐶1-regularity salts or small organic molecules (glucose, for example) can be improved to 퐶1,훼 by means of an epiperimetric-type from a solvent. In the natural process of osmosis, a mem- approach, thus recovering an earlier result of L. Caffarelli brane is used to separate two chambers of equal volume, and N. Rivière. Very recently, A. Figalli and J. Serra ob- for instance one filled with water and the other one with tained in [FS19] an exhaustive characterization of the sin- a dilute salt solution. Water is transferred from the sol- gular set: in two , Σ(푢) is locally contained in vent side of the membrane to the other chamber, until an a 퐶2 curve, whereas in dimensions 푛 ≥ 3 singular points osmotic equilibrium is reached. The hydrostatic pressure (except anomalous ones) can be covered by 퐶1,1—and in which has built up in the salt solution side of the mem- some cases 퐶2—manifolds. Additionally, they showed brane is known as osmotic pressure. that the highest order stratum Σ푛−1 is always contained As the name implies, reverse osmosis is the reversal in a 퐶1,훼 manifold, thus generalizing Weiss’s results. It of this process. The application of a pressure, exceeding should be noted that O. Savin and H. Yu have established the osmotic one, on the salt solution in contact with the in [SY19] essentially the same result for the obstacle prob- membrane forces water to flow from the chamber contain- lem associated to a convex fully nonlinear . ing the salt solution to the other side (see Figure 7). The They developed a novel method, based not on monotonic- semipermeable nature of the membrane prevents the wa- ity formulas but on an iterative scheme, which provides an ter from flowing back in, as would occur by natural osmo- insightful alternative also for the classical problem. sis. The water flux is proportional to the concentration dif- ference across the membrane. This proportionality is ex- 3. Thin Obstacles pressed in terms of the permeability constant of the mem- A problem in , first proposed by A. Sig- brane, which in turn is inversely proportional to the mem- norini [Sig59] and now bearing his name, has been one brane thickness. of the mainsprings of the study of Variational Inequalities. Following [DL76], we now formulate the correspond- In its simplest formulation, it consists of determining the ing mathematical model. We let Ω be a region in ℝ푛 occu- displacements in a heavy, linearly elastic body resting on pied by a viscous fluid which is only slightly compressible, a rigid, frictionless horizontal plane (see Figure 6). with its pressure field denoted by 푢(푥). We assume that a smooth portion Γ of 휕Ω, with exterior unit 휈, con- sists of a semipermeable membrane of finite thickness. In correlation with the process of reverse osmosis, Ω repre- sents the chamber containing the solvent (water), with Γ separating Ω from another chamber filled with the dilute salt solution. Combining the law of conservation of mass

Figure 6. The classical .

The quintessential difficulty is the fact that the region of contact between the body and the plane is not known a priori, and it is thus part of the problem itself. The same mathematical model finds applications, e.g., in the study Figure 7. The process of reverse osmosis. of semipermeable media, in problems of temperature

