Almost minimizers for the thin obstacle problem
aa r X i v : . [ m a t h . A P ] M a y ALMOST MINIMIZERS FOR THE THIN OBSTACLE PROBLEM
SEONGMIN JEON AND ARSHAK PETROSYAN
Abstract.
We consider Anzellotti-type almost minimizers for the thin obsta-cle (or Signorini) problem with zero thin obstacle and establish their C ,β regularity on the either side of the thin manifold, the optimal growth awayfrom the free boundary, the C ,γ regularity of the regular part of the freeboundary, as well as a structural theorem for the singular set. The analysisof the free boundary is based on a successful adaptation of energy methodssuch as a one-parameter family of Weiss-type monotonicity formulas, Almgren-type frequency formula, and the epiperimetric and logarithmic epiperimetricinequalities for the solutions of the thin obstacle problem. Contents
1. Introduction and main results 21.1. The thin obstacle (or Signorini) problem 21.2. Almost minimizers 31.3. Main results 41.4. Notation 62. Almost harmonic functions 63. Almost Lipschitz regularity of almost minimizers 94. C ,β regularity of almost minimizers 125. Weiss- and Almgren-type monotonicity formulas 236. Almgren rescalings and blowups 267. Growth estimates 298. 3 / Mathematics Subject Classification.
Primary 49N60, 35R35.
Key words and phrases.
Almost minimizers, thin obstacle problem, regualr set, singularset, Signorini problem, Weiss-type monotonicity formula, Almgren’s frequency, epiperimetricinequality.The second author is supported in part by NSF Grant DMS-1800527. Introduction and main results
The thin obstacle (or Signorini) problem.
Let D ⊂ R n be an open setand M ⊂ R n a smooth ( n − thin space ) and considerthe problem of minimizing the Dirichlet energy(1.1) J D ( u ) := Z D |∇ u ( x ) | dx among all functions u ∈ W , ( D ) satisfying u = g on ∂D, u ≥ ψ on M ∩ D, where ψ : M → R is the so-called thin obstacle and g : ∂D → R is the prescribedboundary data with g ≥ ψ on M ∩ ∂D . This problem is known as the thin obstacleproblem . In other words, it is a constrained minimization problem for the energyfunctional J D on a closed convex set K ψ,g ( D, M ) := { u ∈ W , ( D ) : u = g on ∂D , u ≥ ψ on M ∩ D } . This problem can be viewed as a scalar version of the Signorini problem with uni-lateral constraint from elastostatics [Sig59] and is often referred to as the Signoriniproblem . It goes back to the origins of variational inequalities and is consideredas one of the prototypical examples of such problems, see [DL76]. An equivalentformulation is given in the form ∆ u = 0 on D \ M ,u = g on ∂D,u ≥ ψ, ∂ ν + u + ∂ ν − u ≥ , ( ∂ ν + u + ∂ ν − u )( u − ψ ) = 0 on M ∩ D, where the conditions on M ∩ D are known as the Signorini complementarity (or ambiguous ) conditions . Here, ∂ ν ± are the exterior normal derivatives from theeither side of M . In particular, at points on M ∩ D we must have one of the twoboundary conditions satisfied: either u = ψ or ∂ ν + u + ∂ ν − u = 0. The set(1.2) Γ( u ) := ∂ M { x ∈ M ∩ D : u ( x ) = ψ ( x ) } , which separates the regions where different boundary conditions are satisfied, isknown as the free boundary and plays a central role in the analysis of the problem.Because of the presence of the thin obstacle, it is not hard to realize that the solu-tions u of the Signorini problem are at most Lipschitz across M , even if both M and ψ are smooth, as we may have ∂ ν + u + ∂ ν − u > M . However, ithas been known since the works [Caf79, Kin81, Ura85] that the solutions of the thinobstacle problem are C ,β on M and consequently on the either side of M , up to M . In recent years, there has been a renewed interest in this problem, following thebreakthrough result of Athanasopoulos and Caffarelli [AC04] on the optimal C , / regularity of the minimizers (on the either side of M ) as well as its relation to theobstacle-type problems for the fractional Laplacian through the Caffarelli-Silvestreextension [CS07]. There has also been a significant effort in understanding the struc-ture and the regularity of the free boundary. The results have been obtained inmany settings, such as for the equations with variable coefficients, time-dependentversions, problems for fractional Laplacian and other nonlocal equations, both re-garding the regularity of minimizers, as well as the properties of the free boundary; This can be seen on the explicit example u ( x ) = Re( x + i | x n | ) / , which is a solution of theobstacle problem with ψ = 0 on M = { x n = 0 } . LMOST MINIMIZERS FOR THE THIN OBSTACLE PROBLEM 3 see e.g., [Sil07, ACS08, CSS08, GP09, GSVG14, KPS15, PP15, PZ15, DS16, GPS16,KRS16,BSZ17,CRS17,DGPT17,KRS17a,KRS17b,CSV17,ACM18,DPP18,FS18a]and many others.1.2.
Almost minimizers.
In [Anz83], Anzellotti introduced the notion of almostminimizers for energy functionals. Given r >
0, we say that ω : (0 , r ) → [0 , ∞ ) isa modulus of continuity or a gauge function , if ω ( r ) is monotone nondecreasing in r and ω (0+) = 0. Definition 1.1 (Almost minimizers) . Given r > ω ( r ) on(0 , r ), we say that u ∈ W , ( D ) is an almost minimizer (or ω - minimizer ) for thefunctional J D , if, for any ball B r ( x ) ⋐ D with 0 < r < r , we have(1.3) J B r ( x ) ( u ) ≤ (1 + ω ( r )) J B r ( x ) ( v ) for any v ∈ u + W , ( B r ( x )) . The idea is that the Dirichlet energy of u on the ball B r ( x ) is not necessarilyminimal among all competitors v ∈ u + W , ( B r ( x )) but almost minimal in thesense that it cannot decrease more than by a factor of 1 + ω ( r ). In the specificcase of the energy functional J D in (1.1), i.e., the Dirichlet energy, we refer to thealmost minimizers of J D as almost harmonic functions in D .Results on almost minimizers for more general energy functionals can be foundin [DGG00, ELM04, EM99, Min06]. Similar notions were considered earlier in thecontext of the geometric measure theory [Alm76,Bom82], see also [Amb97]. Almostminimizers are also related to quasiminimizers, introduced in [GG82,GG84], see also[Giu03]. For energy functionals exhibiting free boundaries, almost minimizers havebeen considered only recently in [DT15, DET17, dQT18, DS18, DS19].Almost minimizers can be viewed as perturbations of minimizers of various na-ture, but their study is motivated also by the observation that the minimizers withcertain constrains, such as the ones with fixed volume or solutions of the obsta-cle problem, are realized as almost minimizers of unconstrained problems, see e.g.[Anz83]. Yet another motivation is that the study of almost minimizers reveals aunique perspective on the problem and leads to the development of methods relyingon less technical assumptions, thus allowing further generalization.In this paper we extend the notion of almost minimizers to the thin obstacleproblem. Essentially, in (1.3), we restrict the function u and its competitors v tostay above the thin obstacle ψ on M . Definition 1.2 (Almost minimizer for the thin obstacle (or Signorini) problem) . Given r > ω ( r ) on (0 , r ), we say that u ∈ W , ( D ) is an almost minimizer for the thin obstacle ( or Signorini) problem , if u ≥ ψ on M ∩ D and, for any ball B r ( x ) ⋐ D with 0 < r < r , we have(1.4) J B r ( x ) ( u ) ≤ (1 + ω ( r )) J B r ( x ) ( v ) , for any v ∈ K ψ,u ( B r ( x ) , M ) . Note that in the case when M ∩ B r ( x ) = ∅ , the condition (1.4) is the same as(1.3) and thus almost minimizers of the Signorini problem are almost harmonic in D \ M . As in the case of the solutions of the Signorini problem, we are interestedin the regularity properties of almost minimizers as well as the structure and theregularity of the free boundary Γ( u ) ⊂ M as defined in (1.2).Some examples of almost minimizers are given in Appendix A. We would alsolike to mention here that a related notion of almost minimizers for the fractionalobstacle problem has been considered by the authors in [JP19]. SEONGMIN JEON AND ARSHAK PETROSYAN
Main results.
Because of the technical nature of the problem, in this paperwe restrict ourselves only to the case when ω ( r ) = r α for some α > M is flat,specifically M = R n − × { } , and the thin obstacle ψ = 0. As we are mainlyinterested in local properties of almost minimizers and their free boundaries, weassume that D is the unit ball B , u ∈ W , ( B ), and the constant r = 1 inDefinition 1.2. We also assume that u is even in x n -variable: u ( x ′ , x n ) = u ( x ′ , − x n ) , for any x = ( x ′ , x n ) ∈ B . Our first main result is then as follows.
Theorem A ( C ,β -regularity of almost minimizers) . Let u be an almost minimizerfor the Signorini problem in B , under the assumptions above. Then, u ∈ C ,β loc ( B ± ∪ B ′ ) for β = β ( α, n ) and k u k C ,β ( K ) ≤ C k u k W , ( B ) , for any K ⋐ B ± ∪ B ′ and C = C ( n, α, K ) . The proof is obtained by using Morrey and Campanato space estimates, followingthe original idea of Anzellotti [Anz83]. However, in our case the proof is muchmore elaborate and, in a sense, based on the idea that the solutions of the Signoriniproblem are 2-valued harmonic functions, as we have to work with both even andodd extensions of u and ∇ u from B +1 to B .While the optimal regularity for the minimizer (or solutions) of the Signoriniproblem is C , / , we do not expect such regularity for almost minimizers. However,we are able to establish the optimal growth for almost minimizers, which then allowsto study the local properties of the free boundaryΓ( u ) = ∂ { u ( · ,
0) = 0 } ∩ B ′ . Theorem B (Optimal growth near free boundary) . Let u be as in Theorem A.Then, Z ∂B r ( x ) u ≤ C ( n, α ) k u k W , ( B ) r n +2 , for x ∈ B ′ / ∩ Γ( u ) , < r < r ( n, α ) . One of the ingredients in the proof is an Almgren-type monotonicity formula,which we describe below. For an almost minimizer u , Almgren’s frequency [Alm00]is defined by N ( r, u, x ) := r R B r ( x ) |∇ u | R ∂B r ( x ) u , x ∈ Γ( u ) . It is one of the most important monotone quantities in the analysis of the freeboundary for the Signorini problem, see e.g. [PSU12, Chapter 9]. We show that foralmost minimizers a small modification of N is monotone. Theorem C (Monotonicity of the truncated frequency) . Let u be as in Theorem A.Then for any κ ≥ , there is b = b ( n, α, κ ) such that r b N κ ( r, u, x ) := min (cid:26) − br α N ( r, u, x ) , κ (cid:27) is monotone for x ∈ B ′ / ∩ Γ( u ) , and < r < r ( n, α, κ ) . Moreover, we havethat either b N κ (0+ , u, x ) = 3 / or b N κ (0+ , u, x ) ≥ . LMOST MINIMIZERS FOR THE THIN OBSTACLE PROBLEM 5
We give an indirect proof of this fact, based on an one-parametric family ofWeiss-type energy functionals { W κ } <κ<κ , see Theorem 5.1, that go back to thework [GP09] for the solutions of the Signorini problem and Weiss [Wei99b] for theclassical obstacle problem. The fact that b N ≥ / epiperimetric inequality for Weiss’s energyfunctional W / (see [GPS16]), to remove a remaining logarithmic term.Our next result concerns the subset of the free boundary R ( u ) := { x ∈ Γ( u ) : b N (0+ , u, x ) = 3 / } , where Almgren’s frequency is minimal, known as the regular set of u . Theorem D (Regularity of the regular set) . Let u be as in Theorem A. Then R ( u ) is a relatively open subset of the free boundary Γ( u ) and is a ( n − -dimensionalmanifold of class C ,γ . Our proof of this theorem is based on the use of the epiperimetric inequality andis similar to the one for the solutions of the Signorini problem in [GPS16].Finally, we state our main result for the so-called singular set . A free boundarypoint x ∈ Γ( u ) is called singular if the coincidence set Λ( u ) := { u ( · ,
0) = 0 } has H n − -density zero at x , i.e.,lim r → H n − (Λ( u ) ∩ B ′ r ( x )) H n − ( B ′ r ) = 0 . If b N κ (0+ , u, x ) = κ < κ , then x is singular if and only if κ = 2 m , m ∈ N (seeProposition 10.2). For such κ , we then defineΣ κ ( u ) := { x ∈ Γ( u ) : b N κ (0+ , u, x ) = κ } . Theorem E (Structure of the singular set) . Let u be as in Theorem A. Then, forany κ = 2 m < κ , m ∈ N , Σ κ ( u ) is contained in a countable union of ( n − -dimensional manifolds of class C , log . A more refined version of this theorem is given in Theorem 10.13. The proofis based on the logarithmic epiperimetric inequality of Colombo-Spolaor-Velichkov[CSV17] for Weiss’s energy functional W κ , with κ = 2 m , m ∈ N . We also point outthat this inequality is instrumental in the proof of the optimal growth at singularpoints, which is rather immediate for the solutions of the Signorini problem, butfar more complicated for the almost minimizers (see Lemmas 10.5–10.8).1.3.1. Proofs of Theorems A–E.
While we don’t give formal proofs of Theorems A–E, in the main body of the paper, they follow from the combination of results there.More specifically, ◦ Theorem A follows by combining Theorems 3.1 and 4.6. ◦ The statement of Theorem B is contained in that of Lemma 7.4. ◦ Theorem C follows by combining Theorem 5.4 and Corollary 9.4. ◦ The statement of Theorem D is contained in that of Theorem 9.7. ◦ The statement of Theorem E is contained in that of Theorem 10.13.
SEONGMIN JEON AND ARSHAK PETROSYAN
Notation.
Throughout the paper we use the following notation. R n standsfor the n -dimensional Euclidean space. We denote the points of R n by x = ( x ′ , x n ),where x ′ = ( x , . . . , x n − ) ∈ R n − . We routinely identify x ′ ∈ R n − with ( x ′ , ∈ R n − × { } . R n ± stand for open halfspaces { x ∈ R n : ± x n > } .For x ∈ R n , r >
0, we use the following notations for balls of radius r , centeredat x . B r ( x ) = { y ∈ R n : | x − y | < r } , ball in R n , B ± r ( x ′ ) = B r ( x ′ , ∩ {± x n > } , half-ball in R n , B ′ r ( x ′ ) = B r ( x ′ , ∩ { x n = 0 } , ball in R n − , or thin ball.We typically drop the center from the notation if it is the origin. Thus, B r = B r (0), B ′ r = B ′ r (0), etc.Next, for a direction e ∈ R n , we denote ∂ e u = ∇ u · e, the directional derivative of u in the direction e . For the standard coordinatedirections e i , i = 1 , . . . , n , we simply write u x i = ∂ x i u = ∂ e i u. Moreover, by ∂ ± x n u ( x ′ ,
0) we mean the limit of ∂ x n u from within B ± r , specifically, ∂ + x n u ( x ′ ,
0) = lim y → ( x ′ , y ∈ B + r ∂ x n u ( y ) = − ∂ ν + u ( x ′ , ,∂ − x n u ( x ′ ,
0) = lim y → ( x ′ , y ∈ B − r ∂ x n u ( y ) = ∂ ν − u ( x ′ , , where ν ± = ∓ e n are unit outward normal vectors for B ± r on B ′ r .In integrals, we often drop the variable and the measure of integration if it iswith respect to the Lebesgue measure or the surface measure. Thus, Z B r u = Z B r u ( x ) dx, Z ∂B r u = Z ∂B r u ( x ) dS x , where S x stands for the surface measure.We indicate by h u i x,r the integral mean value of a function u over B r ( x ). Thatis, h u i x,r := − Z B r ( x ) u = 1 ω n r n Z B r ( x ) u, where ω n = | B | is the volume of unit ball in R n . Similarly to the other notations,we drop the origin if it is 0 and write h u i r for h u i ,r .2. Almost harmonic functions
In this section we recall some results of Anzellotti [Anz83] on almost harmonicfunctions, i.e., almost minimizers of the Dirichlet integral J D ( v ) = R D |∇ v | . Wealso state here some of the relevant auxiliary results that we will need also in thetreatment of almost minimizers for the Signorini problem. Theorem 2.1.
Let u be an almost harmonic function in an open set D with agauge function ω . Then (i) u is locally almost Lipschitz, i.e., u ∈ C ,σ loc ( D ) for all σ ∈ (0 , . LMOST MINIMIZERS FOR THE THIN OBSTACLE PROBLEM 7 (ii) If ω ( r ) ≤ Cr α for some α ∈ (0 , , then u ∈ C ,α/ ( D ) . While we refer to [Anz83] for the full proof of this theorem, we would like tooutline the key steps in Anzellotti’s argument. The idea to prove C ,σ and C ,α/ regularity of u is through the Morrey and Campanato space estimates, namely, byestablishing that Z B ρ ( x ) |∇ u | ≤ Cρ n − σ (2.1) Z B ρ ( x ) |∇ u − h∇ u i x ,ρ | ≤ Cρ n + α (2.2)for x ∈ K ⋐ D , and 0 < ρ < ρ , with C and ρ depending on n , r , d =dist( K, ∂D ), the gauge function ω , and k u k W , ( D ) .To obtain the estimates above, one starts by choosing a special competitor v in(1.3). Namely, we take v = h which solves the Dirichlet problem∆ h = 0 in B r ( x ) , h = u on ∂B r ( x ) . Equivalently, h is the minimizer of the Dirichlet energy R B r ( x ) |∇ v | among allfunctions in u + W , ( B r ( x )). We call this h the harmonic replacement of u in B r ( x ). We then have the following concentric ball estimates for h . Proposition 2.2.
Let h be harmonic in B r ( x ) and < ρ < r . Then Z B ρ ( x ) |∇ h | ≤ (cid:16) ρr (cid:17) n Z B r ( x ) |∇ h | (2.3) Z B ρ ( x ) |∇ h − h∇ h i x ,ρ | ≤ (cid:16) ρr (cid:17) n +2 Z B r ( x ) |∇ h − h∇ h i x ,r | . (2.4) Proof.
The estimates above follow from the monotonicity in ρ of the quantities1 ρ n Z B ρ ( x ) |∇ h | , ρ n +2 Z B ρ ( x ) |∇ h − h∇ h i x ,ρ | . Noticing that h∇ h i x ,ρ = ∇ h ( x ), an easy proof is obtained by decomposing h intothe sum of the series of homogeneous harmonic polynomials. (cid:3) We next use the almost minimizing property of u to deduce perturbed versionsof the estimates above. Proposition 2.3.
Let u be an almost harmonic function in D . Then for any ball B r ( x ) ⋐ D with r < r and < ρ < r we have Z B ρ ( x ) |∇ u | ≤ h(cid:16) ρr (cid:17) n + ω ( r ) i Z B r ( x ) |∇ u | (2.5) Z B ρ ( x ) |∇ u − h∇ u i x ,ρ | ≤ (cid:16) ρr (cid:17) n +2 Z B r ( x ) |∇ u − h∇ u i x ,r | + 24 ω ( r ) Z B r ( x ) |∇ u | . (2.6) Proof. If h is a harmonic replacement of u in B r ( x ), we first note that Z B r ( x ) |∇ ( u − h ) | = Z B r ( x ) |∇ u | − |∇ h | − Z B r ( x ) ∇ h ∇ ( u − h ) SEONGMIN JEON AND ARSHAK PETROSYAN = Z B r ( x ) |∇ u | − |∇ h | ≤ ω ( r ) Z B r ( x ) |∇ h | ≤ ω ( r ) Z B r ( x ) |∇ u | . Then, combined with (2.3), we estimate Z B ρ ( x ) |∇ u | ≤ Z B ρ ( x ) |∇ h | + 2 Z B ρ ( x ) |∇ ( u − h ) | ≤ h(cid:16) ρr (cid:17) n + ω ( r ) i Z B r ( x ) |∇ u | , which gives (2.5). To obtain (2.6), we argue very similarly by using additionallythat by Jensen’s inequality Z B ρ ( x ) |h∇ u i x ,ρ − h∇ h i x ,ρ | ≤ Z B ρ ( x ) |∇ u − ∇ h | . For more details we refer to the proof of Theorem 4.6, Case 1.1. (cid:3)
From here, one deduces the estimates (2.1)–(2.2) with the help of the followinguseful lemma. The proof can be found e.g. in [HL97].
