An Overview of the Obstacle Problem 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 mathematics 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 Guido Stampacchia, who was motivated by questions in potential theory, and Gae- For permission to reprint this article, please contact: [email protected]. tano Fichera, who instead was interested in problems in elasticity with unilateral constraints (more on this later...). DOI: https://doi.org/10.1090/noti2165 NOVEMBER 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1487 functional 퐽(푣) is continuous and strictly convex on the convex set 풦, the existence and uniqueness of minimizers are guaranteed. The obstacle problem can be reformulated as a Varia- tional Inequality 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 Calculus of Variations, Geomet- we can rewrite the obstacle problem in yet another way, ric Measure Theory, and Mathematical Modeling. One of namely as the main driving forces 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 derivatives. 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 displacement 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 Sobolev space 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.