NOVEMBER 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1491 Figure 9. The Dirichlet-to-Neumann map. Figure 8. The thin obstacle problem. An alternative interpretation of the thin obstacle prob- with Darcy’s law, one finds that the equilibrium configura- lem is given as a “standard” obstacle problem as in (1), but tion for 푢 satisfies the equation with −Δ replaced by the fractional Laplacian Δ푢 = 0 in Ω. 푠 푢(푥) − 푢(푦) (−Δ) 푢(푥) = 퐶(푛, 푠) lim ∫ 푛+2푠 푑푦, 휀→0 푛 |푥 − 푦| When a fluid pressure 휑(푥) for 푥 ∈ Γ is applied on the out- ℝ ⧵퐵휀(푥) side of Ω, one of two cases holds: 휑(푥) < 푢(푥), or 휑(푥) ≥ where 0 < 푠 < 1 and 퐶(푛, 푠) is a constant. The connec- 푢(푥). In the former, the semipermeable wall prevents the tion in the case 푠 = 1/2 comes through the Dirichlet-to- fluid from leaving Ω, so that the flux is null, or 휕휈푢 = 0. In Neumann map; see Figure 9. More precisely, let 푢 ∶ ℝ푛 → the latter case, instead, the fluid enters Ω. Since the flow is ℝ be a smooth function, and consider the extension prob- proportional to the pressure differential, 휕휈푢 = −휆(푢 − ℎ), lem where 휆 > 0 is the permeability constant of the mem- Δ ̃푢= 0 in ℝ푛 × {푥 > 0}, brane. Combining the two instances at once, we obtain 푛+1 − 푛 the boundary condition 휕휈푢 = 휆(푢 − ℎ) . Given that the ̃푢(푥, 0) = 푢(푥) for 푥 ∈ ℝ , typical membrane thickness in the process of reverse osmo- which yields a smooth bounded solution ̃푢(푥, 푥푛+1). Now sis is of the order of 100 휇m, and is thus negligible, we are replace 푢 with −휕푛+1 ̃푢(푥, 0) in the Dirichlet condition, to interested in the infinite permeability case 휆 → ∞, which obtain −휕푛+1 ̃푢(푥, 푥푛+1) as the solution to the problem, in- leads to the conditions stead of ̃푢. If we let

푢 ≥ 휑, 휕휈푢 ≥ 0, (푢 − 휑)휕휈푢 = 0 on Γ. 푇 ∶ 푢 ↦ −휕푛+1 ̃푢(푥, 0), Signorini called them ambiguous boundary conditions, but we thus have they are now known as Signorini boundary conditions. Since 푇(푇(푢))(푥) = 푇(−휕푛+1 ̃푢(푥, 0))(푥) the solution 푢 is constrained to lie above 휑 on Γ (and not in = 휕 ̃푢(푥, 0) = −Δ푢(푥). the whole domain Ω, as in the classical obstacle problem), 푛+1,푛+1 휑 is referred to as the thin obstacle, and the problem is also It is easy to check that 푇 is a positive operator by an integra- called the thin obstacle problem (see Figure 8). tion by parts argument. Therefore, we can conclude that Fichera [Fic63] was the first one to establish existence 푇 = (−Δ)1/2 and and uniqueness in Signorini’s problem by framing it in 1/2 (−Δ) 푢(푥) = −휕푛+1 ̃푢(푥, 0). the setting of a . With Ω, Γ, and 휑 as 1/2 above, for a given function 푔 ∶ 휕Ω → ℝ satisfying the com- Hence, if 푢 is a solution of the obstacle problem for (−Δ) 푛 patibility condition 푔 > 휑 on Γ, the thin obstacle problem in ℝ with smooth obstacle 휑, then its harmonic extension 푛 consists of solving the Variational Inequality ̃푢 to ℝ ×(0, +∞) solves the corresponding Signorini prob- lem ∫ ∇푢 ⋅ ∇(푣 − 푢) ≥ 0 for every 푣 ∈ 풦, Δ ̃푢= 0 in ℝ푛 × (0, +∞), Ω 푛 ̃푢(푥, 0) ≥ 휑, 휕푛+1 ̃푢(푥, 0) ≤ 0 in ℝ , where 푛 ( ̃푢− 휑)휕푛+1 ̃푢(푥, 0) = 0 in ℝ , 풦 = {푢 ∈ 푊 1,2(Ω) ∶ 푢 = 푔 on 휕Ω ⧵ Γ, 푢 ≥ 휑 on Γ}. and vice versa. With the two problems being equivalent, As in the classical obstacle problem, this is equivalent to any regularity result for one of them can be carried over to 2 minimizing the Dirichlet 퐽(푣) = ∫Ω |∇푣| over 풦. the other one. There is a clear advantage, however, in using