Lemma 2.4.
Let r > be a positive number and let ϕ : (0 , r ) → (0 , ∞ ) be anondecreasing function. Let a , β , and γ be such that a > , γ > β > . There existtwo positive numbers ε = ε ( a, γ, β ) , c = c ( a, γ, β ) such that, if ϕ ( ρ ) ≤ a h(cid:16) ρr (cid:17) γ + ε i ϕ ( r ) + b r β for all ρ , r with < ρ ≤ r < r , where b ≥ , then one also has, still for < ρ Proof of Theorem 2.1. (i) Taking r small enough so that ω ( r ) < ε , a direct ap-plication of Lemma 2.4 to (2.5) produces the estimate (2.1), which in turn impliesthat u ∈ C ,σ loc ( D ), by the Morrey space embedding theorem.(ii) Using that ω ( r ) ≤ Cr α , combined with the estimate (2.1), we first obtain Z B ρ ( x ) |∇ u − h∇ u i x ,ρ | ≤ (cid:16) ρr (cid:17) n +2 Z B r ( x ) |∇ u − h∇ u i x ,r | + Cr n − σ + α . If σ is so that α ′ = − σ + α > 0, Lemma 2.4 implies that Z B ρ ( x ) |∇ u − h∇ u i x ,ρ | ≤ Cρ n + α ′ . By the Campanato space embedding, we therefore obtain that ∇ u ∈ C ,α ′ loc ( D ).However, it is easy to bootstrap the regularity up to C ,α/ by noticing that wenow know that ∇ u is locally bounded in D and thus R B r ( x ) |∇ u | ≤ Cr n . Pluggingthat in the last term of (2.6), we obtain that Z B ρ ( x ) |∇ u − h∇ u i x ,ρ | ≤ (cid:16) ρr (cid:17) n +2 Z B r ( x ) |∇ u − h∇ u i x ,r | + Cr n + α and repeating the arguments above conclude that u ∈ C ,α/ . (cid:3) LMOST MINIMIZERS FOR THE THIN OBSTACLE PROBLEM 9 Almost Lipschitz regularity of almost minimizers In this section we prove the first regularity results for the almost minimizersfor the Signorini problem, see Definition 1.2. Recall that we assume D = B , M = R n − × { } , ψ = 0, r = 1, and ω ( r ) = r α for some α > 0. Furthermore weassume that u is even symmetric in x n -variable. Theorem 3.1. Let u be an almost minimizer for the Signorini problem in B .Then u ∈ C ,σ ( B ) for all < σ < . Moreover, for any K ⋐ B , k u k C ,σ ( K ) ≤ C k u k W , ( B ) (3.1) with C = C ( n, α, σ, K ) . The idea of the proof is to follow that of Anzellotti [Anz83] that we outlined inSection 2 and to prove an estimate similar to (2.5). The proof of the latter estimatefollowed by a perturbation argument from a similar estimate for the harmonicreplacement of u . However, in the case of the Signorini problem, the harmonicreplacements are not necessarily admissible competitors. Instead, for B r ( x ) ⋐ B ,we consider the Signorini replacements h of u in B r ( x ), which solve the Signoriniproblem in B r ( x ) with the thin obstacle 0 on M and boundary values h = u on ∂B r ( x ). Equivalently, Signorini replacements are the minimizers of J B r ( x ) on theconstraint set K ,u ( B r ( x ) , M ) and they also satisfy the variational inequality (3.2) Z B r ( x ) ∇ h · ∇ ( v − h ) ≥ v ∈ K ,u ( B r ( x ) , M ) . We then have the following concentric ball estimates for Signorini replacementssimilar to the one for harmonic replacements, at least when the center of the ballsis on M = R n − × { } . Proposition 3.2. Let x ∈ M and let h be a solution of the Signorini problem in B r ( x ) with zero obstacle on M , even in x n -variable. Then, Z B ρ ( x ) |∇ h | ≤ (cid:16) ρr (cid:17) n Z B r ( x ) |∇ h | , < ρ < r. (3.3) Proof. We claim that |∇ h | is subharmonic in B r ( x ). This follows from the factthat h ± x i , i = 1 , . . . , n − 1, are subharmonic in B r ( x ), see [PSU12], and similarly thatthe even extensions e h ± x n of h ± x n in x n -variable from B + R ( x ) to all of B R ( x ) are alsosubharmonic. These are all consequences of the fact that a continuous nonnegativefunction, subharmonic in its positivity set is subharmonic, see [PSU12, Ex. 2.6].The subharmonicity of |∇ h | in B r ( x ) then implies, by the sub-mean valueproperty, that the function ρ ρ n Z B ρ ( x ) |∇ h | is monotone nondecreasing. This readily implies (3.3). (cid:3) We next have the perturbed version of Proposition 3.2. which follows from the inequality R B r ( x ) |∇ h | ≤ R B r ( x ) |∇ ((1 − ε ) h + εv ) | , ε ∈ (0 , 1) by afirst variation argument. Proposition 3.3. Let u be an almost minimizer for the Signorini problem in B ,and B r ( x ) ⊂ B . Then, there is C = C ( n ) > such that (3.4) Z B ρ ( x ) |∇ u | ≤ C h(cid:16) ρr (cid:17) n + r α i Z B r ( x ) |∇ u | , < ρ < r. Proof. By using the continuity argument, we may assume that B r ( x ) ⋐ B . Wefirst prove the estimate when x is in the thin space, i.e., x ∈ B ′ and then extendit to arbitrary x ∈ B . Case 1. Suppose x ∈ B ′ and let h be the Signorini replacement of u in B r ( x ).Recall that h satisfies (3.2). Then, plugging v = u , we obtain(3.5) Z B r ( x ) ∇ h · ∇ u − |∇ h | ≥ . Using this, we can estimate Z B r ( x ) |∇ ( u − h ) | = Z B r ( x ) (cid:16) |∇ u | + |∇ h | − ∇ u · ∇ h (cid:17) ≤ Z B r ( x ) |∇ u | − Z B r ( x ) |∇ h | ≤ (cid:0) r α (cid:1) Z B r ( x ) |∇ h | − Z B r ( x ) |∇ h | = r α Z B r ( x ) |∇ h | ≤ r α Z B r ( x ) |∇ u | , (3.6)where in the very last step we have used that h minimizes the Dirichlet integralamong all functions in K ,u ( B r ( x ) , M ).Next, we use the same perturbation argument as in the proof of (2.5). By using(3.3) and (3.6), we estimate Z B ρ ( x ) |∇ u | ≤ Z B ρ ( x ) |∇ h | + 2 Z B ρ ( x ) |∇ ( u − h ) | ≤ (cid:16) ρr (cid:17) n Z B r ( x ) |∇ h | + 2 r α Z B r ( x ) |∇ u | ≤ h(cid:16) ρr (cid:17) n + r α i Z B r ( x ) |∇ u | . Thus, (3.4) follows in this case. Case 2. Consider now the case x ∈ B +1 . If ρ ≥ r/ 4, then we simply have Z B ρ ( x ) |∇ u | ≤ n (cid:16) ρr (cid:17) n Z B r ( x ) |∇ u | . Thus, we may assume ρ < r/ 4. Then, let d := dist( x , B ′ ) > x ∈ ∂B d ( x ) ∩ B ′ . Case 2.1. If ρ ≥ d , then we use B ρ ( x ) ⊂ B ρ ( x ) ⊂ B r/ ( x ) ⊂ B r ( x ) and theresult of Case 1 to write Z B ρ ( x ) |∇ u | ≤ Z B ρ ( x ) |∇ u | ≤ C (cid:20)(cid:18) ρr/ (cid:19) n + ( r/ α (cid:21) Z B r/ ( x ) |∇ u | ≤ C h(cid:16) ρr (cid:17) n + r α i Z B r ( x ) |∇ u | . Case 2.2. Suppose now d > ρ . If d > r , then B r ( x ) ⋐ B +1 . Since u is almostharmonic in B +1 , we can apply Proposition 2.3 to obtain Z B ρ ( x ) |∇ u | ≤ h(cid:16) ρr (cid:17) n + r α i Z B r ( x ) |∇ u | . Thus, we may assume d ≤ r . Then we note that B d ( x ) ⊂ B +1 and by a limitingargument from the previous estimate, we obtain Z B ρ ( x ) |∇ u | ≤ h(cid:16) ρd (cid:17) n + r α i Z B d ( x ) |∇ u | . Case 2.2.1. If r/ ≤ d , then Z B d ( x ) |∇ u | ≤ n (cid:18) dr (cid:19) n Z B r ( x ) |∇ u | , which immediately implies (3.4). Case 2.2.2. It remains to consider the case ρ < d < r/ 4. Using Case 1 again, wehave Z B d ( x ) |∇ u | ≤ Z B d ( x ) |∇ u | ≤ C (cid:20)(cid:18) dr/ (cid:19) n + ( r/ α (cid:21) Z B r/ ( x ) |∇ u | ≤ C (cid:20)(cid:18) dr (cid:19) n + r α (cid:21) Z B r ( x ) |∇ u | , which also implies (3.4). This concludes the proof of the proposition. (cid:3) We can now give the proof of the almost Lipschitz regularity of almost minimiz-ers. Proof of Theorem 3.1. Let K ⋐ B and x ∈ K . Take δ = δ ( n, α, σ, K ) > δ < dist( K, ∂B ) and δ α ≤ ε ( C , n, n + 2 σ − ε = ε ( C , n, n + 2 σ − < ρ < r < δ , by (3.4), Z B ρ ( x ) |∇ u | ≤ C h(cid:16) ρr (cid:17) n + ε i Z B r ( x ) |∇ u | . By applying Lemma 2.4, we obtain Z B ρ ( x ) |∇ u | ≤ C ( n, σ ) (cid:16) ρr (cid:17) n +2 σ − Z B r ( x ) |∇ u | . Taking r ր δ , we can therefore conclude Z B ρ ( x ) |∇ u | ≤ C ( n, α, σ, K ) k∇ u k L ( B ) ρ n +2 σ − . (3.7)From here, we use the Morrey space embedding to obtain u ∈ C ,σ ( K ) with thenorm estimate k u k C ,σ ( K ) ≤ C ( n, α, σ, K ) k u k W , ( B ) , as required. (cid:3) C ,β regularity of almost minimizers In this section we establish the C ,β regularity of almost minimizers for some β > 0. The idea is again to use Signorini replacements and an appropriate versionof the concentric ball estimate (2.4) for solutions of the Signorini problem.As we saw in the proof of the almost Lipschitz regularity of almost minimizers, itis enough to obtain such estimates when balls are centered at x on the thin space M = R n − × { } . It turns out that to prove a proper version of (2.4), we have towork with both even and odd extensions in x n -variable of Signorini replacements h from B + r ( x ) to B r ( x ). The reason is that even extensions are harmonic acrossthe positivity set { h ( · , > } , while the odd extensions are harmonic across theinterior of the coincidence set { h ( · , 0) = 0 } .We start with the borderline case when the center x of concentric balls is onthe free boundary Γ( h ) = B ′ r ( x ) ∩ ∂ R n − { h ( · , 0) = 0 } . Proposition 4.1. Let h be a solution of the Signorini problem in B r ( x ) with x ∈ M , even in x n , and define e h ( x ′ , x n ) := ( h ( x ′ , x n ) , x n ≥ − h ( x ′ , − x n ) , x n < , the odd extension in x n -variable of h from B + r ( x ) to B r ( x ) .Suppose that x ∈ Γ( h ) . Then, for any < α < , there exists C = C ( n, α ) suchthat for any < ρ < s < (3 / r we have Z B ρ ( x ) |∇ h − h∇ h i x ,ρ | ≤ (cid:16) ρs (cid:17) n + α Z B s ( x ) |∇ h − h∇ h i x ,s | + C (cid:18)Z B r ( x ) h (cid:19) s n +1 r n +3 , (4.1) Z B ρ ( x ) |∇ e h − h∇ e h i x ,ρ | ≤ (cid:16) ρs (cid:17) n + α Z B s ( x ) |∇ e h − h∇ e h i x ,s | + C (cid:18)Z B r ( x ) h (cid:19) s n +1 r n +3 . (4.2) Remark . We note here that h∇ h i x ,ρ has its n -th component zero because ofodd symmetry of h x n , while h∇ e h i x ,ρ has its first ( n − 1) components zero becauseof odd symmetry of e h x i , i = 1 , . . . , n − Proof. Without loss of generality we may assume x = 0. For 0 < t < (3 / r ,define ϕ ( t ) := 1 t n + α Z B t |∇ h − h∇ h i t | . Then ϕ ( t ) = 1 t n + α Z B t |∇ h − h∇ h i t | = 1 t n + α (cid:20)Z B t |∇ h | − h∇ h i t Z B t ∇ h + Z B t h∇ h i t (cid:21) = 1 t n + α (cid:20)Z B t |∇ h | − ω n t n (cid:16)Z B t ∇ h (cid:17) (cid:21) . LMOST MINIMIZERS FOR THE THIN OBSTACLE PROBLEM 13 Thus,(4.3) ϕ ′ ( t ) = 1 t n + α (cid:20) − n + αt Z B t |∇ h | + Z ∂B t |∇ h | + 2 n + αω n t n +1 (cid:16)Z B t ∇ h (cid:17) − ω n t n (cid:16)Z B t ∇ h (cid:17)(cid:16)Z ∂B t ∇ h (cid:17)(cid:21) . We next recall that h is C , / regular in B ± r ∪ B ′ r and we have the estimate k∇ h k C , / (cid:16) B ± (3 / r ∪ B ′ (3 / r (cid:17) ≤ C ( n ) r − n +32 k h k L ( B + r ) , (4.4)see e.g. Theorem 9.13 in [PSU12]. Then, using ∇ h (0) = 0, we obtain n + αt n + α +1 Z B t |∇ h | ≤ C ( n, α ) t α r n +3 (cid:16)Z B r h (cid:17) . We can similarly estimate the other term with a negative sign in (4.3) to obtain ϕ ′ ( t ) ≥ − C ( n, α ) t α r n +3 (cid:16)Z B r h (cid:17) . Thus, ϕ ( s ) − ϕ ( ρ ) ≥ − C (cid:16)Z B r h (cid:17) r n +3 Z sρ t − α dt ≥ − C (cid:16)Z B r h (cid:17) s − α r n +3 . Therefore Z B ρ |∇ h − h∇ h i ρ | = ρ n + α ϕ ( ρ ) ≤ ρ n + α (cid:18) ϕ ( s ) + C (cid:16)Z B r h (cid:17) s − α r n +3 (cid:19) ≤ (cid:16) ρs (cid:17) n + α Z B s |∇ h − h∇ h i s | + C (cid:16)Z B r h (cid:17) s n +1 r n +3 . This proves the first estimate in the statement of the lemma. For the secondestimate, involving ∇ e h , we essentially repeat the above argument with e ϕ ( t ) := 1 t n + α Z B t |∇ e h − h∇ e h i t | . (cid:3) We next consider the case when the center x / ∈ Γ( h ). We have to distinguishthe cases x is in { h ( · , > } or the interior of { h ( · , 0) = 0 } . Proposition 4.3. Let h be a solution of the Signorini problem in B r ( x ) with x ∈ M , even in x n -variable. Suppose x ∈ B ′ r ( x ) \ Γ( h ) . Let d := dist( x , Γ( h )) > and fix α ∈ (0 , . Then there are C = C ( n, α ) , C = C ( n, α ) such that for < ρ < s < r the following inequalities hold. (i) If B ′ d ( x ) ⊂ { h ( · , > } , then Z B ρ ( x ) |∇ h − h∇ h i x ,ρ | ≤ C (cid:16) ρs (cid:17) n + α Z B s ( x ) |∇ h − h∇ h i x ,s | + C (cid:16)Z B r ( x ) h (cid:17) s n +1 r n +3 . (ii) If B ′ d ( x ) ⊂ { h ( · , 0) = 0 } , then Z B ρ ( x |∇ e h − h∇ e h i x ,ρ | ≤ C (cid:16) ρs (cid:17) n + α Z B s ( x ) |∇ e h − h∇ e h i x ,s | + C (cid:16)Z B r ( x ) h (cid:17) s n +1 r n +3 . Proof. Without loss of generality we may assume x = 0.(i) First consider the case B ′ d ⊂ { h ( · , > } . If d ≥ s , then h is harmonic in B s and hence Z B ρ |∇ h − h∇ h i ρ | ≤ (cid:16) ρs (cid:17) n +2 Z B s |∇ h − h∇ h i s | ≤ (cid:16) ρs (cid:17) n + α Z B s |∇ h − h∇ h i s | . We can therefore assume 0 < d < s Case 1. s/ ≤ d < s . Case 1.1. Suppose 0 < ρ < d . We first observe that Z B d |∇ h − h∇ h i d | = min C ∈ R n Z B d |∇ h − C | ≤ Z B d |∇ h − h∇ h i s | ≤ Z B s |∇ h − h∇ h i s | . Now using that h is harmonic in B d , we obtain Z B ρ |∇ h − h∇ h i ρ | ≤ (cid:16) ρd (cid:17) n +2 Z B d |∇ h − h∇ h i d | ≤ (cid:16) ρs (cid:17) n + α Z B s |∇ h − h∇ h i s | = 4 n + α (cid:16) ρs (cid:17) n + α Z B s |∇ h − h∇ h i s | . Case 1.2. If ρ ≥ d , then we use ρ/s ≥ / Z B ρ |∇ h − h∇ h i ρ | ≤ Z B s |∇ h − h∇ h i s | ≤ n + α (cid:16) ρs (cid:17) n + α Z B s |∇ h − h∇ h i s | . Case 2. < d < s/ Case 2.1. If 0 < ρ < d , take x ∈ ∂B ′ d ∩ Γ( h ). We first use the harmonicity of h in B d and then applying Proposition 4.1 for balls B d ( x ) ⊂ B s/ ( x ) ⊂ B (2 / r ( x ),to obtain Z B ρ |∇ h − h∇ h i ρ | ≤ (cid:16) ρd (cid:17) n + α Z B d |∇ h − h∇ h i d | ≤ (cid:16) ρd (cid:17) n + α Z B d ( x ) |∇ h − h∇ h i x , d | ≤ (cid:16) ρd (cid:17) n + α (cid:20)(cid:16) ds (cid:17) n + α Z B s/ ( x ) |∇ h − h∇ h i x ,s/ | + C (cid:16)Z B (2 / r ( x ) h (cid:17) s n +1 r n +3 (cid:21) LMOST MINIMIZERS FOR THE THIN OBSTACLE PROBLEM 15 ≤ C (cid:16) ρs (cid:17) n + α Z B s |∇ h − h∇ h i s | + C (cid:16)Z B r h (cid:17) s n +1 r n +3 , where is the last step we have used that B s/ ( x ) ⊂ B s and B (2 / r ( x ) ⊂ B r . Case 2.2. If d ≤ ρ < s/ 4, then we apply Proposition 4.1 with B ρ ( x ) ⊂ B s/ ( x ) ⊂ B (2 / r ( x ) as in Case 2.1: Z B ρ |∇ h − h∇ h i ρ | ≤ Z B ρ ( x ) |∇ h − h∇ h i x , ρ | ≤ (cid:16) ρs (cid:17) n + α Z B s/ ( x ) |∇ h − h∇ h i x ,s/ | + C (cid:16)Z B (2 / r ( x ) h (cid:17) s n +1 r n +3 ≤ C (cid:16) ρs (cid:17) n + α Z B s |∇ h − h∇ h i s | + C (cid:16)Z B r h (cid:17) s n +1 r n +3 . Case 2.3. If ρ ≥ s/ 4, then ρ/s ≥ / h = 0 in B ′ d ( x ). Notice that in the proof of part (i) onlyharmonicity of h in B d and Proposition 4.1 were used. In the present case, it is theodd reflection e h that is harmonic in B d , and we can repeat the same arguments asin part (i) with ∇ h replaced by ∇ e h . (cid:3) Now we have the following estimate combining the two preceding ones. Proposition 4.4. Let h be a solution of the Signorini problem in B r ( x ) with x ∈ M , even in x n -variable. Define c ∇ h := ( ∇ h ( x ′ , x n ) , x n ≥ ∇ h ( x ′ , − x n ) , x n < , the even extension of ∇ h from B + r ( x ) to B r ( x ) . Then for < α < , there are C = C ( n, α ) , C = C ( n, α ) such that for all < ρ < s ≤ (3 / r , Z B ρ ( x ) | c ∇ h − h c ∇ h i x ,ρ | ≤ C (cid:16) ρs (cid:17) n + α Z B s ( x ) | c ∇ h − h c ∇ h i x ,s | + C (cid:18)Z B r ( x ) h (cid:19) s n +1 r n +3 . Remark . We explicitly warn the reader that ∇ e h should not be confused with c ∇ h .In ∇ e h , the first n − x n -variable,while in c ∇ h all components are even in x n . Proof. Let d := dist( x , Γ( h )). Without loss of generality we may assume that d > 0, as the case d = 0 will follow by continuity. Also, without loss of generality,assume x = 0. (i) First consider the case when B ′ d ⊂ { h > } . By the odd symmetry of h x n in x n -variable, we have h h x n i ρ = 0. Thus, for d h x n ( x ) = ( h x n ( x ′ , x n ) , x n ≥ h x n ( x ′ , − x n ) , x n < , we obtain Z B ρ | d h x n − h d h x n i ρ | = Z B ρ d h x n − | B ρ | (cid:18)Z B ρ d h x n (cid:19) = Z B ρ | h x n − h h x n i ρ | − | B ρ | (cid:18)Z B ρ d h x n (cid:19) . Further, if c h x i denotes the i -th component of c ∇ h , we have c h x i = h x i for i =1 , . . . , n − 1, and hence arrive at Z B ρ | c ∇ h − h c ∇ h i ρ | = Z B ρ |∇ h − h∇ h i ρ | − | B ρ | (cid:18)Z B ρ d h x n (cid:19) . (4.5)Similarly, we have Z B s | c ∇ h − h c ∇ h i s | = Z B s |∇ h − h∇ h i s | − | B s | (cid:18)Z B s d h x n (cid:19) . (4.6)Then, by (4.5), (4.6), and Proposition 4.3, we obtain Z B ρ | c ∇ h − h c ∇ h i ρ | ≤ Z B ρ |∇ h − h∇ h i ρ | ≤ C (cid:16) ρs (cid:17) n + α Z B s |∇ h − h∇ h i s | + C (cid:18)Z B r h (cid:19) s n +1 r n +3 ≤ C (cid:16) ρs (cid:17) n + α Z B s | c ∇ h − h c ∇ h i s | + C | B s | (cid:18)Z B s d h x n (cid:19) + C (cid:18)Z B r h (cid:19) s n +1 r n +3 . From h (0) > 0, we have h x n (0) = 0. Thus, using (4.4), we obtain1 | B s | (cid:18)Z B s d h x n (cid:19) ≤ C (cid:18)Z B r h (cid:19) s n +1 r n +3 . This completes the proof in this case.(ii) Suppose now B ′ d ⊂ { h = 0 } . In this case, we use Proposition 4.1 for ∇ e h , whichdiffers from c ∇ h in the first ( n − 1) components by their symmetry, and has thesame even n -th component. Arguing as above, we obtain error terms1 | B s | (cid:18)Z B s h x i (cid:19) , i = 1 , . . . , n − , up to a factor of C ( n, α ). Then, using that h x i (0) = 0, i = 1 , . . . , n − | B s | (cid:18)Z B s h x i (cid:19) ≤ C (cid:18)Z B r h (cid:19) s n +1 r n +3 . This completes the proof. (cid:3) We now prove the C ,β regularity of almost minimizers. LMOST MINIMIZERS FOR THE THIN OBSTACLE PROBLEM 17 Theorem 4.6. Let u be an almost minimizer of the Signorini problem in B . Define c ∇ u ( x ′ , x n ) := ( ∇ u ( x ′ , x n ) , x n ≥ ∇ u ( x ′ , − x n ) , x n < . Then c ∇ u ∈ C ,β ( B ) with β = α n + α ) . Moreover, for any K ⋐ B there holds (4.7) k c ∇ u k C ,β ( K ) ≤ C ( n, α, K ) k u k W , ( B ) . Proof. Without loss of generality, we may assume that K is a ball centered at 0.Fix a small r = r ( n, α, K ) > R := r n n + α ≤ (1 / 2) dist( K, ∂B ), which will imply that e K := { x ∈ B : dist( x, K ) ≤ R } ⋐ B . Our goal now is to show that for x ∈ K , 0 < ρ < r < r ,(4.8) Z B ρ ( x ) | c ∇ u − h c ∇ u i x ,ρ | ≤ C ( n, α ) (cid:16) ρr (cid:17) n + α Z B r ( x ) | c ∇ u − h c ∇ u i x ,r | + C ( n, α, K ) k u k W , ( B ) r n +2 β , which readily gives the estimate (4.7) by applying Lemma 2.4 and using the Cam-panato space embedding.We first prove (4.8) for x ∈ K ∩ B ′ , by taking the advantage of the symmetry of c ∇ u , and then we argue as in the proof of Proposition 3.3 to extend it to all x ∈ K . Case 1 . Suppose x ∈ K ∩ B ′ . For notational simplicity, we assume x = 0 (byshifting the center of the domain D = B to − x ) and let 0 < r < r be given. Letus also denote α ′ := 1 − α n ∈ (0 , , R := r n n + α . We then split our proof into two cases:sup ∂B R | u | ≤ C R α ′ and sup ∂B R | u | > C R α ′ , with C = 2[ u ] ,α ′ , e K = 2 sup x,y ∈ e Kx = y | u ( x ) − u ( y ) || x − y | α ′ . Case 1.1. Assume first that sup ∂B R | u | ≤ C R α ′ . Let h be the Signorini replacementof u on B R . Then, for any 0 < ρ < r , we have Z B ρ | c ∇ u − h c ∇ u i ρ | ≤ Z B ρ | c ∇ h − h c ∇ h i ρ | + 3 Z B ρ | c ∇ u − c ∇ h | + 3 Z B ρ |h c ∇ u i ρ − h c ∇ h i ρ | . Besides, by Jensen’s inequality, we have Z B ρ |h c ∇ u i ρ − h c ∇ h i ρ | ≤ Z B ρ | c ∇ u − c ∇ h | . Hence, combining the estimates above, we obtain(4.9) Z B ρ | c ∇ u − h c ∇ u i ρ | ≤ Z B ρ | c ∇ h − h c ∇ h i ρ | + 6 Z B ρ | c ∇ u − c ∇ h | . Similarly(4.10) Z B r | c ∇ h − h c ∇ h i r | ≤ Z B r | c ∇ u − h c ∇ u i r | + 6 Z B r | c ∇ u − c ∇ h | . Next, note that if r ≤ (3 / n + αα , then r ≤ (3 / R , and thus by Proposition 4.4,(4.11) Z B ρ | c ∇ h − h c ∇ h i ρ | ≤ C ( n, α ) (cid:16) ρr (cid:17) n + α Z B r | c ∇ h − h c ∇ h i r | + C ( n, α ) (cid:16)Z B R h (cid:17) r n +1 R n +3 . Then, using (4.9), (4.10), and (4.11), we obtain Z B ρ | c ∇ u − h c ∇ u i ρ | ≤ Z B ρ | c ∇ h − h c ∇ h i ρ | + 6 Z B ρ | c ∇ u − c ∇ h | ≤ C ( n, α ) (cid:16) ρr (cid:17) n + α Z B r | c ∇ h − h c ∇ h i r | + C ( n, α ) (cid:16)Z B R h (cid:17) r n +1 R n +3 + 6 Z B ρ | c ∇ u − c ∇ h | ≤ C ( n, α ) (cid:16) ρr (cid:17) n + α Z B r | c ∇ u − h c ∇ u i r | + C ( n, α ) (cid:16)Z B R h (cid:17) r n +1 R n +3 + C ( n, α ) Z B r | c ∇ u − c ∇ h | . (4.12)Now take δ = δ ( n, α, K ) > δ < dist( K, ∂B ) and δ α ≤ ε = ε ( C , n, n +2 α ′ − C is as in Theorem 3.1 and ε is as in Lemma 2.4. If r ≤ δ n + α n ,then R < δ , and therefore by (3.7), Z B R | c ∇ u | ≤ C ( n, α, K ) k∇ u k L ( B ) R n +2 α ′ − . Thus, using the above inequality, combined with (3.5), we obtain Z B r | c ∇ u − c ∇ h | ≤ Z B R | c ∇ u − c ∇ h | ≤ Z B R | c ∇ u | − Z B R | c ∇ h | ≤ R α Z B R | c ∇ h | ≤ R α Z B R | c ∇ u | ≤ C ( n, α, K ) k∇ u k L ( B ) R n + α +2 α ′ − = C ( n, α, K ) k∇ u k L ( B ) r n + α n + α ( n − ) . (4.13)We next use that h is subharmonic in B R . (This can be seen for instance by adirect computation ∆( h ) = 2 (cid:0) |∇ h | + h ∆ h (cid:1) = 2 |∇ h | ≥ 0, or by using the factthat h ± are subharmonic.) Then,(4.14) h h i R ≤ sup B R h = sup ∂B R h = sup ∂B R u ≤ C R α ′ . LMOST MINIMIZERS FOR THE THIN OBSTACLE PROBLEM 19 Also note that by (3.1), C ≤ C ( n, α, K ) k u k W , ( B ) . Hence, (cid:18)Z B R h (cid:19) r n +1 R n +3 = C ( n ) h h i R r n +1 R ≤ C ( n, α, K ) k u k W , ( B ) r n + α n + α ) . (4.15)Now (4.12), (4.13), (4.15) give(4.16) Z B ρ | c ∇ u − h c ∇ u i ρ | ≤ C ( n, α ) (cid:16) ρr (cid:17) n + α Z B r | c ∇ u − h c ∇ u i r | + C ( n, α, K ) k u k W , ( B ) r n + α n + α ) + C ( n, α, K ) k∇ u k L ( B ) r n + α n + α ( n − ) ≤ C ( n, α ) (cid:16) ρr (cid:17) n + α Z B r | c ∇ u − h c ∇ u i r | + C ( n, α, K ) k u k W , ( B ) r n + α n + α ) . Case 1.2. Now suppose sup ∂B R | u | > C R α ′ . By the choice of C = 2[ u ] ,α ′ , e K , wehave either u ≥ ( C / R α ′ in all of B R or u ≤ − ( C / R α ′ in all of B R . However,from the inequality u (0) ≥ 0, the only possibility is u ≥ C R α ′ in B R . Let h again be the Signorini replacement of u in B R . Then from positivity of h = u > ∂B R and superharmonicity of h in B R , it follows that h > B R and is therefore harmonic there. Thus, Z B ρ |∇ h − h∇ h i ρ | ≤ (cid:16) ρr (cid:17) n +2 Z B r |∇ h − h∇ h i r | , < ρ < r. Using (4.5) and (4.6) with r in lieu of s , we have for all 0 < ρ < r Z B ρ | c ∇ h − h c ∇ h i ρ | ≤ Z B ρ |∇ h − h∇ h i ρ | ≤ (cid:16) ρr (cid:17) n +2 Z B r |∇ h − h∇ h i r | ≤ (cid:16) ρr (cid:17) n +2 Z B r | c ∇ h − h c ∇ h i r | + 1 | B r | (cid:18)Z B r d h x n (cid:19) . (4.17)Next, note that if r ≤ (1 / n + αα , then r ≤ R/ 2. Then, for γ := 1 − α n ,sup B R/ | D h | ≤ C ( n ) R sup B (3 / R |∇ h | ≤ C ( n ) R n (cid:18)Z B R |∇ h | (cid:19) / ≤ C ( n ) R n (cid:18)Z B R |∇ u | (cid:19) / ≤ C ( n, α, K ) k∇ u k L ( B ) R γ − , where the last inequality follows from (3.7). Thus, for x = ( x ′ , x n ) ∈ B r , we have | h x n | ≤ | x n | sup B R/ | D h |≤ C ( n, α, K ) k∇ u k L ( B ) rR γ − ≤ C ( n, α, K ) k∇ u k L ( B ) r n n + α ( γ − , and hence (4.18) 1 | B r | (cid:18)Z B r d h x n (cid:19) ≤ C ( n, α, K ) k∇ u k L ( B ) r n +2+ n n + α ( γ − = C ( n, α, K ) k∇ u k L ( B ) r n + α n + α ) . Combining (4.17) and (4.18), we obtain(4.19) Z B ρ | c ∇ h − h c ∇ h i ρ | ≤ (cid:16) ρr (cid:17) n +2 Z B r | c ∇ h − h c ∇ h i r | + C ( n, α, K ) k∇ u k L ( B ) r n + α n + α ) . Finally, (4.9), (4.10), (4.13), and (4.19) give(4.20) Z B ρ | c ∇ u − h c ∇ u i ρ | ≤ Z B ρ | c ∇ h − h c ∇ h i ρ | + 6 Z B ρ | c ∇ u − c ∇ h | ≤ (cid:16) ρr (cid:17) n +2 Z B r | c ∇ h − h c ∇ h i r | + C ( n, α, K ) k∇ u k L ( B ) r n + α n + α ) + 6 Z B ρ | c ∇ u − c ∇ h | ≤ (cid:16) ρr (cid:17) n +2 Z B r | c ∇ u − h c ∇ u i r | + C ( n, α, K ) k∇ u k L ( B ) r n + α n + α ) + 24 Z B r | c ∇ u − c ∇ h | ≤ (cid:16) ρr (cid:17) n +2 Z B r | c ∇ u − h c ∇ u i r | + C ( n, α, K ) k∇ u k L ( B ) r n + α n + α ) + C ( n, α, K ) k∇ u k L ( B ) r n + α n + α ( n − ) ≤ (cid:16) ρr (cid:17) n +2 Z B r | c ∇ u − h c ∇ u i r | + C ( n, α, K ) k∇ u k L ( B ) r n + α n + α ) From (4.16) and (4.20) we obtain (4.8) for x ∈ K ∩ B ′ . Case 2. To extend (4.8) to any x ∈ K , we now assume x ∈ K ∩ B +1 . We use anargument similar to the one in Case 2 in the proof of Proposition 3.3.Now, if ρ ≥ r/ 4, then Z B ρ ( x ) | c ∇ u − h c ∇ u i x ,ρ | ≤ Z B ρ ( x ) | c ∇ u − h c ∇ u i x ,r | ≤ n + α (cid:16) ρr (cid:17) n + α Z B r ( x ) | c ∇ u − h c ∇ u i x ,r | , and thus we may assume ρ < r/ 4. Let d := dist( x , B ′ ) > x ∈ ∂B d ( x ) ∩ B ′ . Note that from the assumption that K is a ball centered at 0, wehave x ∈ K ∩ B ′ . LMOST MINIMIZERS FOR THE THIN OBSTACLE PROBLEM 21 Case 2.1. If ρ ≥ d , then from B ρ ( x ) ⊂ B ρ ( x ) ⊂ B r/ ( x ) ⊂ B r ( x ), we have Z B ρ ( x ) | c ∇ u − h c ∇ u i x ,ρ | ≤ Z B ρ ( x ) | c ∇ u − h c ∇ u i x , ρ | ≤ C ( n, α ) (cid:16) ρr (cid:17) n + α Z B r/ ( x ) | c ∇ u − h c ∇ u i x ,r/ | + C ( n, α, K ) k u k W , ( B ) r n +2 β ≤ C ( n, α ) (cid:16) ρr (cid:17) n + α Z B r ( x ) | c ∇ u − h c ∇ u i x ,r | + C ( n, α, K ) k u k W , ( B ) r n +2 β , which gives (4.8) in this case. Case 2.2. Now we suppose d > ρ . If also d > r , then B r ( x ) ⊂ B +1 and since u is almost harmonic in B +1 , we can apply Proposition 2.3, together with the growthestimate (3.7) in the proof of Theorem 3.1, to conclude Z B ρ ( x ) | c ∇ u − h c ∇ u i x ,ρ | ≤ C ( n, α ) (cid:16) ρr (cid:17) n + α Z B r ( x ) | c ∇ u − h c ∇ u i x ,r | + C ( n, α, K ) k u k W , ( B ) r n +2 β . Thus, we may assume d ≤ r . Then, B d ( x ) ⊂ B +1 , and hence, again by thecombination of Proposition 2.3 and the growth estimate (3.7), we have Z B ρ ( x ) | c ∇ u − h c ∇ u i x ,ρ | ≤ C ( n, α ) (cid:16) ρd (cid:17) n + α Z B d ( x ) | c ∇ u − h c ∇ u i x ,d | + C ( n, α, K ) k u k W , ( B ) d n +2 β . We need to consider further subcases. Case 2.2.1. If r/ ≤ d , then (since also d ≤ r ) Z B d ( x ) | c ∇ u − h c ∇ u i x ,d | ≤ n + α (cid:18) dr (cid:19) n + α Z B r ( x ) | c ∇ u − h c ∇ u i x ,r | and combined with the previous inequality, we obtain (4.8) in this subcase. Case 2.2.2. If d < r/ 4, then we also have Z B d ( x ) | c ∇ u − h c ∇ u i x ,d | ≤ Z B d ( x ) | c ∇ u − h c ∇ u i x , d | ≤ C ( n, α ) (cid:18) dr (cid:19) n + α Z B r/ ( x ) | c ∇ u − h c ∇ u i x ,r/ | + C ( n, α, K ) k u k W , ( B ) r n +2 β ≤ C ( n, α ) (cid:18) dr (cid:19) n + α Z B r ( x ) | c ∇ u − h c ∇ u i x ,r | + C ( n, α, K ) k u k W , ( B ) r n +2 β . Hence, the estimate (4.8) has been established in all possible cases.To complete the proof of the theorem, we now apply Lemma 2.4 to the estimate(4.8) to obtain Z B ρ ( x ) | c ∇ u − h c ∇ u i x ,ρ | ≤ C ( n, α ) (cid:20)(cid:16) ρr (cid:17) n +2 β Z B r ( x ) | c ∇ u − h c ∇ u i x ,r | + C ( n, α, K ) k u k W , ( B ) ρ n +2 β (cid:21) . Taking r ր r = r ( n, α, K ), we have Z B ρ ( x ) | c ∇ u − h c ∇ u i x ,ρ | ≤ C ( n, α, K ) k u k W , ( B ) ρ n +2 β . Then by the Campanato space embedding we conclude that c ∇ u ∈ C ,β ( K )with k c ∇ u k C ,β ( K ) ≤ C ( n, α, K ) k u k W , ( B ) . (cid:3) Having the C ,β regularity of almost minimizers, we can now talk about pointwisevalues of ∂ + x n u ( x ′ , 0) = lim y → ( x ′ , y ∈ B + r ∂ x n u ( y )for x ′ ∈ B ′ . The following complementarity condition is of crucial importance inthe study of the free boundary. Lemma 4.7 (Complementarity condition) . Let u be an almost minimizer for theSignorini problem in B . Then u satisfies the following complementarity condition u ∂ + x n u = 0 on B ′ . Moreover, if x ∈ Γ( u ) then u ( x ) = 0 and | c ∇ u ( x ) | = 0 . Proof. Since u ≥ B ′ , the complementarity condition will follow once we showthat ∂ + x n u vanishes where u > B ′ . To this end, let u ( x ′ , > x ′ ∈ B ′ . By the continuity of u in B , (see Theorem 3.1), we have u > U ⊂ B of ( x ′ , B r ( y ) ⋐ U (not necessarily centered on B ′ ) and v is a harmonic replacement of u in B r ( y ), then by the minimum principle v > B r ( y ), and particularly v > B r ( y ) ∩ B ′ . Then v ∈ K ,u ( B r ( y ) , M )and therefore we must have Z B r ( y ) |∇ u | ≤ (1 + r α ) Z B r ( y ) |∇ v | . This means that u is an almost harmonic function in U . Hence u ∈ C ,α/ ( U )by Theorem 2.1. From the even symmetry of u in x n , it is then immediate that ∂ + x n u ( x ′ , 0) = ∂ x n u ( x ′ , 0) = 0.The second part of the lemma now follows by the C ,β regularity and the com-plementarity condition. (cid:3) LMOST MINIMIZERS FOR THE THIN OBSTACLE PROBLEM 23 Weiss- and Almgren-type monotonicity formulas In the rest of the paper we study the free boundary of almost minimizers. In thissection we introduce important technical tools, so-called Weiss- and Almgren-typemonotonicity formulas , which play a significant role in our analysis.We start with Weiss-type monotonicity formulas. They go back to the worksof Weiss [Wei99b, Wei99a] in the case of the classical obstacle problem and Alt-Caffarelli minimum problem, respectively, and to [GP09] for the solutions of the thinobstacle problems. In the context of almost minimizers, this type of monotonicityformulas has been used in a recent paper [DET17]. Theorem 5.1 (Weiss-type monitonicity formula) . Let u be an almost minimizerfor the Signorini problem in B . Suppose x ∈ B ′ / and u ( x ) = 0 . For < κ < κ with a fixed κ ≥ set W κ ( t, u, x ) := e at α t n +2 κ − "Z B t ( x ) |∇ u | − κ − bt α t Z ∂B t ( x ) u , with a = a κ = n + 2 κ − α , b = n + 2 κ α . Then, for < t < t = t ( n, α, κ ) , ddt W κ ( t, u, x ) ≥ e at α t n +2 κ − Z ∂B t ( x ) (cid:18) u ν − κ (1 − bt α ) t u (cid:19) . In particular, W κ ( t, u, x ) is nondecreasing in t for < t < t .Remark . It is important to observe that while a = a κ depends on κ , the constant b depends only on α , n and κ . We also note that in our version of Weiss’s mono-tonicity formula, perturbations (from the case of the thin obstacle problem) appearin the form of multiplicative factors, rather than additive errors as in [DET17].Because of the multiplicative nature of the perturbations, we can then use the one-parametric family of monotonicity formulas { W κ } <κ<κ to derive an Almgren-typemonotonicity formula, see Theorem 5.4. Remark . To avoid bulky notations, we will write W κ ( t, u ) for W κ ( t, u, x ) when x = 0 or even simply W κ ( t ), when both u and x are clear from the context. Proof. The