1492 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 10 the Signorini type formulation, since it avoids the direct generators integro-differential operators of the form use of the nonlocal pseudodifferential operator (−Δ)1/2 by localizing the problem. This allows the use of purely local 퐿푢(푥) = lim ∫ (푢(푦) − 푢(푥)) 퐾(푥 − 푦) 푑푦. 휀→0+ 푛 PDE methods, such as monotonicity formulas and classi- ℝ ⧵퐵휀(푥) fication of blow-up profiles, as we will discuss below. Such operators were introduced in finance by the Nobel One of the main reasons behind the resurgence of inter- Prize winner R. Merton. est in nonlocal operators, for which the fractional Lapla- In this context, the most important unknowns are the cian serves as a prototype, is that they are the infinitesimal exercise region {푢 = 휑}, in which one should exercise the generators of Markov processes relevant in financial math- option, and the continuation region {푢 > 휑}, in which one ematics. The infinitesimal generator of a discontinuous instead should wait; see Figure 10. The free boundary is Markov process in ℝ푛 is no longer a differential operator, the interface separating the two. In the case of American but rather an integro-differential, and therefore nonlocal, op- put options, as in the case of American call options paying erator. It has been known for a long time that the infin- continuous dividends, it is always optimal to exercise early, itesimal generator of an isotropic 훼-stable L´evyprocess is and therefore the exercise region is nonempty. In general, the fractional Laplacian operator of order 훼/2 (0 < 훼 < 2). solutions in this type of processes do not have closed form More precisely, if 푋(푡) is such a process starting at zero and expressions, and thus it becomes important to determine 푥 ∈ ℝ푛, then the regularity of the boundary of this region, which in turn is closely related to the behavior of the value function for 1 (−Δ)훼/2 푓(푥) = lim 피[푓(푥) − 푓(푥 + 푋(ℎ))]. points on the free boundary. ℎ→0+ ℎ Similarly to the classical obstacle problem, the main Here, for a probability space (푆, Σ, 푃), 푓 a bounded contin- theoretical issues are the optimal regularity of the solution uous function, and 푋 a random variable, 피[푓(푋)] denotes and the geometric analysis of the free boundary the expected value of 푓(푋), that is, ℱ(푢) = 휕{푥 ∈ Γ | 푢(푥) = 휑(푥)}. The first result concerning the regularity of the solution 피[푓(푋)] = ∫푓(푋(휔)) 푑푃(휔). 푆 in the Signorini problem is due to L. Caffarelli [Caf79], who proved that 푢 is 퐶1,훼 up to Γ for some 0 < 훼 ≤ 1/2, If we assume that 푋(푡) models the logarithm of an asset provided 휑 ∈ 퐶1,1. By analogy with the classical obsta- 푛 price and 휑 ∶ ℝ → ℝ is the payoff function (i.e., a profit cle problem, however, one expects the regularity to be pre- of 휑(푠) is generated when trading the stock at 푋(푡) = 푠), cisely 퐶1,1/2. This important fact was established in [AC04] then one wants to maximize the expected profit for flat Γ and vanishing obstacle 휑. To get some intuition behind this result, we recall the observation (first made by 푢(푥) = sup 피[휑(푋(휃))], H. Lewy) that in dimension 푛 = 2 the function where the supremum is taken over all 0 < 휃 < ∞, 휃 stop- 3 푢 (푥) = 휌3/2 cos ( 휃) (6) ping time. The obstacle problem associated to nonlocal 0 2 operators, such as the fractional Laplacian, arises in mathe- matical finance in the valuation of the so-called American option on multiple assets. An American option gives its holder the right to buy or sell a stock at a fixed price at any time prior to the set expiration time 푇. For instance, the S&P 100 index option is an American option on a value- weighted index of 100 stocks. In the perpetual American option (that is, when 푇 → ∞), the value function 푢 is a solution to the obstacle-type problem