proof uses an argument similar to the one in Theorem 1.2 in [Wei99a].Essentially, it follows from a comparison (1.4) with special competitors, describedbelow. Without loss of generality, assume x = 0. Then for t ∈ (0 , / w by w ( x ) := (cid:18) | x | t (cid:19) κ u (cid:18) t x | x | (cid:19) , for x ∈ B t . Note that w is κ -homogeneous in B t , i.e., w ( λx ) = λ κ w ( λx ) for λ > x, λx ∈ B t ,and coincides with u on ∂B t . Also note that w ≥ B ′ t and is therefore a validcompetitor for u in (1.4). We refer to this w as the κ -homogeneous replacement of u in B t .Now, in B t , we have ∇ w ( x ) = (cid:18) | x | t (cid:19) κ − (cid:20) κt u (cid:18) t x | x | (cid:19) x | x | + ∇ u (cid:18) t x | x | (cid:19) − ∇ u (cid:18) t x | x | (cid:19) · x | x | x | x | (cid:21) , which gives Z B t |∇ w | dx = Z t Z ∂B r |∇ w ( x ) | dS x dr = Z t Z ∂B r (cid:16) rt (cid:17) κ − (cid:12)(cid:12)(cid:12)(cid:12) κt u (cid:16) t xr (cid:17) ν − (cid:16) ∇ u (cid:16) t xr (cid:17) · ν (cid:17) ν + ∇ u (cid:16) t xr (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) dS x dr = Z t Z ∂B t (cid:16) rt (cid:17) n +2 κ − (cid:12)(cid:12)(cid:12) κt uν − (cid:0) ∇ u · ν (cid:1) ν + ∇ u (cid:12)(cid:12)(cid:12) dS x dr = tn + 2 κ − Z ∂B t (cid:12)(cid:12)(cid:12) ∇ u − (cid:0) ∇ u · ν (cid:1) ν + κt uν (cid:12)(cid:12)(cid:12) dS x = tn + 2 κ − Z ∂B t (cid:16) |∇ u | − (cid:0) ∇ u · ν (cid:1) + (cid:16) κt (cid:17) u (cid:17) dS x . The latter equality can be rewritten as(5.1) Z ∂B t u dS x = (cid:16) tκ (cid:17) (cid:20) n + 2 κ − t Z B t |∇ w | dx + Z ∂B t (cid:0) u ν − |∇ u | (cid:1) dS x (cid:21) . Since w is a competitor for u , we have Z B t |∇ w | dx ≥ 11 + t α Z B t |∇ u | dx ≥ (1 − t α ) Z B t |∇ u | dx (5.2)and combining (5.1) and (5.2) yields(5.3) Z ∂B t u dS x ≥ (cid:16) tκ (cid:17) (cid:20) ( n + 2 κ − 2) 1 − t α t Z B t |∇ u | dx + Z ∂B t (cid:0) u ν − |∇ u | (cid:1) dS x (cid:21) . Multiplying this by κ e at α t − n − κ and rearranging terms, we obtain(5.4) ddt (cid:16) e at α t − n − κ +2 (cid:17) Z B t |∇ u | dx = − ( n + 2 κ − e at α t − n − κ (cid:0) t − t α +1 (cid:1) Z B t |∇ u | dx ≥ e at α t − n − κ +2 Z ∂B t (cid:0) u ν − |∇ u | (cid:1) dS x − κ e at α t − n − κ Z ∂B t u dS x . Define now an auxiliary function ψ ( t ) = κe at α (1 − bt α ) t n +2 κ − . Then we write W κ ( t, u, 0) = e at α t − n − κ +2 Z B t |∇ u | dx − ψ ( t ) Z ∂B t u dS x and, using (5.4), obtain ddt W κ ( t, u, 0) = ddt (cid:16) e at α t − n − κ +2 (cid:17) Z B t |∇ u | dx + e at α t − n − κ +2 Z ∂B t |∇ u | dS x − ψ ′ ( t ) Z ∂B t u dS x − ψ ( t ) Z ∂B t uu ν dS x − ( n − ψ ( t ) t Z ∂B t u dS x LMOST MINIMIZERS FOR THE THIN OBSTACLE PROBLEM 25 ≥ e at α t − n − κ +2 Z ∂B t (cid:0) u ν − |∇ u | (cid:1) dS x − κ e at α t − n − κ Z ∂B t u dS x + e at α t − n − κ +2 Z ∂B t |∇ u | dS x − ψ ′ ( t ) Z ∂B t u dS x − ψ ( t ) Z ∂B t uu ν dS x − ( n − ψ ( t ) t Z ∂B t u dS x = e at α t − n − κ +2 Z ∂B t u ν dS x − ψ ( t ) Z ∂B t uu ν dS x − (cid:18) κ e at α t − n − κ + ψ ′ ( t ) + ( n − ψ ( t ) t (cid:19) Z ∂B t u dS x . Now observe that ψ ( t ) satisfies the inequality − e at α t n +2 κ − (cid:18) κ e at α t − n − κ + ψ ′ ( t ) + ( n − ψ ( t ) t (cid:19) − ψ ( t ) ≥ < t < t ( n, α, κ ) and 0 < κ < κ . Indeed, a direct computation shows thatthe above inequality is equivalent to2 α (1 + κ − κ ) − ( n + 2 κ )[( n + 2 κ ) κ − α ( n + 2 κ − t α ≥ , which holds for 0 < κ < κ and small t > α − n + 2 κ ) κ t α ≥ . Hence, recalling also the formula for ψ ( t ), we can conclude that ddt W κ ( t, u, ≥ e at α t n +2 κ − (cid:20)Z ∂B t u ν dS x − κ (1 − bt α ) t Z ∂B t uu ν dS x + (cid:16) κ (1 − bt α ) t (cid:17) Z ∂B t u dS x (cid:21) = e at α t n +2 κ − Z ∂B t (cid:18) u ν − κ (1 − bt α ) t u (cid:19) , for 0 < t < t ( n, α, κ ). (cid:3) Next, for an almost minimizer u in B and x ∈ B ′ / , consider the quantity N ( t, u, x ) := t R B t ( x ) |∇ u | R ∂B t ( x ) u , < t < / Almgren’s frequency and goes back to Almgren’s Big RegularityPaper [Alm00]. This kind of quantities have also been used in unique continuationfor a class of elliptic operators [GL86, GL87] and have been instrumental in thinobstacle-type problems, starting with the works [ACS08, CSS08, GP09].Before proceeding, we observe that Almgren’s frequency is well defined when x is a free boundary point, since R ∂B t ( x ) u > 0. Indeed, otherwise u = 0 on ∂B t ( x )and we can use h ≡ B t ( x ) as a competitor, to obtain that R B t ( x ) |∇ u | ≤ (1 + t α )0 = 0, implying u ≡ B t ( x ), contradicting the assumption that x is afree boundary point. Next, we also consider a modification of N : e N ( t, u, x ) := 11 − bt α N ( t, u, x ) , where b is as in Theorem 5.1, as well as b N κ ( t, u, x ) := min { e N ( t ) , κ } , < t < t , which we call the truncated frequency .For the frequencies N , e N , and b N κ , we will follow the same notational conven-tions as outlined in Remark 5.3 for Weiss’s functionals W κ .With the Weiss type monotonicity formula at hand, we easily obtain the followingmonotonicity of b N κ . Theorem 5.4 (Almgren-type monotonicity formula) . Let u , κ , and t be as inTheorem 5.1, and x a free boundary point. Then b N κ ( t, u, x ) is nondecreasing in < t < t .Proof. We assume x = 0. It is quite important to observe that t depends only on n , α , and κ . Then, if e N ( t ) < κ for some t ∈ (0 , t ) and κ ∈ (0 , κ ), then W κ ( t ) = e at α t n +2 κ − (cid:18)Z ∂B t u (cid:19) ( N ( t ) − κ (1 − bt α ))= e at α t n +2 κ − (cid:18)Z ∂B t u (cid:19) (1 − bt α ) (cid:16) e N ( t ) − κ (cid:17) < . By Theorem 5.1 we also have W κ ( s ) ≤ W κ ( t ) < s ∈ (0 , t ), and thus e N ( s ) < κ . This completes the proof. (cid:3) Remark . The proof above is rather indirect and establishes the monotonicity of b N κ from that of Weiss-type formulas in one-parametric family { W κ } <κ<κ . Thiskind of relation has been first observed in [GP09].6. Almgren rescalings and blowups In this section we prove a lower bound on Almgren’s frequency for almost min-imizers at free boundary points. The idea is to consider appropriate rescalingsand blowups of almost minimizers to obtain solutions of the Signorini problem, forwhich a bound N (0+) ≥ / u be an almost minimizer for the Signorini problem in B , and x ∈ B ′ / a free boundary point. For 0 < r < / Almgren rescaling of u at x u Ax ,r ( x ) := u ( rx + x ) (cid:16) r n − R ∂B r ( x ) u (cid:17) , x ∈ B / (2 r ) . When x = 0, we also write u Ar instead of u A ,r . The Almgren rescalings have thefollowing normalization and scaling properties k u Ax ,r k L ( ∂B ) = 1 N ( ρ, u Ax ,r ) = N ( ρr, u, x ) , ρ < / (2 r ) . We will call the limits of u Ax ,r over any sequence r = r j → Almgren blowups of u at x and denote by u Ax , . We use the superscript A to distinguish this rescaling from the other rescalings, namely,homogeneous and almost homogeneous rescalings that we consider later. LMOST MINIMIZERS FOR THE THIN OBSTACLE PROBLEM 27 Proposition 6.1 (Existence of Almgren blowups) . Let x ∈ B ′ / ∩ Γ( u ) be suchthat b N κ (0+ , u, x ) = κ < κ . Then every sequence of Almgren rescalings u Ax ,r j ,with r j → contains a subsequence, still denoted r j , such that for a function u Ax , ∈ W , ( B ) ∩ C ( B ± ∪ B ′ ) u Ax ,r j → u Ax , in W , ( B ) ,u Ax ,r j → u Ax , in L ( ∂B ) ,u Ax ,r j → u Ax , in C ( B ± ∪ B ′ ) . Moreover, u Ax , is a nonzero solution of the Signorini problem in B , even in x n ,and homogeneous of degree κ in B , i.e., u Ax , ( λx ) = λ κ u Ax , ( x ) , for λ > , provided x, λx ∈ B .Proof. Without loss of generality, we assume x = 0. From the fact that b N (0+ , u ) = κ < κ , it follows also that N (0+ , u ) = b N (0+ , u ) = κ . In particular, N ( r j , u ) < κ for large j . Then, for such j Z B |∇ u Ar j | = N (1 , u Ar j ) = N ( r j , u ) ≤ κ and combined with the normalization R ∂B ( u Ar j ) = 1, we see that the sequence u Ar j is bounded in W , ( B ). Hence, there is a function u A ∈ W , ( B ) such that, overa subsequence, u Ar j → u A weakly in W , ( B ) ,u Ar j → u A strongly in L ( ∂B ) . In particular, R ∂B ( u A ) = 1, implying that u A B .Next, we observe that since u is an almost minimizer in B with gauge function ω ( t ) = t α , u Ar is also an almost minimizer in B / (2 r ) with gauge function ω r ( t ) =( rt ) α . This is rather easy to see, since u Ar ( x ) up to a positive constant factor is u ( rx )and the multiplication (or the division) by a positive number preserves the almostminimizing property. Since ω r ( t ) ≤ ω ( t ), Theorem 4.6 is applicable to rescalings u Ar j , from where we can deduce that over yet another subsequence, u Ar j → u A in C ( B ± ∪ B ′ ) . (6.1)Now, we claim that since the gauge functions ω r ( t ) = ( rt ) α → r → 0, theblowup u A is a solution of the Signorini problem in B . Indeed, for a fixed r j , let h r j be the Signorini replacement of u Ar j in B . Then, by repeating the argument asin the proof of Proposition 3.3 Z B |∇ ( u Ar j − h r j ) | ≤ r αj Z B |∇ u Ar j | . This implies that h r j → u A weakly in W , ( B ). On the other hand, by the bound-edness of the sequence h r j in W , ( B ), we have also boundedness in C , / ( B ± ∪ B ′ ) and hence, over a subsequence, h r j → u A in C ( B ± ∪ B ′ ). By this convergencewe then conclude that u A satisfies ∆ u A = 0 in B \ B ′ u A ≥ , − ∂ + x n u A ≥ , u A ∂ + x n u A = 0 on B ′ , and hence u A itself solves the Signorini problem in B .Using the C convergence again, we have that for any 0 < ρ < N ( ρ, u A ) = lim r j → N ( ρ, u Ar j ) = lim r j → N ( ρr j , u ) = N (0+ , u ) = κ. Thus, the Almgren frequency of u A is constant κ , which is possible only if u A is a κ -homogeneous solution of the Signorini problem in B , see [PSU12, Theorem 9.4]. (cid:3) In what follows, it will be sufficient for us to fix κ ≥ κ = 2), in thedefinition of b N κ and we will simply write b N = b N κ . Lemma 6.2 (Minimal frequency) . Let u be an almost minimizer for the Signoriniproblem in B . If x ∈ B ′ / ∩ Γ( u ) , then b N (0+ , u, x ) = lim r → b N ( r, u, x ) ≥ . Consequently, we also have b N ( t, u, x ) ≥ / for < t < t . Proof. As before, let x = 0. Assume to the contrary that b N (0+ , u ) = κ < / κ < κ we can apply Proposition 6.1 to obtain that over a sequence r j → u Ar j → u A in C ( B ± ∪ B ′ ), where u A is a nonzero κ -homogeneous solution of theSignorini problem in B , even in x n . Moreover, since 0 ∈ Γ( u ), by Lemma 4.7 wehave that u (0) = | c ∇ u (0) | = 0, implying that u Ar j (0) = | [ ∇ u Ar j (0) | = 0 and, by passingto the limit, u A (0) = | [ ∇ u A (0) | = 0. Now, to arrive at a contradiction, we argueas in the proof of [PSU12, Proposition 9.9] to reduce the problem to dimension n = 2, where we can classify all possible homogeneous solutions of the Signoriniproblem, even in x n . The only nonzero homogeneous solutions with κ < / n = 2 are possible for κ = 1 and have the form u A ( x ) = − cx n for some c > 0, but they fails to satisfy the condition | [ ∇ u A (0) | = 0. Thus, we arrived atcontradiction, implying that b N (0+ , u ) ≥ / 2. Finally, applying Theorem 5.4, weobtain b N ( t, u ) ≥ b N (0+ , u ) ≥ / 2, for 0 < t < t . (cid:3) Corollary 6.3. Let u be an almost minimizer for the Signorini problem in B and x a free boundary points. Then W / ( t, u, x ) ≥ for < t < t . Proof. We simply observe that e N ( t ) ≥ b N ( t ) ≥ / < t < t and hence W / ( t, u, x ) = e at α t n +2 κ − (cid:18)Z ∂B t u (cid:19) (1 − bt α ) (cid:18) e N ( t ) − (cid:19) ≥ . (cid:3) LMOST MINIMIZERS FOR THE THIN OBSTACLE PROBLEM 29 Growth estimates An important step in the study of the free boundary in the Signorini problem(and in many other free boundary problems) is the proof of the optimal regularity ofsolutions, which in this case is C , / on each side of the thin space. This allows tomake proper blowup arguments to establish the regularity of the so-called regularpart of the free boundary. However, in the case of almost minimizers, we only know C ,β regularity for some small β > optimal growth of the almost minimizers atfree boundary points with the help of the Weiss-type monotonicity formula and theepiperimetric inequality.Finally, we want to point out that the results in this section are rather immediatein the case of minimizers, as they follow easily from the differentiation formulas forthe quantities involved in the Almgren’s frequency formula. This is completelyunavailable for almost minimizers.We start by defining a new type of rescalings. Fix κ ≥ / 2. For a free boundarypoint x in B ′ / and r > 0, we define the κ -homogeneous rescaling by u x ,r ( x ) := u ( κ ) x ,r ( x ) = u ( rx + x ) r κ , x ∈ B / (2 r ) . To take advantage of the Weiss-type monotonicity formula, we need a slight modi-fication of this rescaling. With the help of an auxiliary function φ ( r ) = φ κ ( r ) := e − ( κb/α ) r α r κ , r > , which is a solution of the differential equation φ ′ ( r ) = κ φ ( r ) 1 − br α r , r > κ -almost homogeneous rescalings by u φx ,r ( x ) := u ( rx + x ) φ ( r ) , x ∈ B / (2 r ) . Lemma 7.1 (Weak growth estimate) . Let u be an almost minimizer of the Sig-norini problem in B and x ∈ B ′ / ∩ Γ( u ) be such that b N (0+ , u, x ) ≥ κ for κ ≤ κ .Then Z ∂B t ( x ) u ≤ C ( n, α, κ ) k u k W , ( B ) (cid:18) log 1 t (cid:19) t n +2 κ − , Z B t ( x ) |∇ u | ≤ C ( n, α, κ ) k u k W , ( B ) (cid:18) log 1 t (cid:19) t n +2 κ − , for < t < t = t ( n, α, κ ) .Proof. Without loss of generality, assume x = 0. We first note that the condition b N (0+ , u ) ≥ κ implies that b N ( t, u ) ≥ κ for 0 < t < t = t ( n, α, κ ). Then also e N ( t, u ) ≥ κ for such t and consequently, W κ ( t, u ) = e at α t n +2 κ − (cid:18)Z ∂B t u (cid:19) (1 − bt α ) (cid:16) e N ( t, u ) − κ (cid:17) ≥ . Next, for φ = φ κ , we have that ddr u φr ( x ) = ∇ u ( rx ) · xφ ( r ) − u ( rx )[ φ ′ ( r ) /φ ( r )] φ ( r ) = 1 φ ( r ) (cid:18) ∇ u ( rx ) · x − κ (1 − br α ) r u ( rx ) (cid:19) . Now let m ( r ) = (cid:18)Z ∂B ( u φr ( ξ )) dS ξ (cid:19) / , r > . Then, m ′ ( r ) = (cid:18)Z ∂B u φr ( ξ ) ddr u φr ( ξ ) dS ξ (cid:19) (cid:18)Z ∂B ( u φr ( ξ )) dS ξ (cid:19) − / and consequently, by Cauchy-Schwarz, | m ′ ( r ) | ≤ Z ∂B (cid:20) ddr u φr ( ξ ) (cid:21) dS ξ ! / . Hence, | m ′ ( r ) | ≤ φ ( r ) Z ∂B (cid:18) ∇ u ( rξ ) · ξ − κ (1 − br α ) r u ( rξ ) (cid:19) dS ξ ! / = 1 φ ( r ) r n − Z ∂B r (cid:18) ∂ ν u ( x ) − κ (1 − br α ) r u ( x ) (cid:19) dS x ! / ≤ φ ( r ) (cid:18) r n − r n +1 e ar α ddr W κ ( r ) (cid:19) / = e cr α r / (cid:18) ddr W κ ( r ) (cid:19) / , c = κ bα − a , for 0 < r < t = t ( n, α, κ ). Thus, we have shown | m ′ ( r ) | ≤ e cr α r / (cid:18) ddr W κ ( r ) (cid:19) / , < r < t . Integrating in r over the interval ( s, t ) ⊂ (0 , t ), we obtain | m ( t ) − m ( s ) | ≤ Z ts e cr α r / (cid:18) ddr W κ ( r ) (cid:19) / dr ≤ (cid:18)Z ts e cr α r dr (cid:19) / (cid:18)Z ts ddr W κ ( r ) (cid:19) / ≤ C (cid:18) log ts (cid:19) / [ W κ ( t ) − W κ ( s )] / . In particular (recalling that W κ ( s ) ≥ m ( t ) ≤ m ( t ) + C (cid:18) log t t (cid:19) / [ W κ ( t )] / . Varying t by an absolute factor, we can guarantee that m ( t ) ≤ C ( n, α, κ ) k u k L ( B ) , W κ ( t ) ≤ C ( n, α, κ ) k u k W , ( B ) . Hence, we can conclude Z ∂B t u ≤ C ( n, α, κ ) k u k W , ( B ) (cid:18) log 1 t (cid:19) t n +2 κ − , LMOST MINIMIZERS FOR THE THIN OBSTACLE PROBLEM 31 for 0 < t < t = t ( n, α, κ ). This implies the first bound. The second boundfollows immediately from the first one by using that W κ ( t, u ) ≤ W κ ( t , u ):1 t n +2 κ − Z B t |∇ u | ≤ κ (1 − bt α ) t n +2 κ − Z ∂B t u + e − at α W κ ( t , u ) ≤ C ( n, α, κ ) k u k W , ( B ) (cid:18) log 1 t (cid:19) + e at α t n +2 κ − Z B t |∇ u | ≤ C ( n, α, κ ) k u k W , ( B ) (cid:18) log 1 t (cid:19) . (cid:3) The logarithmic term in Lemma 7.1 does not allow to conclude that the sequenceof κ -homogeneous or almost homogeneous rescaling is uniformly bounded say in W , ( B ). In the rest of this section we show that in the case of the minimalfrequency κ = 3 / W / ( w ) := Z B |∇ w | − Z ∂B w . To state this result, we let(7.1) A := { w ∈ W , ( B ) : w ≥ B ′ , w ( x ′ , x n ) = w ( x ′ , − x n ) } Theorem 7.2 (Epiperimetric inequality) . There exists η ∈ (0 , such that if w ∈ A is homogeneous of degree / in B , then there exists v ∈ A with v = w on ∂B such that W / ( v ) ≤ (1 − η ) W / ( w ) . This kind of inequalities go back to the work of Weiss [Wei99b], in the case ofthe classical obstacle problem. For the Signorini problem, a version of this theoremwas proved in [GPS16] and [FS16]. In fact, the theorem above is the version in[RS17]. The inequality in [GPS16] and [FS16] requires w to be close to the blowupprofile, but this can be easily removed by a scaling argument (see [RS17]). We alsorefer to [CSV17], for a more direct proof of this inequality with an explicit constant η = 1 / (2 n + 3).Now, with the help of the epiperimetric inequality, we can prove a decay estimatefor the Weiss-type energy functional W / . For the rest of the section, we willassume κ = 2 , which will make some of the constants independent of κ , but the results hold alsofor any other value of κ ≥ 2, with possible added dependence of constants on κ . Lemma 7.3. Let x ∈ B ′ / be a free boundary point. Then, there exist δ = δ ( n, α ) > such that ≤ W / ( t, u, x ) ≤ Ct δ , < t < t = t ( n, α ) , with C = C ( n, α ) k u k W , ( B ) .Proof. As before, without loss of generality we assume that x = 0. The proof will follow from a differential inequality that we derive by using ourearlier computations and the epiperimetric inequality. Recalling the proof of theWeiss-type monotonicity formula (Theorem 5.1), for small t > 0, we have ddt W / ( t, u ) = e at α t n +1 Z ∂B t |∇ u | − ( n + 1)(1 − t α ) e at α t n +2 Z B t |∇ u | − ψ ′ ( t ) Z ∂B t u − ( n − ψ ( t ) t Z ∂B t u − ψ ( t ) Z ∂B t u∂ ν u = − ( n + 1)(1 − t α ) t W / ( t, u ) + e at α t n +1 Z ∂B t |∇ u | − (cid:18) [( n + 1)(1 − t α ) + ( n − ψ ( t ) t + ψ ′ ( t ) (cid:19) Z ∂B t u − ψ ( t ) Z ∂B t u∂ ν u ≥ − ( n + 1)(1 − t α ) t W / ( t, u )+ e at α (1 − bt α ) t n +1 Z ∂B t (cid:18) |∇ u | − t u∂ ν u − t (cid:20) ( n + 1)(1 − t α ) + ( n − t + ψ ′ ( t ) ψ ( t ) (cid:21) u (cid:19) . To proceed, note that( n + 1)(1 − t α ) + ( n − t + ψ ′ ( t ) ψ ( t ) = ( n − 2) + O ( t α ) t . Now, for the homogeneous rescalings u t ( x ) = u ( tx ) t / , we can write Z ∂B t |∇ u | − t u∂ ν u − 32 ( n − 2) + O ( t α ) t u = t n Z ∂B |∇ u t | − u t ∂ ν u t − 32 [( n − 2) + O ( t α )] u t = t n Z ∂B (cid:18) ∂ ν u t − u t (cid:19) + ( ∂ τ u t ) − (cid:20)(cid:18) n − (cid:19) + O ( t α ) (cid:21) u t , where ∂ τ u t is the tangential component of ∇ u t on the unit sphere. We can summa-rize for now that ddt W / ( t, u ) ≥ − ( n + 1)(1 − t α ) t W / ( t, u )+ e at α (1 − bt α ) t Z ∂B "(cid:18) ∂ ν u t − u t (cid:19) + ( ∂ τ u t ) − (cid:18) n − (cid:19) u t + O ( t α − ) Z ∂B u t . LMOST MINIMIZERS FOR THE THIN OBSTACLE PROBLEM 33 On the other hand, if w t is a 3 / u t in B , i.e., w t ( x ) = | x | / u t ( x/ | x | )then Z ∂B ( ∂ τ u t ) − (cid:18) n − (cid:19) u t = Z ∂B ( ∂ τ w t ) − (cid:18) n − (cid:19) w t = ( n + 1) W / ( w t ) , where W / ( w t ) = Z B |∇ w t | − Z ∂B w t . The last equality follows by repeating the arguments in the beginning of the proofof Theorem 5.1 with κ = 3 / 2. Let v t be the solution of the Signorini problem in B with v t = u t = w t on ∂B . Then by the epiperimetric inequality W / ( v t ) ≤ (1 − η ) W / ( w t ) . On the other hand, since u is an almost minimizer, we have Z B |∇ u t | ≤ (1 + t α ) Z B |∇ v t | and since also u t = v t on ∂B , we have W / ( t, u ) = e at α t n +1 (cid:20)Z B t |∇ u | − (3 / − bt α ) t Z ∂B t u (cid:21) ≤ (1 + O ( t α )) W / ( v t ) + O ( t α ) Z ∂B u t ≤ (cid:16) − η (cid:17) W / ( w t ) + O ( t α ) Z ∂B u t , for 0 < t < t = t ( n, α ) . We can therefore write ddt W / ( t, u ) ≥ − ( n + 1)(1 − t α ) t W / ( t, u )+ ( n + 1) e at α (1 − bt α ) t W / ( w t ) + O ( t α − ) Z ∂B u t ≥ n + 1 t (cid:18) − − η/ O ( t α ) (cid:19) W / ( t, u ) + O ( t α ) t n +3 Z ∂B t u ≥ η t W / ( t, u ) − Ct α/ − , for small t , where we have also used the growth estimate in Lemma 7.1. Takingnow δ such that 0 < δ < min n η , α o , we have ddt (cid:20) W / ( t, u ) t − δ + Cα/ − δ t α/ − δ (cid:21) = t − δ (cid:18) ddt W / ( t, u ) − δt W / ( t, u ) (cid:19) + Ct α/ − δ − ≥ t − δ − h η − δ i W / ( t, u ) − Ct α/ − δ − + Ct α/ − δ − ≥ , for small t , where we have used again that W / ( t, u ) ≥ 0. Thus, we can concludethat 0 ≤ W / ( t, u ) ≤ Ct δ , < t < t = t ( n, α ) , with C = C ( n, α ) k u k W , ( B ) . (cid:3) Using the estimate on W / ( t, u ) in Lemma 7.3, we can improve on Lemma 7.1in the case κ = 3 / Lemma 7.4 (Optimal growth estimate) . Let x ∈ B ′ / be a free boundary point.Then, for < t < t = t ( n, α ) , Z ∂B t ( x ) u ≤ C ( n, α ) k u k W , ( B ) t n +2 , Z B t ( x ) |∇ u | ≤ C ( n, α ) k u k W , ( B ) t n +1 . Proof. We proceed as in the proof of Lemma 7.1 up to the estimate | m ( t ) − m ( s ) | ≤ C (cid:18) log ts (cid:19) / [ W / ( t, u ) − W / ( s, u )] / . From there, using Lemma 7.3, we now have an improved bound | m ( t ) − m ( s ) | ≤ C (cid:18) log ts (cid:19) / t δ/ , s < t < t , with C = C ( n, α ) k u k W , ( B ) . Then, by a dyadic argument, we can conclude that | m ( t ) − m ( s ) | ≤ Ct δ/ . Indeed, let k = 0 , , , . . . be such that t/ k +1 ≤ s < t/ k . Then, | m ( t ) − m ( s ) | ≤ k X j =1 | m ( t/ j − ) − m ( t/ j ) | + | m ( t/ k ) − m ( s ) |≤ C (log 2) / k +1 X j =1 ( t/ j − ) δ/ ≤ C (log 2) / t δ/ − − δ/ = Ct δ/ . In particular, we have m ( t ) ≤ m ( t ) + Ct δ/ ≤ C ( n, α ) k u k W , ( B ) , t < t . This implies the first bound. The second bound follows immediately from the firstone by using that W / ( t, u ) ≤ W / ( t , u ):1 t n +1 Z B t |∇ u ( x ) | dx ≤ (3 / − bt α ) t n +2 Z ∂B t u ( x ) dS x + e − at α W / ( t , u ) LMOST MINIMIZERS FOR THE THIN OBSTACLE PROBLEM 35 ≤ C ( n, α ) k u k W , ( B ) + e at α t n +10 Z B t |∇ u ( x ) | dx ≤ C ( n, α ) k u k W , ( B ) . (cid:3) 8. 3 / -Homogeneous blowups For a free boundary point x ∈ B ′ / , we consider again the 3 / u φx ,t ( x ) = u ( tx + x ) φ ( t ) , x ∈ B / (2 t ) , with φ = φ / . We now observe that the optimal growth estimates in Lemma 7.4implies the boundedness of this family of rescalings in W , ( B R ) for any R > B R if t < / (2 R ), and by Lemma 7.4,we will have Z B R |∇ u φx ,t | = e bα t α t n +1 Z B Rt ( x ) |∇ u | ≤ C ( n, α ) k u k W , ( B ) R n +1 , Z ∂B R ( u φx ,t ) = e bα t α t n +2 Z ∂B Rt ( x ) u ≤ C ( n, α ) k u k W , ( B ) R n +2 , for 0 < t < t /R . Arguing as in the proof of Proposition 6.1, we have for a sequence t = t j → u φx ,t j → u φx , in C ( B ± R ∪ B ′ R ) . By letting R → ∞ and using Cantor’s diagonal argument, we therefore have thatover a subsequence t = t j → u φx ,t j → u φx , in C ( R n ± ∪ R n − ) . We call such u φx , a 3 / -homogeneous blowup of u at x . The name is explained bythe fact that lim t → φ ( t ) t / = 1 , which implies that if we consider the 3 / u (3 / x ,t ( x ) = u ( tx + x ) t / , then we will have u φx , = lim t j → u φx ,t j = lim t j → u (3 / x ,t j =: u (3 / x , and thus u φx , = u (3 / x , . Remark . Because of the logarithmic term in the weak growth estimates inLemma 7.1, at the moment we are unable to consider κ -homogeneous blowups asabove for frequencies other than κ = 3 / 2. However, once the logarithmic term isremoved, the same construction as for κ = 3 / κ = 2 m < κ , m ∈ N , enabling us to consider the κ -homogeneous blowups for these values of κ .We show next that the 3 / u φx ,r ( x ). Lemma 8.2 (Rotation estimate) . Let u be an almost minimizer for the Signoriniproblem in B , x ∈ B ′ / a free boundary point, and δ as in Lemma 7.3. Then for κ = 3 / and φ = φ / Z ∂B | u φx ,t − u φx ,s | ≤ Ct δ/ , s < t < t = t ( n, α ) , for C = C ( n, α ) k u k W , ( B ) .Proof. The proof uses computations similar to the proof of Lemma 7.1 combinedwith the growth estimated for W / ( t, u ) in Lemma 7.3. We assume x = 0, andhave Z ∂B | u φt − u φs | ≤ Z ∂B Z ts (cid:12)(cid:12)(cid:12)(cid:12) ddr u φr (cid:12)(cid:12)(cid:12)(cid:12) dr = Z ts Z ∂B (cid:12)(cid:12)(cid:12)(cid:12) ddr u φr (cid:12)(cid:12)(cid:12)(cid:12) dr ≤ C n Z ts Z ∂B (cid:12)(cid:12)(cid:12)(cid:12) ddr u φr (cid:12)(cid:12)(cid:12)(cid:12) ! / ≤ C n (cid:18)Z ts r dr (cid:19) / Z ts r Z ∂B (cid:12)(cid:12)(cid:12)(cid:12) ddr u φr (cid:12)(cid:12)(cid:12)(cid:12) ! / ≤ C n e ct α (cid:18) log ts (cid:19) / (cid:18)Z ts ddr W / ( r, u ) dr (cid:19) / , c = 3 b α − a , where we have re-used the computation made in the proof of Lemma 7.1. Thus, weobtain Z ∂B | u φt − u φs | ≤ C ( n, α ) (cid:18) log ts (cid:19) / ( W / ( t, u ) − W / ( s, u )) / ≤ C (cid:18) log ts (cid:19) / t δ/ . Then, using a dyadic argument as Lemma 7.4, we can conclude that Z ∂B | u φt − u φs | ≤ Ct δ/ , s < t < t , as required. Indeed, let k = 0 , , , . . . be such that t/ k +1 ≤ s < t/ k . Then Z ∂B | u φt − u φs | ≤ k X j =1 Z ∂B (cid:12)(cid:12)(cid:12) u φt/ j − − u φt/ j (cid:12)(cid:12)(cid:12) + Z ∂B (cid:12)(cid:12)(cid:12) u φt/ k − u φs (cid:12)(cid:12)(cid:12) ≤ C (log 2) / k +1 X j =1 ( t/ j − ) δ/ ≤ C (log 2) / t δ/ − − δ/ . This completes the proof. (cid:3) The uniqueness of 3 / Lemma 8.3. Let u φx , be a blowup at a free boundary point x ∈ B ′ / . Then for κ = 3 / Z ∂B | u φx ,t − u φx , | ≤ Ct δ/ , < t < t , LMOST MINIMIZERS FOR THE THIN OBSTACLE PROBLEM 37 where C = C (cid:0) n, α, k u k W , ( B ) (cid:1) and δ = δ ( n, α ) > are as in Lemma 8.2. Inparticular, the blowup u φx , is unique.Proof. If u x , is the limit of u φx ,t j for t j → 0, then first part of the lemma followsimmediately from Lemma 8.2, by taking s = t j → u φx , is a solution of theSignorini problem in B , by arguing as in the proof of Lemma 6.2 for Almgrenblowups. Now, if ˜ u φx , is another blowup, then from the first part of the lemma wewill have Z ∂B | ˜ u φx , − u φx , | = 0 , implying that both ˜ u φx , and u φx , are solutions of the Signorini problem in B withthe same boundary values on ∂B . By the uniqueness of such solutions, we have˜ u φx , = u φx , in B . The equality propagates to all of R n by the unique continuationof harmonic functions in R n ± . (cid:3) We next show that not only the blowups are unique, but also depend continuouslyon a free boundary point. Lemma 8.4 (Continuous dependence of blowups) . There exists ρ = ρ ( n, α ) > such that if x , y ∈ B ′ ρ are free boundary points, then Z ∂B | u φx , − u φy , | ≤ C | x − y | γ , with C = C (cid:0) n, α, k u k W , ( B ) (cid:1) and γ = γ ( n, α ) > .Proof. Let d = | x − y | and d µ ≤ r ≤ d µ with µ ∈ (0 , 1] to be determined. ByLemma 8.3 we have Z ∂B | u φx , − u φy , | ≤ Cr δ/ + Z ∂B | u φx ,r − u φy ,r |≤ Cd µδ/ + Cd µ ( n +1 / Z ∂B r | u ( x + z ) − u ( y + z ) | dS z and taking the average over d µ ≤ r ≤ d µ , we have Z ∂B | u φx , − u φy , | ≤ Cd µδ/ + Cd µ ( n +3 / Z B dµ \ B dµ | u ( x + z ) − u ( y + z ) | dz. On the other hand, by using Lemma 7.4, Z B dµ \ B dµ | u ( x + z ) − u ( y + z ) | dz ≤ Z B dµ \ B dµ (cid:12)(cid:12)(cid:12)(cid:12)Z dds u ( z + x (1 − s ) + y s ) ds (cid:12)(cid:12)(cid:12)(cid:12) dz ≤ | x − y | Z Z B dµ |∇ u ( z + x (1 − s ) + y s ) | dzds ≤ d Z (cid:18)Z B dµ ( x (1 − s )+ y s ) |∇ u | (cid:19) ds ≤ d Z B dµ + d ( x ) |∇ u | ≤ d Z B dµ ( x ) |∇ u |≤ Cd µn/ (cid:18)Z B dµ ( x ) |∇ u | (cid:19) / ≤ Cd µn/ d µ ( n +1) / ≤ Cd µ ( n +1 / , provided 3 d µ < t , which will hold if d < ρ ( n, α ).Combining the estimates, we infer that Z ∂B | u φx , − u φy , | ≤ Cd µδ/ + Cd − µ . Now choosing µ so that µδ/ − µ , that is µ = 1 / (1 + δ/ Z ∂B | u φx , − u φy , | ≤ C | x − y | γ , x , y ∈ B ′ ρ with γ = δδ + 2 . (cid:3) Regularity of the regular set In this section we establish one of the main result of this paper, the C ,γ regu-larity of the regular set. In fact, the most technical part of the proof has alreadybeen done in the previous section, where we proved the uniqueness of the 3 / Definition 9.1 (Regular points) . For an almost minimizer u for the Signoriniproblem in B , we say that a free boundary point x is regular if b N (0+ , u, x ) = 3 / . Note that since 3 / < ≤ κ , we will have that b N ( r ) < κ for small r > 0, implyingthat e N ( r ) = b N ( r ) for such r and consequently that N (0+) = e N (0+) = b N (0+) = 3 / . In particular, the condition above does not depend on the choice of κ ≥ u by R ( u ) and call it the regular set .An important ingredient in the analysis of the regular set is the following non-degeneracy lemma. LMOST MINIMIZERS FOR THE THIN OBSTACLE PROBLEM 39 Lemma 9.2 (Nondegeneracy at regular points) . Let x ∈ B ′ / ∩ R ( u ) for analmost minimizer u for the Signorini problem in B . Then, for κ = 3 / , lim inf t → Z ∂B ( u φx ,t ) = lim inf t → t n +2 Z ∂B t ( x ) u > . Proof. As before, assume x = 0. In terms of the quantities defined in the proofsof Lemmas 7.1 and 7.4, we want to prove thatlim inf t → m ( t ) > . Assume, towards a contradiction, that m ( t j ) → t j → 0. Recallthat by the proof of Lemma 7.4, we have | m ( t ) − m ( s ) | ≤ Ct δ/ , < s < t < t . Now, setting s = t j → 0, we conclude that | m ( t ) | ≤ Ct δ/ , < t < t . Equivalently, we can rewrite this as Z ∂B t u ≤ Ct n +2+ δ . Next, take ˜ κ = 3 / δ/ W ˜ κ ( t, u ) = e a ˜ κ t α t n +2˜ κ − (cid:20)Z B t |∇ u | − ˜ κ − bt α t Z ∂B t u (cid:21) . Now observe that 1 t n +2˜ κ − Z ∂B t u ≤ Ct δ/ → , which readily implies that W ˜ κ (0+ , u ) ≥ . In particular, by monotonicity, W ˜ κ ( t, u ) ≥ 0, for small t > 0, which also impliesthat e N ( t, u ) ≥ ˜ κ . But then N (0+ , u ) = e N (0+ , u ) ≥ ˜ κ = 3 / δ/ (cid:3) The next result provides two important facts: a gap in possible values of Alm-gren’s frequency N (0+) as well as the classification of 3 / Proposition 9.3. If b N (0+ , u, x ) = κ < , then κ = 3 / and u φx , ( x ) = a x Re( x ′ · ν x + i | x n | ) / for some a x > , ν x ∈ ∂B ′ .Proof. Without loss of generality, we may assume x = 0. Let r j → 0+ be asequence such that u φr j → u φ in C ( R n ± ∪ R n − ). Comparing 3 / u φr ( x ) = u Ar ( x ) µ ( r ) , µ ( r ) := (cid:16) r n − R ∂B r u (cid:17) / φ ( r ) . By the optimal growth estimate (Lemma 7.4) and the nondegeneracy at regularpoints (Lemma 9.2) we have0 < lim inf r → µ ( r ) ≤ lim sup r → µ ( r ) < ∞ . Thus, we may assume that, over a subsequence of r j , µ ( r j ) → µ ∈ (0 , ∞ ), andtherefore u φr j → µ u A in C ( B ± ∪ B ′ ) , where u A is an Almgren blowup of u at x = 0. Now, since κ < κ , we canapply Proposition 6.1 to obtain that u A is a nonzero κ -homogeneous solution of theSignorini problem in B , even in x n -variable. Next, applying Lemma 6.2, we have3 / ≤ κ < κ = 3 / u A ( x ) = C n Re( x ′ · ν + i | x n | ) / for some C n > ν ∈ ∂B ′ . (The constant C n comes from the normalization R ∂B ( u A ) = 1.) Thus, u φ ( x ) = a Re( x ′ · ν + i | x n | ) / in B with a = C n µ . By the unique continuation of harmonic functions in R n ± , weobtain that the above formula for u φ propagates to all of R n . (cid:3) Proposition 9.3 has an immediate corollary. Corollary 9.4 (Almgren frequency gap) . Let u be an almost minimizer for theSignorini problem in B and x a free boundary point. Then either b N (0+ , u ) = 3 / or b N (0+ , u ) ≥ . Yet another important fact is as follows. Corollary 9.5. The regular set R ( u ) is a relatively open subset of the free boundary.Proof. For a fixed 0 < t < t , the mapping x b N ( t, u, x ) is continuous on Γ( u ).Then, by the monotonicity of b N , the mapping x b N (0+ , u, x ) is upper semicon-tinuous on Γ( u ). Moreover, by Proposition 9.3, R ( u ) = { x ∈ Γ( u ) : b N (0+ , u, x ) < } , which implies that R ( u ) is relatively open in Γ( u ). (cid:3) The combination of Proposition 9.3 and Lemma 8.4 implies the following lemma. Lemma 9.6. Let u be an almost minimizer for the Signorini problem in B , and x ∈ R ( u ) . Then there exists ρ > , depending on x such that B ′ ρ ( x ) ∩ Γ( u ) ⊂ R ( u ) and if u φ ¯ x, ( x ) = a ¯ x Re( x ′ · ν ¯ x + i | x n | ) / is the unique / -homogeneous blowupat ¯ x ∈ B ′ ρ ( x ) ∩ Γ( u ) , then | a ¯ x − a ¯ y | ≤ C | ¯ x − ¯ y | γ , | ν ¯ x − ν ¯ y | ≤ C | ¯ x − ¯ y | γ , for any ¯ x, ¯ y ∈ B ′ ρ ( x ) ∩ Γ( u ) with a constant C depending on x .Proof. The proof follows by repeating the argument in Lemma 7.5 in [GPS16]. (cid:3) Now we are ready to prove the main result on the regularity of the regular set. LMOST MINIMIZERS FOR THE THIN OBSTACLE PROBLEM 41 Theorem 9.7 ( C ,γ regularity of the regular set) . Let u be an almost minimizerfor the Signorini problem in B . Then, if x ∈ B ′ / ∩ R ( u ) , there exists ρ > ,depending on x such that, after a possible rotation of coordinate axes in R n − , onehas B ′ ρ ( x ) ∩ Γ( u ) ⊂ R ( u ) , and B ′ ρ ( x ) ∩ Γ( u ) = B ′ ρ ( x ) ∩ { x n − = g ( x , . . . , x n − ) } , for g ∈ C ,γ ( R n − ) with an exponent γ = γ ( n, α ) ∈ (0 , .Proof. The proof of the theorem is similar to that of Theorem 1.2 in [GPS16].However, we provide full details since there are technical differences. Step 1. By relative openness of R ( u ) in Γ( u ), for small ρ > B ′ ρ ( x ) ∩ Γ( u ) ⊂ R ( u ). We then claim that for any ε > 0, there is r ε > x ∈ B ′ ρ ( x ) ∩ Γ( u ), r < r ε , we have that for φ = φ / k u φ ¯ x,r − u φ ¯ x, k C ( B ± ) < ε. Assuming the contrary, there is a sequence of points ¯ x j ∈ B ′ ρ ( x ) ∩ Γ( u ) and radii r j → k u φ ¯ x j ,r j − u φ ¯ x j , k C ( B ± ) ≥ ε for some ε > 0. Taking a subsequence, if necessary, we may assume ¯ x j → ¯ x ∈ B ′ ρ ( x ) ∩ Γ( u ). Using estimates (3.1), (4.7) and Lemma 7.4, we can see that u φ ¯ x j ,r j are uniformly bounded in C ,β ( B ± ∪ B ′ ). Thus, we may assume that for some wu φ ¯ x j ,r j → w in C ( B ± ) . By arguing as in the proof of Proposition 6.1, we see that the limit w is a solutionof the Signorini problem in B . Further, by Lemma 8.3, we have k u φ ¯ x j ,r j − u φ ¯ x j , k L ( ∂B ) → . On the other hand, by Lemma 9.6, we have u φ ¯ x j , → u φ ¯ x , in C ( B ± ) , and thus w = u φ ¯ x , on ∂B . Since both w and u φ ¯ x , are solutions of the Signorini problem, they must coincidealso in B . Therefore u φ ¯ x j ,r j → u φ ¯ x , in C ( B ± ) , implying also that k u φ ¯ x j ,r j − u φ ¯ x j , k C ( B ± ) → , which contradicts our assumption. Step 2. As [GPS16], for a given ε > ν ∈ R n − define the cone C ε ( ν ) = { x ′ ∈ R n − : x ′ · ν > ε | x ′ |} . By Lemma 9.6, we may assume a ¯ x ≥ a x for ¯ x ∈ B ′ ρ ( x ) ∩ Γ( u ) by taking ρ small.For such ρ we then claim that for any ε > r ε > x ∈ B ′ ρ ( x ) ∩ Γ( u ) we have¯ x + (cid:0) C ε ( ν ¯ x ) ∩ B ′ r ε (cid:1) ⊂ { u ( · , > } . Indeed, denoting K ε ( ν ) = C ε ∩ ∂B ′ / , we have for some universal C ε > K ε ( ν ¯ x ) ⋐ { u φ ¯ x, ( · , > } ∩ B ′ and u φ ¯ x, ( · , ≥ a ¯ x C ε ≥ a x C ε on K ε ( ν ¯ x ) . Since a x C ε is independent of ¯ x , by Step 1 we can find r ε > r < r ε , u φ ¯ x,r ( · , > K ε ( ν ¯ x ) . This implies that for r < r ε , u ( · , > x + r K ε ( ν ¯ x ) = ¯ x + (cid:16) C ε ( ν ¯ x ) ∩ ∂B ′ r/ (cid:17) . Taking the union over all r < r ε , we obtain u ( · , > x + (cid:0) C ε ( ν ¯ x ) ∩ B ′ r ε (cid:1) . Step 3. We claim that for given ε > 0, there exists r ε > x ∈ B ′ ρ ( x ) ∩ Γ( u ) we have ¯ x − (cid:0) C ε ( ν ¯ x ) ∩ B ′ r ε (cid:1) ⊂ { u ( · , 0) = 0 } .Indeed, we first note that − ∂ + x n u φ ¯ x, ≥ a ¯ x C ε > (cid:16) a x (cid:17) C ε on − K ε ( ν ¯ x )for a universal constant C ε > 0. From Step 1, there exists r ε > r < r ε , − ∂ + x n u φ ¯ x,r ( · , > − K ε ( ν ¯ x ) . By arguing as in Step 2, we obtain − ∂ + x n u ( · , > x − (cid:0) C ( ν ¯ x ) ∩ B ′ r ε (cid:1) . By the complementarity condition in Lemma 4.7, we therefore conclude that¯ x − (cid:0) C ( ν ¯ x ) ∩ B ′ r ε (cid:1) ⊂ {− ∂ + x n u ( · , > } ⊂ { u ( · , 0) = 0 } . Step 4. By rotation in R n − we may assume ν x = e n − . For any ε > 0, byLemma 9.6 again, we can take ρ ε = ρ ( x , ε ), possibly smaller than ρ in the previoussteps, such that C ε ( e n − ) ∩ B ′ r ε ⊂ C ε ( ν ¯ x ) ∩ B ′ r ε for ¯ x ∈ B ′ ρ ε ( x ) ∩ Γ( u ) . By Step 2 and Step 3, for ¯ x ∈ B ′ ρ ε ( x ) ∩ Γ( u ),¯ x + (cid:0) C ε ( e n − ) ∩ B ′ r ε (cid:1) ⊂ { u ( · , > } , ¯ x − (cid:0) C ε ( e n − ) ∩ B ′ r ε (cid:1) ⊂ { u ( · , 0) = 0 } . Now, fixing ε = ε , by the standard arguments, we conclude that there exists aLipschitz function g : R n − → R with |∇ g | ≤ C n /ε such that B ′ ρ ε ( x ) ∩ { u ( · , 0) = 0 } = B ′ ρ ε ( x ) ∩ { x n − ≤ g ( x ′′ ) } ,B ′ ρ ε ( x ) ∩ { u ( · , > } = B ′ ρ ε ( x ) ∩ { x n − > g ( x ′′ ) } . Step 5. Taking ε → u ) is differentiable at x with normal ν x .Recentering at any ¯ x ∈ B ′ ρ ε ( x ) ∩ Γ( u ), we see that Γ( u ) has a normal ν ¯ x at ¯ x . ByLemma 9.6, we conclude that g in Step 4 is C ,γ . This completes the proof of thetheorem. (cid:3) LMOST MINIMIZERS FOR THE THIN OBSTACLE PROBLEM 43 Singular points In this section we study the set of so-called singular free boundary points. An im-portant technical tool to accomplish this is the logarithmic epiperimetric inequalityof [CSV17]. We use it for two purposes: to establish the optimal growth at singu-lar points as well as the rate of convergence of rescalings to blowups, ultimatelyimplying a structural theorem for the singular set. Definition 10.1 (Singular points) . Let u be an almost minimizer for the Signoriniproblem in B . We say that a free boundary point x is singular if the coincidenceset Λ( u ) = { u ( · , 0) = 0 } ⊂ B ′ has zero H n − -density at x , i.e.,lim r → H n − (Λ( u ) ∩ B ′ r ( x )) H n − ( B ′ r ( x )) = 0 . By using Almgren’s rescalings u Ax ,r , we can rewrite this condition aslim r → H n − (Λ( u Ax ,r ) ∩ B ′ ) = 0 . We denote the set of all singular points by Σ( u ) and call it the singular set .Throughout the section we will assume that κ > . We can take κ as large as we like, however, we have to remember that the constantsin b N = b N κ and W κ do depend on κ .We then have the following characterization of singular points, similar to Propo-sition 9.22 in [PSU12] for the solutions of the Signorini problem. Proposition 10.2 (Characterization of singular points) . Let u be an almost mini-mizer for the Signorini problem in B , and x ∈ B ′ / ∩ Γ( u ) be such that b N (0+ , u, x ) = κ < κ . Then the following statements are equivalent. (i) x ∈ Σ( u ) . (ii) any Almgren blowup of u at x is a nonzero polynomial from the class Q κ = { q : q is homogeneous polynomial of degree κ such that ∆ q = 0 , q ( x ′ , ≥ , q ( x ′ , x n ) = q ( x ′ , − x n ) } . (iii) κ = 2 m for some m ∈ N . Note that for κ < κ , the condition b N (0+) = κ is equivalent to N (0+) = κ . Proof. Without loss of generality we may assume x = 0. By Proposition 6.1, anyAlmgren blowup u A of u at 0 is a nonzero global solution of the Signorini problem,homogeneous of degree κ . Moreover u A is a C limit of Almgren rescalings u Ar j in B ± ∪ B ′ . Because of that, most parts of the proof of this proposition are just therepetitions of Proposition 9.22 in [PSU12]. Thus, by following Proposition 9.22 in[PSU12], we can easily see the implications (ii) ⇒ (iii), (iii) ⇒ (ii), (ii) ⇒ (i). More-over, in the proof of the remaining implication (i) ⇒ (ii), the only nontrivial part isthat any blowup u A is harmonic in B . But this comes from the complementaritycondition in Lemma 4.7. Indeed, assuming (i), we claim that ∂ + x n u A = 0 in B ′ . Otherwise, H n − (cid:0) {− ∂ + x n u A ( · , > } ∩ B ′ (cid:1) ≥ δ for some δ > 0. Then using the continuity from the below we also have that forsome ρ > H n − (cid:0) {− ∂ + x n u A ( · , > ρ } ∩ B ′ − ρ (cid:1) ≥ δ/ . Using C convergence u Ar j → u A in B ± ∪ B ′ and applying the complementaritycondition in Lemma 4.7 to rescalings u Ar j , we obtain that for small r j , H n − (cid:16) Λ( u Ar j ) ∩ B ′ (cid:17) ≥ H n − (cid:16) {− ∂ + x n u Ar j ( · , > } ∩ B ′ (cid:17) ≥ δ/ , which contradicts (i). Now recalling that u A is a solution of the Signorini problem,even in x n -variable, it satisfies∆ u A = 2( ∂ + x n u A ) H n − | Λ( u A ) = 0 in B . By homogeneity, we obtain that u A is harmonic in all of R n , and we complete theproof as in [PSU12]. (cid:3) In order to study the singular set, in view of Proposition 10.2, we need to refinethe growth estimate in Lemma 7.1 by removing the logarithmic term in the casewhen κ = 2 m < κ , m ∈ N . In the case κ = 3 / W / with the help of the epiperimetric inequality. In the case κ = 2 m we will use the so-called logarithmic epiperimetric inequality for the Weissenergy W κ ( w ) = Z B |∇ w | − κ Z ∂B w , κ = 2 m, m ∈ N that first appeared in [CSV17]. To state this result, we recall the notation A = { w ∈ W , ( B ) : w ≥ B ′ , w ( x ′ , x n ) = w ( x ′ , − x n ) } . Theorem 10.3 (Logarithmic epiperimetric inequality) . Let κ = 2 m , m ∈ N and w ∈ A be homogeneous of degree κ in B such that w ∈ W , ( ∂B ) and Z ∂B w ≤ , | W κ ( w ) | ≤ . There is constant ε = ε ( n, κ ) > and a function v ∈ A with v = w on ∂B suchthat W κ ( v ) ≤ W κ ( w )(1 − ε | W κ ( w ) | γ ) , where γ = n − n . To simplify the notations, in the results below all constants will depend on n , α , κ , κ , as well as k u k W , ( B ) , unless stated otherwise, in addition to other quantities.Thus, when we write C = C ( σ ), we mean C = C ( n, α, κ, κ , k u k W , ( B ) , σ ).The next lemma allows to apply the logarithmic epiperimetric inequality, withoutthe constraints. Lemma 10.4. Let u be an almost minimizer for the Signorini problem in B suchthat ∈ Γ( u ) and b N (0+ , u ) = κ < κ , κ = 2 m , m ∈ N . For < r < , let u r ( x ) = u ( κ ) r ( x ) = u ( rx ) r κ , w r ( x ) = | x | κ u r (cid:18) x | x | (cid:19) . LMOST MINIMIZERS FOR THE THIN OBSTACLE PROBLEM 45 Suppose that for a given ≤ σ ≤ , there is C = C ( σ ) such that Z ∂B r u ≤ C (cid:18) log 1 r (cid:19) σ r n +2 κ − . Then there is a constant ε = ε ( σ ) > and h ∈ A with h = w r on ∂B such that (i) If | W κ ( w r ) | ≥ R ∂B w r , then W κ ( h ) ≤ (1 − ε ) W κ ( w r )(ii) If | W κ ( w r ) | ≤ R ∂B w r , then W κ ( h ) ≤ W κ ( w r ) − ε (cid:18) log 1 r (cid:19) − σγ | W κ ( w r ) | γ ! , where γ = n − n . Proof. Let A = R ∂B w r + | W κ ( w r ) | . Then by Theorem 10.3 applied to w r /A / ,there is h ∈ A such that h = w r on ∂B and W κ ( h ) ≤ W κ ( w r ) (cid:0) − εA − γ | W κ ( w r ) | γ (cid:1) . If | W κ ( w r ) | ≥ R ∂B w r , then A ≤ | W κ ( w r ) | , implying W κ ( h ) ≤ W κ ( w r ) (cid:0) − ε − γ (cid:1) . If | W κ ( w r ) | ≤ R ∂B w r , then A ≤ Z ∂B w r = 3 r n +2 κ − Z ∂B r u ≤ C ( σ ) (cid:18) log 1 r (cid:19) σ . This completes the proof. (cid:3) Now we show that the logarithmic epiperimetric inequality, combined with agrowth estimate for u , implies a growth estimate on W κ ( t, u ). This is the first partof a bootstrapping argument that gradually decreases the power of log(1 /t ) in thebound for u . Lemma 10.5. Let u be an almost minimizer for the Signorini problem in B suchthat ∈ Γ( u ) and b N (0+ , u ) = κ < κ , κ = 2 m , m ∈ N . Suppose that for some ≤ σ ≤ Z ∂B r u ≤ C ( σ ) (cid:18) log 1 r (cid:19) σ r n +2 κ − , < r < r ( σ ) . Then, ≤ W κ ( t, u ) ≤ C ( σ ) (cid:18) log 1 t (cid:19) − − σγγ , < t < t ( σ ) . Proof. We first observe that W κ ( t, u ) ≥ < t < t , which follows easily fromthe condition b N (0+ , u ) = κ < κ , see the beginning of the proof of Lemma 7.1.Next, recall that in the proof of Lemma 7.3, we have used epiperimetric inequalityto show that 0 ≤ W / ( t, u ) ≤ Ct δ . This followed by obtaining a differentialinequality for W / . Thus, if for 0 < t < t , if alternative (i) holds in Lemma 10.4,i.e., W κ ( h ) ≤ (1 − ε ) W κ ( w t ), by arguing in the same way, we can show that(10.1) ddt W κ ( t, u ) ≥ ε/ t W κ ( t, u ) − Ct α/ − , for C = C ( σ ). Suppose now the alternative (ii) holds in Lemma 10.4 for some 0 < t < t . Then,following the computations in Lemma 7.3, we have ddt W κ ( t, u ) ≥ − ( n + 2 κ − − t α ) t W κ ( t, u )+ e at α (1 − bt α ) t Z ∂B ( ∂ ν u t − κu t ) + ( ∂ τ u t ) − κ ( n + κ − u t + (2 κ + n ) t α − Z ∂B u t . For w t as in the statement of Lemma 10.4, by following the computations in theproof of Theorem 5.1, we have the identity Z ∂B ( ∂ τ u t ) − κ ( n + κ − u t = ( n + 2 κ − W κ ( w t ) . This gives(10.2) ddt W κ ( t, u ) ≥ − ( n + 2 κ − − t α ) t W κ ( t, u )+ e at α (1 − bt α ) t ( n + 2 κ − W κ ( w t ) + (2 κ + n ) t α − Z ∂B u t . Let now v t be the solution of the Signorini problem in B with v t = u t = w t on ∂B . Then(10.3) (1 + t α ) W κ ( w t ) ≥ (1 + t α ) W κ ( v t ) ≥ Z B |∇ u t | − κ (1 + t α ) Z ∂B u t = W κ ( u t ) − κt α Z ∂B u t = e − at α W κ ( t, u ) − κ ( b + 1) t α Z ∂B u t . Now, if e − at α W κ ( t, u ) − κ ( b + 1) t α Z ∂B u t ≤ , then by Lemma 7.1 we have W κ ( t, u ) ≤ e at α κ ( b + 1) t α Z ∂B u t (10.4) ≤ Ct α (cid:18) log 1 t (cid:19) ≤ Ct α/ . We then proceed under the assumption e − at α W κ ( t, u ) − κ ( b + 1) t α Z ∂B u t > , which also implies W κ ( w t ) > . LMOST MINIMIZERS FOR THE THIN OBSTACLE PROBLEM 47 Now, applying Lemma 10.4, we have(10.5) W κ ( w t ) ≥ W κ ( v t ) + ε (cid:18) log 1 t (cid:19) − σγ W κ ( w t ) γ +1 ≥ 11 + t α (cid:20) e − at α W κ ( t, u ) − κ ( b + 1) t α Z ∂B u t (cid:21) + ε (cid:18) log 1 t (cid:19) − σγ (cid:18) 11 + t α (cid:19) γ +1 ×× (cid:20) e − at α W κ ( t, u ) − κ ( b + 1) t α Z ∂B u t (cid:21) γ +1 ≥ (1 − t α ) (cid:20) e − at α W κ ( t, u ) − κ ( b + 1) t α Z ∂B u t (cid:21) + ε (cid:18) log 1 t (cid:19) − σγ (1 − t α ) γ +1 ×× " (cid:0) e − at α W κ ( t, u ) (cid:1) γ +1 γ − (cid:18) κ ( b + 1) t α Z ∂B u t (cid:19) γ +1 = (1 − t α ) e − at α W κ ( t, u )+ ε γ (cid:18) log 1 t (cid:19) − σγ (1 − t α ) γ +1 e − a ( γ +1) t α W κ ( t, u ) γ +1 − (1 − t α ) κ ( b + 1) t α Z ∂B u t − ε (cid:18) log 1 t (cid:19) − σγ (1 − t α ) γ +1 κ γ +1 ( b + 1) γ +1 t α ( γ +1) (cid:18)Z ∂B u t (cid:19) γ +1 , where we used (10.3) in the second inequality and the convexity of x x γ +1 on R + in the third inequality. Now (10.2) and (10.5), together with Lemma 7.1, yield(10.6) ddt W κ ( t, u ) ≥ − C t α − W κ ( t, u )+ C t − (cid:18) log 1 t (cid:19) − σγ W κ ( t, u ) γ +1 − C t α/ − , where C i = C i ( σ ). Summarizing, we have that at every 0 < t < t ( σ ), either (10.1),(10.6), or the bound (10.4) holds. Further note that by the growth estimate inLemma 7.1, the bound (10.1) implies (10.6) for sufficiently small t and thus we mayassume that (10.6) holds for all 0 < t < t for which W κ ( t, u ) > Ct α/ .To proceed, let 0 < t < t be such that W κ ( t, u ) ≥ t α/ . Then the bound (10.6)holds and we can derive that for C = γC − σγ ) , we have ddt − W κ ( t, u ) − γ e − t α/ + C (cid:18) log 1 t (cid:19) − σγ ! = W κ ( t, u ) − γ − e − t α/ (cid:18) γ ddt W κ ( t, u ) + α W κ ( t, u ) t α/ − (cid:19) − C (1 − σγ ) t − (cid:18) log 1 t (cid:19) − σγ ≥ W κ ( t, u ) − γ e − t α/ t α/ − (cid:18) α − γC t α/ − γC t α/ W κ ( t, u ) (cid:19) + (cid:18) log 1 t (cid:19) − σγ t − (cid:16) e − t α/ γC − C (1 − σγ ) (cid:17) ≥ , < t < t = t ( σ ). Since also the function − t − γ ( α/ e − t α/ + C (cid:0) log t (cid:1) − σγ isnondecreasing for small t , denoting c W κ ( t, u ) = max { W κ ( t, u ) , t α/ } , we obtain that the function − c W κ ( t, u ) − γ e − t α/ + C (cid:18) log 1 t (cid:19) − σγ is nondecreasing on (0 , t ). Hence, − c W κ ( t, u ) − γ e − t α/ + C (cid:18) log 1 t (cid:19) − σγ ≤ − c W κ ( t , u ) − γ e − t α/ + C (cid:18) log 1 t (cid:19) − σγ ≤ C (cid:18) log 1 t (cid:19) − σγ . If 0 < t < t , then (cid:16) log t (cid:17) − σγ < (cid:0) (cid:1) − σγ (cid:0) log t (cid:1) − σγ , implying that − c W κ ( t, u ) − γ e − t α/ ≤ C (cid:16) (1 / − σγ − (cid:17) (cid:18) log 1 t (cid:19) − σγ and hence W κ ( t, u ) ≤ c W κ ( t, u ) ≤ C (cid:16) − (1 / − σγ (cid:17) − γ (cid:18) log 1 t (cid:19) − − σγγ . (cid:3) Lemma 10.6. If u is as in Lemma 10.5 with n − < σ ≤ , then there exist positive C = C ( σ ) , t = t ( σ ) such that Z ∂B t u ≤ C (cid:18) log 1 t (cid:19) σ − n − t n +2 κ − , < t < t . Proof. Going back to the proof and notations of Lemma 7.1, we have that for0 < s < t < t | m ( t ) − m ( s ) | ≤ C (cid:18) log ts (cid:19) / ( W κ ( t ) − W κ ( s )) / . Let now 0 ≤ j ≤ i be such that 2 − i +1 < t ≤ − i , 2 − j +1 < t ≤ − j . Then LMOST MINIMIZERS FOR THE THIN OBSTACLE PROBLEM 49 | m ( t ) − m ( t ) |≤ | m ( t ) − m (2 − j +1 ) | + | m (2 − i ) − m ( t ) | + i − X k = j +1 | m (2 − k ) − m (2 − k +1 ) |≤ i X k =0 C h log (cid:16) − k (cid:17) − log (cid:16) − k +1 (cid:17)i / h W κ (cid:16) − k (cid:17) − W κ (cid:16) − k +1 (cid:17)i / ≤ C i X k =0 k/ W κ (cid:16) − k (cid:17) / ≤ C i X k =0 (1 − − σγγ ) k/ ≤ C ( σ − n − ) i/ ≤ C (cid:18) log 1 t (cid:19) ( σ − n − ) . Note that in the fifth inequality we have used that 1 − − σγγ = σ − n − > 0. Thus m ( t ) ≤ m ( t ) + C (cid:18) log 1 t (cid:19) ( σ − n − ) ≤ C (cid:18) log 1 t (cid:19) ( σ − n − ) . This implies the desired result. (cid:3) Lemma 10.5 and Lemma 10.6 imply the following. Corollary 10.7 (Bootstraping) . Let u be an almost minimizer for the Signoriniproblem in B such that ∈ Γ( u ) and b N (0+ , u ) = κ < κ , κ = 2 m , m ∈ N .Suppose that for n − < σ ≤ Z ∂B t u ≤ C ( σ ) (cid:18) log 1 t (cid:19) σ t n +2 κ − , < t < t ( σ ) . Then Z ∂B t u ≤ C ′ ( σ ) (cid:18) log 1 t (cid:19) σ − n − t n +2 κ − , < t < t ′ ( σ ) . Lemma 10.8 (Optimal growth estimate at sigular points) . Let u be an almostminimizer for the Signorini problem in B such that ∈ Γ( u ) and b N (0+ , u ) = κ <κ , κ = 2 m , m ∈ N . Then, for < t < t , Z ∂B t u ≤ Ct n +2 κ − , Z B t |∇ u | ≤ Ct n +2 κ − . Proof. Starting with σ = 1 in Lemma 7.1 and repeatedly applying Corollary 10.7,we find 0 < σ ≤ min { n − , } such that Z ∂B t u ≤ C (cid:18) log 1 t (cid:19) σ t n +2 κ − , < t < t . In fact, we can make σ to be strictly less than n − by noticing that in Lemma 10.6we can replace n − by any smaller positive number. Then by Lemma 10.50 ≤ W κ ( t, u ) ≤ C (cid:18) log 1 t (cid:19) − − σγγ . Recall also that for 0 < s < t < t | m ( t ) − m ( s ) | ≤ C (cid:18) log ts (cid:19) / ( W κ ( t ) − W κ ( s )) / . We then again consider the exponentially dyadic decomposition as in the proof ofLemma 10.6. Let 0 ≤ j ≤ i be such that 2 − i +1 ≤ s/t < − i and 2 − j +1 ≤ t/t < − j . Then,(10.7) | m ( t ) − m ( s ) | ≤ C i X k = j k/ W κ (2 − k t ) / ≤ C ∞ X k = j − − σγγ ) k/ ≤ C σ − n − ) j/ ≤ C (cid:18) log 1 t (cid:19) ( σ − n − ) / . Particularly, m ( t ) ≤ m ( t ) + C (cid:18) log 1 t (cid:19) ( σ − n − ) / . This gives the first bound. The second bound is obtained from the first one byarguing as at the end of Lemma 7.1. (cid:3) Remark . The growth estimates in Lemma 10.8 enable us to consider κ -homogeneousblowups u φt j → u φ in C ( R n ± ∪ R n − ) . for t = t j → / Proposition 10.10. Let u be an almost minimizer for the Signorini problem in B such that ∈ Γ( u ) and b N (0+ , u ) = κ < κ , κ = 2 m , m ∈ N . Then there exist C > and t > such that Z ∂B | u φt − u φs | ≤ C (cid:18) log 1 t (cid:19) − − γ γ , < t < t . In particular the blowup u φ is unique.Proof. Using Lemma 10.8, we apply Lemma 10.5 with σ = 0 to obtain0 ≤ W κ ( t, u ) ≤ C (cid:18) log 1 t (cid:19) − γ . LMOST MINIMIZERS FOR THE THIN OBSTACLE PROBLEM 51 Recall now the estimate Z ∂B | u φt − u φs | ≤ C (cid:18) log ts (cid:19) / ( W κ ( t ) − W κ ( s )) / , for 0 < s < t < t , that we proved in Lemma 8.2 in the case κ = 3 / < κ < κ . Then, applying the exponentially dyadicargument as in the proof of Lemma 10.8, we obtain Z ∂B | u φt − u φs | ≤ C (cid:18) log 1 t (cid:19) − − γ γ . (cid:3) Lemma 10.11 (Nondegeneracy) . Let be a free boundary point of u such that b N (0+ , u ) = κ , κ = 2 m , m ∈ N . Then lim inf t → Z ∂B ( u φt ) = lim inf t → t n +2 κ − Z ∂B t u > . Proof. We use the approach of [CSV17, Lemma 7.2]. Assume to the contrary thatfor some r j ց 0+ lim j →∞ r n +2 κ − j Z ∂B rj u = 0 . Consider then the corresponding Almgren rescalings u Ar j ( x ). By Proposition 6.1,over a subsequence, u Ar j → q for some blowup q . By a characterization of singularpoints in Proposition 10.2, q is κ -homogeneous and is normalized by k q k L ( ∂B ) = 1.Next, for each Almgren rescaling u Ar j consider its κ -almost homogeneous rescalings[ u Ar j ] φt := u Ar j ( tx ) φ ( t ) . Since u Ar j is an almost minimizer in B /r j with gauge function ω ( t ) = ( r j t ) α , wehave N (0+ , u Ar j ) = lim s → N ( s, u Ar j ) = lim s → N ( r j s, u ) = N (0+ , u ) = κ. Thus, by Proposition 10.10, over subsequences, [ u Ar j ] φt converges to a unique blowup q r j and Z ∂B (cid:12)(cid:12)(cid:12) [ u Ar j ] φt − q r j (cid:12)(cid:12)(cid:12) ≤ C (cid:18) log 1 t (cid:19) − − γ γ , < t < t . Notice that since k u Ar j k W , ( B ) is uniformly bounded, the constant C is independentof r j , t . Now we fix r j , and consider a sequence { ρ i } ∞ i =1 = { r i /r j } ∞ i =1 . Note thatup to subsequence, [ u Ar j ] φρ i → q r j as ρ i → 0, by the uniqueness. Then Z ∂B q r j = lim ρ i → ρ n +2 κ − i Z ∂B ρi ( u Ar j ) = r n +2 κ − j R ∂B rj u lim i →∞ r j ρ i ) n +2 κ − Z ∂B rjρi u = r n +2 κ − j R ∂B rj u lim i →∞ r n +2 κ − i Z ∂B ri u = 0 by the contradiction assumption. Thus, q r j = 0 on ∂B , and hence Z ∂B (cid:12)(cid:12)(cid:12) [ u Ar j ] φt (cid:12)(cid:12)(cid:12) ≤ C (cid:18) log 1 t (cid:19) − − γ γ . Now for any ρ > r j ,1 = 1 ρ n +2 κ − Z ∂B ρ q ≤ k q k L ∞ ( ∂B ρ ) ρ κ ρ n + κ − Z ∂B ρ | q |≤ k q k L ∞ ( ∂B ) " ρ n + κ − Z ∂B ρ | q − u Ar j | + 1 ρ n + κ − Z ∂B ρ | u Ar j | ≤ k q k L ∞ ( ∂B ) ρ n + κ − C n ρ n − Z ∂B ρ | q − u Ar j | ! / + e − ( κbα ) ρ α Z ∂B (cid:12)(cid:12)(cid:12) [ u Ar j ] φρ (cid:12)(cid:12)(cid:12) ≤ C k q k L ∞ ( ∂B ) ρ n +2 κ − Z ∂B ρ | q − u Ar j | ! / + (cid:18) log 1 ρ (cid:19) − − γ γ . Note that u Ar j → q in C ( B ± ∪ B ′ ). We choose first ρ > r j = r j ( ρ ) > (cid:3) The nondegeneracy implies the following important fact, which enables the useof the Whitney Extension Theorem in the proof of the structural theorem on thesingular set (Theorem 10.13 below).For κ = 2 m < κ , m ∈ N , we denoteΣ κ ( u ) := { x ∈ Σ( u ) : N (0+ , u, x ) = κ } . Lemma 10.12. The set Σ κ ( u ) is of topological type F σ ; i.e., it is a countable unionof closed sets.Proof. For j ∈ N , j ≥ 2, let E j := n x ∈ Σ κ ( u ) ∩ B − /j : 1 j ≤ ρ n +2 κ − Z ∂B ρ ( x ) u ≤ j for 0 < ρ < j o . Then by Lemma 10.8 and Lemma 10.11, Σ κ ( u ) = S ∞ j =2 E j . We now claim that E j is closed for any j ≥ 2. Indeed, take a sequence x i ∈ E j such that x i → x as i → ∞ . Then x ∈ B − /j and for every 0 < ρ < / (2 j ), by the local uniformcontinuity of u ,(10.8) 1 ρ n +2 κ − Z ∂B ρ ( x ) u = lim i →∞ ρ n +2 κ − Z ∂B ρ ( x i ) u ∈ (cid:20) j , j (cid:21) . Next, since Γ( u ) is relatively closed in B ′ , we also know that x ∈ Γ( u ). Moreover,since N (0+ , u, x i ) = κ and the function x b N (0+ , u, x ) is upper semicontinuous,we have κ = lim sup i →∞ b N (0+ , u, x i ) ≤ b N (0+ , u, x ) . LMOST MINIMIZERS FOR THE THIN OBSTACLE PROBLEM 53 If b N (0+ , u, x ) = κ ′ > κ , then by Lemma 7.1,1 ρ n +2 κ − Z ∂B ρ ( x ) u ≤ Cρ κ ′ − κ ) (cid:18) log 1 ρ (cid:19) → ρ → , which contradicts (10.8). Therefore, b N (0+ , u, x ) = κ and consequently x ∈ E j .Hence, E j is closed, j = 2 , , . . . , implying that Σ κ ( u ) is F σ . (cid:3) To state the main result of this paper concerning the singular points, we needto introduce the following notations. For κ = 2 m < κ , m ∈ N and x ∈ Σ κ ( u ), wedefine d ( κ ) x := dim { ξ ∈ R n − : ξ · ∇ x ′ u φx ( x ′ , ≡ R n − } , which has the meaning of the dimension of Σ κ ( u ) at x , and where u φx is the unique κ -homogeneous blowup at x . In fact, d ( κ ) x is the dimension of the linear subspaceΣ κ ( u φx ) ⊂ R n − . Since u φx is a nonzero solution of the Signorini problem, it cannotvanish identically on R n − (see [GP09]) and therefore d ( κ ) x < n − d = 0 , , . . . , n − 2, we denoteΣ dκ ( u ) := { x ∈ Σ κ ( u ) : d ( κ ) x = d } . Theorem 10.13 (Structure of the singular set) . Let u be an almost minimizerfor the Signorini problem in B . Then for every κ = 2 m < κ , m ∈ N , and d = 0 , , . . . , n − , the set Σ dκ ( u ) is contained in the union of countably manysubmanifolds of dimension d and class C , log .Proof. Let κ = 2 m , m ∈ N . For x ∈ Σ κ ( u ) ∩ B ′ / , let q x ∈ Q κ denote theunique κ -homogeneous blowup of u . By the optimal growth (Lemma 10.8) and thenondegeneracy (Lemma 10.11), we can write q x = λ x q Ax , λ x > , k q Ax k L ( ∂B ) = 1 , where q Ax ∈ Q κ is the corresponding Almgren blowup. We want to show that the q x , q Ax , λ x depend continuously on x ∈ Σ κ , with a logarithmic modulus of continuity.Let x , x ∈ Σ κ ( u ) ∩ B / . Then for t > 0, to be chosen below, we can write(10.9) k q x − q x k L ( ∂B ) ≤ k q x − u φx ,t k L ( ∂B ) + k u φx ,t − u φx ,t k L ( ∂B ) + k u φx ,t − q x k L ( ∂B ) . By Proposition 10.10, we have(10.10) k q x − u φx,t k L ( ∂B ) ≤ C (cid:18) log 1 t (cid:19) − n − for x ∈ Σ κ ( u ) ∩ B ′ / . This controls the first and third term on the right hand sideof (10.9) To estimate the middle term, we observe that k u φx ,t − u φx ,t k L ( ∂B ) ≤ e ( κbα ) t α t κ Z ∂B Z |∇ u ( x + tz + r ( x − x )) | | x − x | dr dS z for any 0 < t < / 2. Recalling that ∇ u ( x ) = 0 and u ∈ C ,β ( B ± ∪ B ′ ), we have |∇ u ( x + tz + r ( x − x ) |≤ C | tz + r ( x − x ) | β ≤ C ( t + | x − x | ) β ≤ C | x − x | β κ − β ) if we choose t = | x − x | κ − β ) and have | x − x | < (1 / κ − β ) . This gives(10.11) k u φx ,t − u φx ,t k L ( ∂B ) ≤ Ct κ | x − x | β κ − β ) | x − x | ≤ C | x − x | / . Combining (10.9), (10.11), and (10.10), we obtain(10.12) k q x − q x k L ( ∂B ) ≤ C (cid:18) log 1 | x − x | (cid:19) − n − . Next, by Lemma 10.8, for any x ∈ Σ κ ( u ) ∩ B ′ / and small t Z ∂B ( u φx,t ) ≤ C with C independent of x , and passing to the limit as t → ∞ obtain the bound λ x = Z ∂B q x ≤ C Moreover, since q x is a κ -homogeneous harmonic polynomial, we also have(10.13) k q x k L ∞ ( B ) ≤ C ( n, κ ) k q x k L ( ∂B ) ≤ C. Then, by combining (10.12) and (10.13), we have(10.14) | λ x − λ x | ≤ | λ x − λ x | / ≤ (cid:18)Z ∂B | q x − q x | (cid:19) / ≤ k q x + q x k / L ∞ ( B ) k q x − q x k / L ( ∂B ) ≤ C (cid:18) log 1 | x − x | (cid:19) − n − . Finally, we want to estimate q Ax − q Ax . By writing k q x − q x k L ( ∂B ) = Z ∂B | λ x q Ax − λ x q Ax | = Z ∂B | λ x ( q Ax − q Ax ) + ( λ x − λ x ) q Ax |≥ λ x Z ∂B | q Ax − q Ax | − | λ x − λ x | Z ∂B | q Ax | , we estimate(10.15) λ x Z ∂B | q Ax − q Ax | ≤ k q x − q x k L ( ∂B ) + | λ x − λ x | Z ∂B | q Ax |≤ k q x − q x k L ( ∂B ) + C ( n ) | λ x − λ x |≤ C (cid:18) log 1 | x − x | (cid:19) − n − , where we used k q Ax k L ( ∂B ) = 1 in the second inequality and (10.12) and the bound(10.14) in the third inequality. Next, using that q Ax are κ -homogeneous harmonicpolynomials, we have k q Ax − q Ax k L ∞ ( B ) ≤ C k q Ax − q Ax k L ( ∂B ) , LMOST MINIMIZERS FOR THE THIN OBSTACLE PROBLEM 55 which combined with (10.15) gives(10.16) λ x k q Ax − q Ax k L ∞ ( B ) ≤ C (cid:18) log 1 | x − x | (cid:19) − n − . Now we fix x ∈ Σ κ ( u ) ∩ B ′ / . Then by (10.14), there exists δ = δ ( x ) ∈ (cid:16) , (1 / κ − β )+1 (cid:17) such that λ x ≥ / λ x if x ∈ Σ κ ( u ) ∩ B ′ δ ( x ). Then by (10.16),we conclude that(10.17) k q Ax − q Ax k L ∞ ( B ) ≤ C (cid:18) log 1 | x − x | (cid:19) − n − , x , x ∈ Σ κ ( u ) ∩ B δ ( x ) . Notice that the constant C does not depend on x , x , but both C and δ do dependon x .Once we have the estimates (10.14) and (10.17), as well as Lemma 10.12, wecan apply the Whitney Extension Theorem of Fefferman’s [Fef09], to