min{퐿푢, 푢 − 휑} = 0 in ℝ푛, where 퐿 is the infinitesimal generator of 푋(푡). When the option has a finite expiration 푇, we note that the relevant operator 퐿 is instead of parabolic type. If the stochastic pro- cess is Brownian motion, then 퐿 = −Δ and 푢 will satisfy the classical obstacle problem (1). However, modeling the Figure 10. The free boundary separates the exercise region behavior of stocks with possible discontinuities requires from the continuation region. the use of jump processes, which have as infinitesimal

NOVEMBER 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1493 is a homogeneous global solution to the Signorini prob- lem. The main ingredients in the proof in [AC04] are a quasi-convexity property of the solution and a monotonic- ity formula for an appropriately weighted average of the local energy of the normal of the solution. This result opened the way to study regularity properties of the free boundary using geometric PDEs techniques. In [ACS08], I. Athanasopoulos, L. Caffarelli, and S. Salsa, un- der the same assumptions of flat boundary and zero obsta- cle, proved the existence of a basic global nondegenerate blow-up profile. Additionally, at points attaining such pro- 1,훼 file, they showed that the free boundary isa 퐶 -“curve,” 3/2 3 1,훼 Figure 11. The graphs of 푢0(푥) = 휌 cos ( 휃) (left) and or, more precisely, a 퐶 (푛 − 2)-dimensional graph on 2 2 the (푛 − 1)-dimensional boundary. The groundbreaking 푢0(푥) = 휌 cos (2휃) (right). idea in [ACS08] consists in the novel use of a classical re- The next step in the program, following the roadmap sult, Almgren’s monotonicity formula, to control a fam- provided by the classical obstacle problem, is the classi- ily of rescalings tailor-made for the Signorini problem. In fication of free boundary points by analyzing the corre- its original formulation [Alm79], Almgren’s result reads as sponding blow-ups. We introduce the family of Almgren- follows. For a function 푢 harmonic in 퐵 (0) and 0 < 푟 < 1, 1 rescalings define the quantities 푢(푟푥) 푢 (푥) = . 푟 1/2 (8) 퐷(푟; 푢) = ∫ |∇푢|2 푑푥 (푟1−푛퐻(푟; 푢)) 퐵푟(0) In order to apply compactness arguments to extract a con- and vergent subsequence, it is necessary to control the 퐿2- of ∇푢푟, uniformly in 푟 > 0. To this end, we observe that 퐻(푟; 푢) = ∫ 푢2 푑퐻푛−1, 휕퐵푟(0) 2 ∫ |∇푢푟| = 푁(1, 푢푟) = 푁(푟, 푢) ≤ 푁(1, 푢), 푛−1 where 푑퐻 denotes the Hausdorff measure on 휕퐵푟(0). 퐵1(0) Then the frequency functional where we have used, in order, the definition of 푁(푟; 푢), a 푟퐷(푟; 푢) simple change of variable, and the monotonicity of 푁(푟; 푢) 푁(푟; 푢) = 퐻(푟; 푢) for 0 < 푟 ≤ 1. This inequality gives the desired control, and allows one to pass to the limit on a subsequence 푟푗 → 0 is nondecreasing in 푟. The reason for the name is that, 푢 ∈ 푊 1,2(퐵+) 푘 2 to obtain a blow-up profile 0 1 . Thanks to given the function 푢푘(휌, 휃) = 푎푘휌 sin(푘휃) in ℝ , it is read- the 퐶1,훼 regularity of solutions, the convergence is actually ily checked that 푁(푟) = 푘. Remarkably, it was shown in 1 + ′ in 퐶 (퐵1 ∪ 퐵1). This fact, in turn, coupled with the ob- [ACS08] that the same monotonicity property continues 푙표푐 servation that ‖푢푟‖퐿2(휕퐵 ) = 1, allows one to prove that to hold when the function 푢 is a solution to the Signorini 1 푢0 is a nonzero global solution of the Signorini problem problem in + (i.e., in 퐵푅 for all 푅 > 0). In addition, even if the blow-up + may not be unique, depending on the choice of the subse- Ω = 퐵1 ∶= 퐵1(0) ∩ {푥푛 > 0}, quence chosen in the limiting process, it is characterized ′ with Γ = 퐵1 ∶= 퐵1(0) ∩ {푥푛 = 0} and 휑 ≡ 0 (assuming by the fact that it is homogeneous of degree 휇 ∶= 푁(0+; 푢). that 푢 is extended to the whole of 퐵1(0) by even symmetry This process can be repeated at any free boundary point in 푥푛). An important—albeit immediate—consequence of 푥0 ∈ ℱ(푢) by simply translating the point to the origin, this fact is that there exists that is, centering the balls in the definition of 푁(푟; 푢) in 푥 instead of 0. We will denote by 푁(0+; 푢, 푥 ) the corre- 푁(0+; 푢) ∶= lim 푁(푟; 푢). 0 0 푟→0+ sponding value. The precise value of 푁(0+; 푢, 푥 ) plays a preeminent role Another corollary, to keep in mind for future purposes, is 0 in the study of the structure of the free boundary. As a pre- a growth estimate of the solution at free boundary points: liminary observation, from the optimal 퐶1,1/2-regularity of if 0 ∈ ℱ(푢) and 휇 ≤ 푁(0+; 푢), then there exists a positive global solutions, one deduces that necessarily 휇 ≥ 3/2. In constant 퐶 such that addition, it can be shown that if 3/2 ≤ 휇 < 2, then nec- 1 sup |푢| ≤ 퐶푟휇 for 0 < 푟 < . (7) essarily 휇 = 3/2 and up to a multiplicative constant and 2 퐵푟 in a suitable system of coordinates, 푢0 is in the form given

1494 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 10 2 2 in (6), with 휌 = 푥푛−1 + 푥푛 and tan 휃 = 푥푛/푥푛−1 (see Fig- At this point it suffices to observe that if 푔 is as in (9), then ure 11). We will soon see that points of minimal frequency (at least heuristically) 휇 = 3/2 share many similarities with the regular points in 휕푒푖 푢 the classical obstacle problem, and therefore we introduce 휕푒 푔 = , 푖 = 1, … , 푛 − 2. 푖 휕 푢 the regular set as 푒푛−1 Hence, we have shown that the regular part of the free ℛ(푢) = {푥0 ∈ ℱ(푢) | 푁(0+; 푢, 푥0) = 3/2}. boundary ℛ(푢) is a locally 퐶1,훼 (푛 − 2)-dimensional graph, To begin with, we note that the free boundary is locally as stated above. It is important to note that this result Lipschitz continuous around regular points, i.e. (after pos- (together with the optimal regularity of solutions) has 2,1 sibly a rotation in ℝ푛−1), been generalized to nonzero obstacles 휑 ∈ 퐶 and to all fractional Laplacians (−Δ)푠 for 0 < 푠 < 1 by L. Caf- ′ ″ ″ ℛ(푢) ∩ 퐵휌 = {푥푛−1 = 푔(푥 ) | 푥 ∈ 퐵휌 }, (9) farelli, S. Salsa, and L. Silvestre in [CSS08]. Their ap- proach is based on a PDE realization of (−Δ)푠 developed with 푔 ∈ 퐶0,1(퐵″). Here 푥″ = (푥 , … , 푥 ), and 퐵″ is the 휌 1 푛−2 휌 by Caffarelli and Silvestre in [CS07], which generalizes the ℝ푛−2 휌 ball in centered at the origin and of radius . The Dirichlet-to-Neumann map for the case 푠 = 1/2. Indeed, idea of the proof is analogous to the one for the classical in a weak sense it holds that obstacle problem, and it relies on the explicit knowledge (−Δ)푠푢(푥) = −휅 lim 푥푎 휕 ̃푢(푥, 푥 ) of the blow-up limit and the existence of a thin cone of 푎 + 푛+1 푛+1 푛+1 푥푛+1→0 directions for a suitable constant 휅푎, 푎 = 1 − 2푠, and ̃푢 solution to the ′ ″ 푛−1 ″ 퐶훿 = {푥 = (푥 , 푥푛−1) ∈ ℝ | 푥푛−1 > 훿|푥 |} problem 푎 푛 along which the solution is monotone. ℒ푎 ̃푢= div(푥푛+1∇ ̃푢)= 0 in ℝ × (0, +∞), To establish further regularity properties, the following ̃푢(푥, 0) = 푢(푥) in ℝ푛. result, known as the Boundary Harnack Principle, is crucial. It states that the quotient of two positive harmonic func- This extension procedure allows one to transform the ob- 푠 tions 푢1 and 푢2 in a Lipschitz domain Ω, both vanishing stacle problem for (−Δ) into a Signorini-type problem for on 휕Ω, satisfies the following Harnack-type inequality: the degenerate operator ℒ푎, which can be studied with 푢 푢 proper generalizations of the techniques we have just de- sup 1 ≤ 퐶 inf 1 . scribed. 푢 퐵 ∩Ω 푢 퐵푅∩Ω 2 푅 2 More recently, higher regularity properties of ℛ(푢) have From this follows, in a rather standard fashion, the Hölder been proved in [KPS15] and [DSS16]. Specifically, in the continuity of the quotient itself: former paper H. Koch, A. Petrosyan, and W. Shi show that the regular part of the free boundary is real analytic by us- 푢 휌 훼 푢 osc 1 ≤ 퐶 ( ) osc 1 ing a partial hodograph-Legendre transformation, whereas 퐵휌∩Ω 푢2 푟 퐵푟∩Ω 푢2 in the latter D. De Silva and O. Savin obtain the 퐶∞ regular- ity of ℛ(푢) by means of a higher-order boundary Harnack for 0 < 휌 < 푟. We now note that 휕푒푢 is nonnegative in ′ inequality. a small ball 퐵푟 (0) for any unit vector 푒 ∈ 퐶훿, and it is 훿 This shows that regular points are aptly named, as it so harmonic in 퐵푟 (0) ⧵ {푢 = 0}. One would like to apply the 훿 happens in the classical obstacle problem. A substantial Boundary Harnack Principle to the ratio 휕푒푢/휕푒푛−1 푢 to infer 훼 difference from the latter, however, rests in the fact that that it is of class 퐶 in a small ball 퐵휌(0). However, this is 퐵 (0)⧵{푢 = 0} there is no dichotomy in the subclassification of ℱ(푢). Al- not immediately possible since the domain 푟훿 is not quite Lipschitz, being lower-dimensional. It is pos- ready in the 2-dimensional case, in fact, one observes that sible, though, to transform it into one by means of a bi- the Signorini problem admits global homogeneous solu- tions of the form 휌휅 cos(휅휃) for 휅 = 2푚 or 휅 = 2푚 + 1/2, Lipschitz transformation 푇 that maps 퐵푟 (0)⧵{푢 = 0} onto, 훿 with 푚 ∈ ℕ (see Figure 11). Proceeding by analogy with say, the upper half-ball {|푧| < 1, 푧푛−1 > 0}. This mapping will transform the Laplacian into a uniformly elliptic oper- the classical theory, we now focus our attention on the free ator in divergence form. Applying an appropriate version boundary points at which the contact set has vanishing of the Boundary Harnack Principle for this class of equa- density. We define the singular set Σ(푢) as the collection tions first, and the inverse transformation 푇−1 afterwards, of points 푥0 ∈ ℱ(푢) such that 푛−1 ′ gives the desired Hölder continuity of 휕푒푢/휕푒푛−1 푢. From ℋ (퐵푟(푥0) ∩ {푢 = 0}) lim sup 푛−1 = 0. this it is easy to deduce that 푟→0+ 푟

휕푒 푢 Following the ideas introduced by N. Garofalo and A. Pet- 푖 ∈ 퐶훼(퐵 (0)), 푖 = 1, … , 푛 − 2. 휌 Σ(푢) 휕푒푛−1 푢 rosyan in [GP09], we shall see that is contained in the

NOVEMBER 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1495 union of 퐶1-manifolds of suitable dimension. Before do- to control a suitable family of rescaled functions. For regu- ing so, though, we observe that there are situations where lar points, we have seen that Almgren’s frequency function the entire free boundary could be composed by singular works in pair with the namesake family of rescalings de- points, as the example fined in (8). Said family, however, is not well suited tode- 1 scribe the behavior of the free boundary at singular points, 푝(푥 , 푥 , 푥 ) = 푥2푥2 − (푥2 + 푥2)푥2 + 푥4 1 2 3 1 2 1 2 3 3 3 nor is it fitting to be paired with Monneau’s monotonic- ity property. We turn, once more, to the classical obstacle shows. This is, in fact, a solution of the zero-obstacle Sig- + problem for inspiration, in particular to (2) and (3). Ob- norini problem in 퐵1 , with ℱ(푢) = {푝 = 0} given by the serving that, if 푥0 ∈ Σ휇, it follows from (7) and (10) that union of the lines 휇 푢(푥) ≃ 푟 in 퐵푟(푥0), the logical choice of rescalings is given 푥1 = 푥3 = 0 and 푥2 = 푥3 = 0. by The starting point is a fundamental characterization of 푢(푥 + 푟푥) 푢(휇)(푥) = 0 . singular points, given by the three following equivalent 푟 푟휇 properties: With this notion at hand, it is possible to proceed as illus- (i) 푥0 ∈ Σ(푢). trated in the previous section to prove a structural theorem (ii) Any blow-up limit of 푢 at 푥0 is a nonnegative, non- for Σ휇(푢). Namely, for 푥0 ∈ Σ휇 and 푝 ∈ 풫휇 as in (ii) above, identically zero polynomial, homogeneous of de- define the dimension of Σ휇 at 푥0 as gree 휇, harmonic, and even in 푥푛. We will denote by 풫휇 the class of such polynomials. 휇 푛−1 ′ ′ 푛−1 푑푥 = dim{휉 ∈ ℝ | 휉 ⋅ ∇푥′ 푝(푥 , 0) = 0 for all 푥 ∈ ℝ }. (iii) 휇 = 2푚, 푚 ∈ ℕ. 0 Given the prominent role occupied by the homogeneity of Next, for 푑 = 0, 1, … , 푛 − 2, let the blow-ups in describing the structure of the free bound- 푑 휇 ary, it becomes natural to classify points in Σ(푢) accord- Σ휇 = {푥0 ∈ Σ휇 | 푑푥0 = 푑}. ingly. We thus let Then, every set Σ푑 , with 휇 = 2푚 (푚 ∈ ℕ) and 푑 = 0, … , 푛 − Σ (푢) = {푥 ∈ ℱ(푢) | 푁(0+; 푢, 푥 ) = 휇}. 휇 휇 0 2, is contained in a countable union of 푑-dimensional 퐶1- Next, following the blueprint provided by the classical ob- manifolds. This result has been very recently improved stacle problem, we introduce two one-parameter families upon in [CSV20], where M. Colombo, L. Spolaor, and of monotonicity formulas. The first result is for function- B. Velichkov prove an explicit logarithmic modulus of con- als of Weiss-type, and it consists in the nondecreasing char- tinuity. acter, for 0 < 푟 < 1, of the functional 1 4. Concluding Remarks 풲 (푟; 푢) = ∫ |∇푢|2 휇 푟푛−2+2휇 The presentation in this article focused only on two basic 퐵푟 models of obstacle problems. Its purpose is only to give a 휇 − ∫ 푢2. bird’s-eye view of the topic, and of the new mathematical 푟푛−1+2휇 휕퐵푟 tools and ideas whose developments it has fostered. Many It should be noted that Weiss’s functional is closely related generalizations to other classes of operators (quasi-linear to Almgren’s frequency function, as they are both con- and fully nonlinear, with variable coefficients, integro- structed with the same building blocks 퐷(푢) = ∫ |∇푢|2 differential, and time-dependent, just to name a few) have 퐵푟 been extensively studied over the years, and continue to and 퐻(푟) = ∫ 푢2. The second family of functionals is of 휕퐵푟 be developed. Due to the restriction on the number of Monneau-type, given by references, the author could not acknowledge several im- 휇 ℳ (푟; 푢, 푝) = ∫ (푢 − 푝)2. portant contributions, both old and new. For instance, a 휇 푟푛 − 1 + 2휇 more refined analysis of the structure and regularity ofthe 휕퐵푟 free boundary in the Signorini problem, correlated with If 0 ∈ Σ and 푝 ∈ 풫 (휇 = 2푚 for some 푚 ∈ ℕ), then 휇 휇 the study of admissible frequencies, has been the object of ℳ (푟; 푢, 푝) is nondecreasing for 0 < 푟 < 1. A first conse- 휇 active investigation as of late. This has been made possible quence of this monotonicity formula is a nondegeneracy by the introduction of delicate epiperimetric inequalities, property for solutions at singular points 푥 ∈ Σ , i.e., 0 휇 modeled on the one for the classical obstacle problem es- sup |푢(푥)| ≥ 푟휇. (10) tablished by Weiss [Wei99]. For a more comprehensive 휕퐵푟(푥0) treatment of some of these aspects, the author refers to the In order to proceed, we should recall that the power of monograph [PSU12] and to the survey article [DS18], and monotonicity formulas lies in the fact that they allow one to the references therein.

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