complete theproof, see e.g., the proof of Theorem 5 in [CSV17]. (cid:3) Appendix A. Some examples of almost minimizers Example A.1 . If u is a minimizer of the functional Z D a ( x ) |∇ u | over the set K ψ,g ( D, M ) with strictly positive a ∈ C ,α ( D ), 0 < α ≤ 1, then u isan almost minimizer for the Signorini problem with a gauge function ω ( r ) = Cr α . Proof. This is rather immediate. (cid:3) Example A.2 . Let u be a solution of the Signorini problem for the Laplacian withdrift with the velocity field b ∈ L p ( B ), p > n : − ∆ u + b ( x ) ∇ u = 0 in B ± − ∂ x n u ≥ , u ≥ , u∂ x n u = 0 on B ′ , even in x n -variable. We understand this in the weak sense that u satisfies thevariational inequality Z B ∇ u ∇ ( w − u ) + ( b ( x ) ∇ u )( w − u ) ≥ , for any competitor w ∈ K ,u ( B , B ′ ), i.e. w ∈ u + W , ( B ) such that w ≥ B ′ in the sense of traces. Then u is an almost minimizer for the Signorini problemwith ψ = 0 on M = R n − × { } and a gauge function ω ( r ) = Cr − n/p . Proof. Let B r ( x ) ⋐ B and w ∈ K ,u ( B r ( x ) , B ′ ). Extending w as equal to u in B \ B r ( x ), and applying the variational inequality for u , we obtain(A.1) Z B r ( x ) ∇ u ∇ ( w − u ) + b ( x ) ∇ u ( w − u ) ≥ . Let v be the Signorini replacement of u on B r ( x ). Then v satisfies the variationalinequality(A.2) Z B r ( x ) ∇ v ∇ ( w − v ) ≥ , for all w as above. Now, taking w = u ± ( u − v ) + in (A.1) we will have Z B r ( x ) ∇ u ∇ ( u − v ) + + ( b ( x ) ∇ u )( u − v ) + = 0 . Next, taking w = v + ( u − v ) + in (A.2), we have Z B r ( x ) ∇ v ∇ ( u − v ) + ≥ . Taking the difference, we then obtain Z B r ( x ) |∇ ( u − v ) + | ≤ − Z B r ( x ) b ( x ) ∇ u ( u − v ) + . Similarly, taking w = v ± ( v − u ) + in (A.2) and w = u + ( v − u ) + in (A.1) andsubtracting the resulting inequalities, we obtain Z B r ( x ) |∇ ( v − u ) + | ≤ Z B r ( x ) b ( x ) ∇ u ( v − u ) + . Hence, combining the inequalities above, we arrive at Z B r ( x ) |∇ ( v − u ) | ≤ Z B r ( x ) | b ( x ) ||∇ u || v − u | . Then, applying H¨older’s inequality, we have for p > n Z B r ( x ) |∇ ( v − u ) | ≤ k b k L p ( B r ( x )) k∇ u k L ( B r ( x )) k v − u k L p ∗ ( B r ( x )) , with p ∗ = 2 p/ ( p − v − u ∈ W , ( B ), from the Sobolev’s inequalitywe have k v − u k L p ∗ ( B r ( x )) ≤ C n,p r − n/p k∇ ( v − u ) k L ( B r ( x )) and hence we can conclude that Z B r ( x ) |∇ ( v − u ) | ≤ Cr − n/p ) Z B r ( x ) |∇ u | with C = C n,p k b k L p ( B ) . This implies Z B r ( x ) |∇ u | − Z B r ( x ) |∇ v | = Z B r ( x ) ( ∇ u + ∇ v )( ∇ u − ∇ v ) ≤ Cr γ Z B r ( x ) ( |∇ u | + |∇ v | ) + Cr − γ Z B r ( x ) |∇ ( v − u ) | ≤ Cr γ Z B r ( x ) ( |∇ u | + |∇ v | ) + Cr − n/p ) − γ Z B r ( x ) |∇ u | , where we have used Young’s inequality in the second line. Choosing γ = 1 − n/p we then deduce that for small enough 0 < r < r ( n, p, k b k L p ( B ) ) Z B r ( x ) |∇ u | ≤ (1 + Cr − n/p ) Z B r ( x ) |∇ v | with C = C n,p k b k L p ( B ) . (cid:3) LMOST MINIMIZERS FOR THE THIN OBSTACLE PROBLEM 57 References [Alm76] F. J. Almgren Jr., Existence and regularity almost everywhere of solutions to ellip-tic variational problems with constraints , Mem. Amer. Math. Soc. (1976), no. 165,viii+199, doi:10.1090/memo/0165. MR0420406[Alm00] Frederick J. Almgren Jr., Almgren’s big regularity paper , World Scientific MonographSeries in Mathematics, vol. 1, World Scientific Publishing Co., Inc., River Edge, NJ,2000. Q -valued functions minimizing Dirichlet’s integral and the regularity of area-minimizing rectifiable currents up to codimension 2; With a preface by Jean E. Taylorand Vladimir Scheffer. MR1777737[Amb97] Luigi Ambrosio, Corso introduttivo alla teoria geometrica della misura ed alle superficiminime , Appunti dei Corsi Tenuti da Docenti della Scuola. [Notes of Courses Given byTeachers at the School], Scuola Normale Superiore, Pisa, 1997 (Italian). MR1736268[Anz83] Gabriele Anzellotti, On the C ,α -regularity of ω -minima of quadratic functionals ,Boll. Un. Mat. Ital. C (6) (1983), no. 1, 195–212 (English, with Italian summary).MR718371[AC04] I. Athanasopoulos and L. A. Caffarelli, Optimal regularity of lower dimensional ob-stacle problems , Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) (2004), no. Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 35 [34], 49–66,226, doi:10.1007/s10958-005-0496-1 (English, with English and Russian summaries);English transl., J. Math. Sci. (N.Y.) (2006), no. 3, 274–284. MR2120184[ACM18] Ioannis Athanasopoulos, Luis Caffarelli, and Emmanouil Milakis, On the regularity ofthe non-dynamic parabolic fractional obstacle problem , J. Differential Equations (2018), no. 6, 2614–2647, doi:10.1016/j.jde.2018.04.043. MR3804726[ACS08] I. Athanasopoulos, L. A. Caffarelli, and S. Salsa, The structure of the free boundaryfor lower dimensional obstacle problems , Amer. J. Math. (2008), no. 2, 485–498,doi:10.1353/ajm.2008.0016. MR2405165[Bom82] Enrico Bombieri, Regularity theory for almost minimal currents , Arch. Rational Mech.Anal. (1982), no. 2, 99–130, doi:10.1007/BF00250836. MR648941[BSZ17] Agnid Banerjee, Mariana Smit Vega Garcia, and Andrew K. Zeller, Higher regularityof the free boundary in the parabolic Signorini problem , Calc. Var. Partial DifferentialEquations (2017), no. 1, Art. 7, 26, doi:10.1007/s00526-016-1103-7. MR3592762[Caf79] L. A. Caffarelli, Further regularity for the Signorini problem , Comm. Partial Differen-tial Equations (1979), no. 9, 1067–1075, doi:10.1080/03605307908820119. MR542512[CRS17] Luis Caffarelli, Xavier Ros-Oton, and Joaquim Serra, Obstacle problems for integro-differential operators: regularity of solutions and free boundaries , Invent. Math. (2017), no. 3, 1155–1211, doi:10.1007/s00222-016-0703-3. MR3648978[CS07] Luis Caffarelli and Luis Silvestre, An extension problem related to the fractionalLaplacian , Comm. Partial Differential Equations (2007), no. 7-9, 1245–1260,doi:10.1080/03605300600987306. MR2354493[CSS08] Luis A. Caffarelli, Sandro Salsa, and Luis Silvestre, Regularity estimates for the solu-tion and the free boundary of the obstacle problem for the fractional Laplacian , Invent.Math. (2008), no. 2, 425–461, doi:10.1007/s00222-007-0086-6. MR2367025[CSV17] Maria Colombo, Luca Spolaor, and Bozhidar Velichkov, Direct epiperimetric inequal-ities for the thin obstacle problem and applications (September 2017), preprint, avail-able at arXiv:1709.03120.[DGPT17] Donatella Danielli, Nicola Garofalo, Arshak Petrosyan, and Tung To, Optimal regular-ity and the free boundary in the parabolic Signorini problem , Mem. Amer. Math. Soc. (2017), no. 1181, v + 103, doi:10.1090/memo/1181. MR3709717[DPP18] Donatella Danielli, Arshak Petrosyan, and Camelia Pop, Obstacle problems for nonlo-cal operators , Contemp. Math. (2018), 22 pp., to appear, available at arXiv:1709.10384.[DET17] Guy David, Max Engelstein, and Tatiana Toro, Free boundary regularity for almost-minimizers (February 2017), preprint, available at arXiv:1702.06580.[DT15] G. David and T. Toro, Regularity of almost minimizers with free bound-ary , Calc. Var. Partial Differential Equations (2015), no. 1, 455–524,doi:10.1007/s00526-014-0792-z. MR3385167 [DS16] Daniela De Silva and Ovidiu Savin, Boundary Harnack estimates in slit domains andapplications to thin free boundary problems , Rev. Mat. Iberoam. (2016), no. 3,891–912, doi:10.4171/RMI/902. MR3556055[DS18] , Thin one-phase almost minimizers (December 2018), preprint, available atarXiv:1812.03094.[DS19] , Almost minimizers of the one-phase free boundary problem (January 2019),preprint, available at arXiv:1901.02007.[dQT18] Olivaine S. de Queiroz and Leandro S. Tavares, Almost minimizers for semilinearfree boundary problems with variable coefficients , Math. Nachr. (2018), no. 10,1486–1501, doi:https://doi.org/10.1002/mana.201600103.[DEF96] A. Dolcini, L. Esposito, and N. Fusco, C ,α regularity of ω -minima , Boll. Un. Mat.Ital. A (7) (1996), no. 1, 113–125 (English, with Italian summary). MR1386250[DL76] G. Duvaut and J.-L. Lions, Inequalities in mechanics and physics , Springer-Verlag,Berlin-New York, 1976. Translated from the French by C. W. John; Grundlehren derMathematischen Wissenschaften, 219. MR0521262[DGG00] Frank Duzaar, Andreas Gastel, and Joseph F. Grotowski, Partial regularity for almostminimizers of quasi-convex integrals , SIAM J. Math. Anal. (2000), no. 3, 665–687,doi:10.1137/S0036141099374536. MR1786163[ELM04] Luca Esposito, Francesco Leonetti, and Giuseppe Mingione, Sharp regularity forfunctionals with ( p, q ) growth , J. Differential Equations (2004), no. 1, 5–55,doi:10.1016/j.jde.2003.11.007. MR2076158[EM99] L. Esposito and G. Mingione, A regularity theorem for ω -minimizers of integral func-tionals , Rend. Mat. Appl. (7) (1999), no. 1, 17–44 (English, with English and Italiansummaries). MR1710133[Fef09] Charles Fefferman, Extension of C m,ω -smooth functions by linear operators , Rev. Mat.Iberoam. (2009), no. 1, 1–48, doi:10.4171/RMI/568. MR2514337[FS16] Matteo Focardi and Emanuele Spadaro, An epiperimetric inequality for the thin ob-stacle problem , Adv. Differential Equations (2016), no. 1-2, 153–200. MR3449333[FS18a] , On the measure and the structure of the free boundary of the lower di-mensional obstacle problem , Arch. Ration. Mech. Anal. (2018), no. 1, 125–184,doi:10.1007/s00205-018-1242-4. MR3840912[FS18b] , Correction to: on the measure and the structure of the free boundary of thelower dimensional obstacle problem , Arch. Ration. Mech. Anal. (2018), no. 2,783–784, doi:10.1007/s00205-018-1273-x. MR3842059[GL86] Nicola Garofalo and Fang-Hua Lin, Monotonicity properties of variational integrals, A p weights and unique continuation , Indiana Univ. Math. J. (1986), no. 2, 245–268,doi:10.1512/iumj.1986.35.35015. MR833393[GL87] , Unique continuation for elliptic operators: a geometric-variational approach ,Comm. Pure Appl. Math. (1987), no. 3, 347–366, doi:10.1002/cpa.3160400305.MR882069[GP09] Nicola Garofalo and Arshak Petrosyan, Some new monotonicity formulas and thesingular set in the lower dimensional obstacle problem , Invent. Math. (2009),no. 2, 415–461, doi:10.1007/s00222-009-0188-4. MR2511747[GPS16] Nicola Garofalo, Arshak Petrosyan, and Mariana Smit Vega Garcia, An epiperimet-ric inequality approach to the regularity of the free boundary in the Signorini prob-lem with variable coefficients , J. Math. Pures Appl. (9) (2016), no. 6, 745–787, doi:10.1016/j.matpur.2015.11.013 (English, with English and French summaries).MR3491531[GPPS17] Nicola Garofalo, Arshak Petrosyan, Camelia A. Pop, and Mariana Smit Vega Gar-cia, Regularity of the free boundary for the obstacle problem for the fractional Lapla-cian with drift , Ann. Inst. H. Poincar´e Anal. Non Lin´eaire (2017), no. 3, 533–570,doi:10.1016/j.anihpc.2016.03.001. MR3633735[GSVG14] Nicola Garofalo and Mariana Smit Vega Garcia, New monotonicity formulas and theoptimal regularity in the Signorini problem with variable coefficients , Adv. Math. (2014), 682–750, doi:10.1016/j.aim.2014.05.021. MR3228440[GG82] Mariano Giaquinta and Enrico Giusti, On the regularity of the minima of variationalintegrals , Acta Math. (1982), 31–46, doi:10.1007/BF02392725. MR666107 LMOST MINIMIZERS FOR THE THIN OBSTACLE PROBLEM 59 [GG84] , Quasiminima , Ann. Inst. H. Poincar´e Anal. Non Lin´eaire (1984), no. 2,79–107. MR778969[Giu03] Enrico Giusti, Direct methods in the calculus of variations , World Scientific PublishingCo., Inc., River Edge, NJ, 2003. MR1962933[HL97] Qing Han and Fanghua Lin, Elliptic partial differential equations , Courant LectureNotes in Mathematics, vol. 1, New York University, Courant Institute of Mathe-matical Sciences, New York; American Mathematical Society, Providence, RI, 1997.MR1669352[JP19] Seongmin Jeon and Arshak Petrosyan, Almost minimizers for certain fractional vari-ational problems (2019), 26pp, preprint.[Kin81] David Kinderlehrer, The smoothness of the solution of the boundary obstacle problem ,J. Math. Pures Appl. (9) (1981), no. 2, 193–212. MR620584[KPS15] Herbert Koch, Arshak Petrosyan, and Wenhui Shi, Higher regularity of the freeboundary in the elliptic Signorini problem , Nonlinear Anal. (2015), 3–44,doi:10.1016/j.na.2015.01.007. MR3388870[KRS16] Herbert Koch, Angkana R¨uland, and Wenhui Shi, The variable coefficientthin obstacle problem: Carleman inequalities , Adv. Math. (2016), 820–866,doi:10.1016/j.aim.2016.06.023. MR3539391[KRS17a] , The variable coefficient thin obstacle problem: higher regularity , Adv. Differ-ential Equations (2017), no. 11-12, 793–866. MR3692912[KRS17b] , The variable coefficient thin obstacle problem: optimal regularity and regular-ity of the regular free boundary , Ann. Inst. H. Poincar´e Anal. Non Lin´eaire (2017),no. 4, 845–897, doi:10.1016/j.anihpc.2016.08.001. MR3661863[Min06] Giuseppe Mingione, Regularity of minima: an invitation to the dark side of the calculusof variations , Appl. Math. (2006), no. 4, 355–426, doi:10.1007/s10778-006-0110-3.MR2291779[PP15] Arshak Petrosyan and Camelia A. Pop, Optimal regularity of solutions to the obstacleproblem for the fractional Laplacian with drift , J. Funct. Anal. (2015), no. 2,417–472, doi:10.1016/j.jfa.2014.10.009. MR3283160[PSU12] Arshak Petrosyan, Henrik Shahgholian, and Nina Uraltseva, Regularity of free bound-aries in obstacle-type problems , Graduate Studies in Mathematics, vol. 136, AmericanMathematical Society, Providence, RI, 2012. MR2962060[PZ15] Arshak Petrosyan and Andrew Zeller, Boundedness and continuity of the time deriva-tive in the parabolic Signorini problem , Math. Res. Let. (December 2015), 8, to appear,available at arXiv:1512.09173.[RS17] Angkana R¨uland and Wenhui Shi, Optimal regularity for the thin obstacle problemwith C ,α coefficients , Calc. Var. Partial Differential Equations (2017), no. 5, Art.129, 41, doi:10.1007/s00526-017-1230-9. MR3689152[Sig59] A. Signorini, Questioni di elasticit`a non linearizzata e semilinearizzata , Rend. Mat. eAppl. (5) (1959), 95–139 (Italian). MR0118021[Sil07] Luis Silvestre, Regularity of the obstacle problem for a fractional power of the Laplaceoperator , Comm. Pure Appl. Math. (2007), no. 1, 67–112, doi:10.1002/cpa.20153.MR2270163[Ura85] N. N. Ural ′ tseva, H¨older continuity of gradients of solutions of parabolic equationswith boundary conditions of Signorini type , Dokl. Akad. Nauk SSSR (1985), no. 3,563–565 (Russian). MR775926[Wei99a] Georg Sebastian Weiss, Partial regularity for a minimum problem with free boundary ,J. Geom. Anal. (1999), no. 2, 317–326, doi:10.1007/BF02921941. MR1759450[Wei99b] Georg S. Weiss, A homogeneity improvement approach to the obstacle problem , Invent.Math. (1999), no. 1, 23–50, doi:10.1007/s002220050340. MR1714335 Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA E-mail address , S.J.: [email protected] Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA E-mail address , A.P.:, A.P.: