Almost minimizers for the thin obstacle problem with variable coefficients

We study almost minimizers for the thin obstacle problem with variable H\"older continuous coefficients and zero thin obstacle and establish their $C^{1,\beta}$ regularity on the either side of the thin space. Under an additional assumption of quasisymmetry, we establish the optimal growth of almost minimizers as well as the regularity of the regular set and a structural theorem on the singular set. The proofs are based on the generalization of Weiss- and Almgren-type monotonicity formulas for almost minimizers established earlier in the case of constant coefficients.


Introduction and Main Results
1.1. The thin obstacle (or Signorini) problem with variable coefficients. Let D be a domain in R n , n ≥ 2, and Π a smooth hypersurface (the thin space), that splits D into two subdomains D ± : D \ Π = D + ∪ D − . Let ψ : Π → R be a certain (smooth) function (the thin obstacle) and g : ∂D → R (the boundary values). Let also A(x) = (a ij (x)) be an n × n symmetric uniformly elliptic matrix, α-Hölder continuous as a function of x ∈ D, for some 0 < α < 1, with ellipticity constants 0 < λ ≤ 1 ≤ Λ < ∞: Then consider the minimizer U of the energy functional Because of the unilateral constraint on the thin space Π, the problem is known as the thin obstacle problem. Away from Π, the minimizer solves a uniformly elliptic divergence form equation with variable coefficients div(A(x)∇U ) = 0 in D + ∪ D − .
On the thin space, the minimizers satisfy U ≥ ψ, A∇U, ν + + A∇U, ν − ≥ 0, (U − ψ)( A∇U, ν + + A∇U, ν − ) = 0 on D ∩ Π, in a certain weak sense, where ν ± are the exterior normals to D ± on Π and A∇U, ν ± are understood as the limits from inside D ± . These are known as the Signorini complementarity conditions and therefore the problem is often referred to as the Signorini problem with variable coefficients (or A-Signorini problem, for short). One of the main objects of the study is the free boundary which separates the coincidence set {U = ψ} from the noncoincidence set {U > ψ} in D ∩ Π. The set Γ(U ) is also called a thin free boundary as it lives in Π and is expected to be of codimension two with respect to the domain D.
These types of problems go back to the original Signorini problem in elastostatics [Sig59], but also appear in many applications ranging from math biology (semipermeable membranes) to boundary heat control [DL76] or more recently in math finance, with connection to the obstacle problem for the fractional Laplacian, through the Caffarelli-Silvestre extension [CS07]. The presence of the free boundary makes the problem particularly challenging and while the C 1,β regularity of the minimizers (on the either side of the thin space) was known already in [Caf79,Kin81,Ura85], the study of the free boundary became possible only after the breakthrough work of [AC04] on the optimal C 1,1/2 regularity of the minimizers. Since then there has been a significant effort in the literature to understand the structure and regularity properties of the free boundary in many different settings including equations with variable coefficients, problems for the fractional Laplacian, as well as the timedependent problems, see e.g. [Sil07, ACS08, CSS08, GP09, GSVG14, KPS15, PP15,  DS16, GPS16, KRS16, BSZ17, CRS17, DGPT17, GPPS17, KRS17a, KRS17b, RS17,  ACM18, DPP18, FS18a, GPS18, PZ19, CSV20], and many others.
1.2. Almost minimizers. The approach we take in this paper is by considering the so-called almost minimizers of the functional J A,D in the sense of Anzellotti [Anz83]. For this we need a gauge function ω : (0, r 0 ) → [0, ∞), r 0 > 0, which is a nondecreasing function with ω(0+) = 0, as well as a family {E r (x 0 )} 0<r<r0 of open sets for any x 0 ∈ D, comparable to balls centered at x 0 (in what comes next, we will take it to be a family of ellipsoids).

Er(x0)
A(x 0 )∇V, ∇V , by replacing the gauge ω(r) with C(ω(r) + r α ) if necessary. An example of an almost minimizer is given in Appendix. Generally, we view almost minimizers as perturbations of minimizers in a certain sense, but in the case of variable coefficients there are even some advantages of treating minimizers themselves as almost minimizers, particularly in the sense of frozen coefficients (1.2).
Almost minimizers for the Signorini problem have already been studied in [JP19] in the case A(x) ≡ I, where their C 1,β -regularity (on the either side of the thin space) has been established and a number of technical tools such as Weiss-and Almgrentype monotonicity formulas were proved. In combination with the epiperimetric and log-epiperimetric inequalities these tools allowed to establish the optimal growth and prove the C 1,γ -regularity of the regular set and a structural theorem on the singular set. The aim of this paper is to extend these results to the variable coefficient case. It is noteworthy that the results that we obtain (see Theorems I-IV below) for almost minimizers improve even on some of the results available for the minimizers. For example, we only need the coefficients A(x) to be C 0,α with arbitrary 0 < α < 1 in order to study the free boundary, compared to W 1,p , p > n, in [KRS17b] or C 0,α , 1/2 < α < 1, in [RS17] for the regular part of the free boundary and C 0,1 in [GPS18] for the singular set.
A related notion of almost minimizers has been considered recently in [JP20] for the obstacle problem for the fractional Laplacian, with the help of the Caffarelli-Silvestre extension. While the C 1,β regularity of almost minimizers holds for the fractional orders 1/2 ≤ s < 1, the study of the free boundary still remains open.
Almost minimizers have been studied also for other free boundary problems, particularly Alt-Caffarelli-type (or Bernoulli-type) problems [DT15, DET19,DS19], their thin counterpart [DS18], as well as the variable coefficient versions [dQT18,DEST19]. We have to mention that the Signorini problem is quite different from Alt-Caffarelli-type problems, as the solutions may grow at different rates near the free boundary (such as 3/2, 2, 7/2, 4, . . . , powers of the distance), as opposed to a specific rate in Alt-Caffarelli-type problems (linear in the classical case and the square root of the distance in the thin counterpart). Therefore, it is quite important that the almost minimizing property that we impose for the Signorini problem is multiplicative, to allow the capture of all possible rates, while the almost minimizing property in the Alt-Caffarelli-type problems can be also imposed in an additive way, see [DEST19].
1.3. Main results. Since we are interested in local regularity results, we will assume that D = B 1 , the unit ball in R n , and that Π = R n−1 × {0} after a local diffeomorphism. In this paper, we will consider only the case when the thin obstacle is identically zero: ψ ≡ 0.
Further, we will assume r 0 = 1 in Definition 1.1 and take {E r (x 0 )} to be the family of ellipsoids associated with the positive symmetric matrix A(x 0 ): By the ellipticity of A(x 0 ), we have B λ 1/2 r (x 0 ) ⊂ E r (x 0 ) ⊂ B Λ 1/2 r (x 0 ).
Then we can go between almost minimizing properties (1.1) and (1.2) by changing M if necessary.
Then our first result is as follows.
Theorem I (C 1,β -regularity of almost minimizers). Let U ∈ W 1,2 (B 1 ) be an almost minimizer for the A-Signorini problem in B 1 , under the assumptions above. Then, U ∈ C 1,β loc (B ± 1 ∪ B 1 ) for β = β(α, n) ∈ (0, 1) and U C 1,β (K) ≤ C U W 1,2 (B1) , for any K B ± 1 ∪ B 1 and C = C(n, α, M, K). The proof is obtained by using Morrey and Campanato space estimates, following the original idea of Anzellotti [Anz83] that was successfully used in the constant coefficient case of our problem in [JP19]. We explicitly mention, however, that in the above theorem we do not require the even symmetry of the almost minimizer in the x n -variable, so Theorem I extends the corresponding result in [JP19] also in that respect.
To state our results related to the free boundary, we need to assume the following quasisymmetry condition. For x 0 ∈ B 1 = B 1 ∩ Π, let A(x 0 )e n ⊗ e n a nn (x 0 ) be a matrix corresponding to the reflection with respect to Π in the conormal direction A(x 0 )e n at x 0 . Note that P x0 x = x for any x ∈ Π and P x0 E r (x 0 ) = E r (x 0 ). Then, for a function U in B 1 define Note that U * x0 may not be defined in all of B 1 , but is defined in any ellipsoid E r (x 0 ) as long as it is contained in B 1 . Note also that U = U * x0 on Π. Definition 1.2 (Quasisymmetry). We say that U ∈ W 1,2 (B 1 ) is A-quasisymmetric with respect to Π, if there is a constant Q such that for any ellipsoid E r (x 0 ) B 1 centered at any x 0 ∈ B 1 . We will assume Q ≤ M throughout the paper, in addition to (1.3).
Note that when A(x) ≡ I and U is even in x n , then it is automatically quasisymmetric in the sense of the above definition. The quasisymmetry condition will also hold for even minimizers if e n is an eigenvector of A(x 0 ) for any x 0 ∈ B 1 , i.e., when This condition is typically imposed in the existing literature and can be satisfied with an application of a local C 1,α -diffeomorphism that preserves Π, see [Ura86,GSVG14,RS17]. The reason for a quasisymmetry condition is that the growth rate of the symmetrization U * x0 over the ellipsoids E r (x 0 ) captures that of U = U * x0 on the thin space Π at x 0 ∈ Γ(U ), while in the nonsymmetric case there could be a mismatch in those rates caused by the odd component of U , vanishing on Π.
More specifically, the growth rate of U on Π at x 0 ∈ Γ(U ) is determined by the following quantity which is a version of Almgren's frequency functional [Alm00] written in the geometric terms determined by A(x 0 ), where µ x0 (z) = |A −1/2 (x0)z| |A −1 (x0)z| is the conformal factor. As in the constant coefficient case, this quantity is of paramount importance for the classification of free boundary points.
Theorem II (Monotonicity of the truncated frequency). Let U be as in Theorem I and assume additionally that U is A-quasisymmetric with respect to Π. Then for any κ 0 ≥ 2, there is b = b(n, α, M, κ 0 ) such that the truncated frequency is monotone increasing for x 0 ∈ B 1/2 ∩ Γ(u), and 0 < r < r 0 (n, α, M, κ 0 ). Moreover, if we define κ(x 0 ) := N A κ0 (0+, U * x0 , x 0 ), the frequency of U at x 0 , then we have that either The monotonicity of the truncated frequency follows from that of an oneparametric family of so-called Weiss-type energy functionals {W A κ } 0<κ<κ0 , see Section 7, which also play a fundamental role in the analysis of the free boundary.
The theorem above gives the following decomposition of the free boundary The set Γ 3/2 (U ), where the frequency is minimal is known as the regular set and is also denoted R(U ).
Theorem III (Regularity of the regular set). Let U be as in Theorem II. Then R(U ) is a relatively open subset of the free boundary Γ(U ) and is an (n − 2)-dimensional manifold of class C 1,γ .
Finally, we state our main result for the so-called singular set. A free boundary point We denote the set of all singular points by Σ(U ) and call it the singular set. It can be shown that if κ(x 0 ) < κ 0 , then x 0 ∈ Σ(U ) if and only if κ(x 0 ) = 2m, m ∈ N (see Proposition 12.2). For such values of κ, we then define Theorem IV (Structure of the singular set). Let U be as in Theorem II. Then, for any κ = 2m < κ 0 , m ∈ N, Σ κ (U ) is contained in a countable union of (n − 2)dimensional manifolds of class C 1,log .
A more refined version of this result is given in Theorem 12.8. Theorems III and IV follow by establishing the uniqueness and continuous dependence of almost homogeneous blowups with Hölder modulus of continuity in the case of regular free boundary points and a logarithmic one in the case of the singular points. These follow from optimal growth and rotation estimates which are based on the use of Weiss-type monotonicity formulas in conjunction with so-called epiperimetric [GPS16] and log-epiperimetric [CSV20] inequalities for the solutions of the Signorini problem.
1.3.1. Proofs of Theorems I-IV. While we don't give formal proofs of the theorems above in the main body of the paper, they are contained in the following results proved there: • Theorem I is essentially the same as Theorem 5.2. • Theorem II follows by combining Theorem 7.2 and Corollary 11.4. • The statement of Theorem III is contained in that of Theorem 11.7.
• The statement of Theorem IV is contained in that of Theorem 12.8. 1.4. Notation. We use the following notation throughout the paper.
For x ∈ R n , r > 0, we denote We typically drop the center from the notation if it is the origin. Thus, B r := B r (0), For a function f in R n , ∇f denotes its gradient (in the classical or weak sense) where ∂ xi f are the partial derivatives in the variables x i , i = 1, 2, . . . , n.
In integrals, we often drop the variable and the measure of integration if it is with respect to the Lebesgue measure or the surface measure. Thus, where S x stands for the surface measure.
If E is a set of positive and finite Lebesgue measure, we indicate by f E the integral mean value of a function u over E. That is,

Coordinate transformations
In order to use the results available for almost minimizers in the case of A ≡ I, proved in [JP19], in this section we describe a "deskewing procedure" or coordinate transformations to straighten A(x 0 ), x 0 ∈ B 1 .
For the notational convenience, we will denote Then a x0 is a symmetric positive definite matrix, with eigenvalues between λ 1/2 and Λ 1/2 and the mapping x 0 → a x0 is α-Hölder continuous for x 0 ∈ B 1 . For every x 0 ∈ B 1 , we define an affine transformation T x0 by Then Π x0 is a hyperplane parallel to a linear subspace a −1 x0 Π spanned by the vectors a −1 x0 e 1 , a −1 x0 e 2 , . . . , a −1 x0 e n−1 and with a normal a x0 e n . Generally, this hyperplane will be tilted with respect to Π, unless a x0 e n is a multiple of e n , or equivalently that e n is an eigenvector of the matrix A(x 0 ), or that a in (x 0 ) = 0 for i = 1, . . . , n − 1 for its entries. To rectify that, we construct a family of orthogonal transformations O x0 , x 0 ∈ B 1 , by applying the Gram-Schmidt process to the ordered basis {a −1 x0 e 1 , a −1 x0 e 2 , . . . , a −1 x0 e n−1 } of a −1 x0 Π. Namely, let we obtain an ordered orthonormal basis {e x0 1 , . . . , e x0 n−1 , e x0 n } of R n . Then consider the rotation O x0 of R n that takes the standard basis {e 1 , e 2 , . . . , e n } to the one above, i.e., O x0 : R n → R n , O x0 (e i ) = e x0 i , i = 1, 2, . . . , n. Note that the Gram-Schmidt process above guarantees that x 0 → O x0 is α-Hölder continuous. We also have that by construction O −1 x0 a −1 x0 Π = Π. In particular, when x 0 ∈ Π, we have Π x0 = a −1 x0 Π and therefore O −1 x0 (Π x0 ) = Π. Because of this property, we also define the modifications of the matrices a x0 and the transformations T x0 as follows: Fig. 1.
Next, for a function U : B 1 → R and a point x 0 ∈ B 1 , we define its "deskewed" version at x 0 by u x0 = U •T −1 x0 . As we will see, if U is an almost minimizer, the transformed function u x0 will satisfy an almost minimizing property with the identity matrix I at the origin. Before we state and prove that fact, we need the following basic change of variable formulas: We also have the following modified version of (2.2).
We will also need the following estimate for the conformal factor µ x0 : Definition 2.1 (Almost Signorini property at a point). We say that a function u ∈ W 1,2 (B R ) satisfies the almost Signorini property at 0 in B R if Br |∇u| 2 ≤ (1 + ω(r)) Br |∇v| 2 , for all 0 < r < R and v ∈ K 0,u (B r , Π).
Lemma 2.2. Suppose U is an almost minimizer of the A-Signorini problem in B 1 .

Almost A-harmonic functions
We start our analysis of almost minimizers in the absence of the thin obstacle. We call such functions almost A-harmonic functions. In this section, we establish their C 1,α/2 regularity (Theorem 3.6). A similar result has already been proved by Anzellotti [Anz83], but for almost minimizers over balls {B r (x 0 )} rather than ellipsoids {E r (x 0 )}; nevertheless, the proofs are similar. The proofs in this section also illustrate how we are going to use the results available for "deskewed" functions u x0 = U •T −1 x0 to infer the corresponding results for almost minimizers U . Definition 3.1 (Almost A-harmonic functions). We say that U is an almost Aharmonic function in B 1 if U ∈ W 1,2 (B 1 ) and Er(x0) A∇U, ∇U ≤ (1 + ω(r)) Er(x0) A∇V, ∇V , whenever E r (x 0 ) B 1 and V ∈ K U (E r (x 0 )) := U + W 1,2 0 (E r (x 0 )). Note that similarly to the case of A-Signorini problem, we can write the almost minimizing property above in the form with frozen coefficients Definition 3.2 (Almost harmonic property at a point). We say that a function u ∈ W 1,2 (B R ) satisfies almost harmonic property at 0 in B R if Br |∇u| 2 ≤ (1 + ω(r)) Br |∇v| 2 , for all 0 < r < R and v ∈ K u (B r ).  . Let U be an almost A-harmonic function in B 1 . Then for any B r (x 0 ) B 1 and 0 < ρ < r, we have with C = C(n, α, M ).
Proof. Since u x0 satisfies the almost harmonic property at 0, if h is the harmonic replacement of u x0 in B r (i.e., h is harmonic in B r with h = u x0 on ∂B r ), then This is enough to repeat the arguments in [JP19, Proposition 2.3], to obtain Then, by the change of variables formulas (2.2) and (2.5), we have A(x 0 )∇U, ∇U . (3.4) To show now that (3.3)-(3.4) imply (3.1)-(3.2), we first consider the case 0 < ρ < (λ/Λ) 1/2 r.

Almost Lipschitz regularity of almost minimizers
In this section, we make the first step towards the regularity of almost minimizers for the A-Signorini problem and show that they are almost Lipschitz, i.e., C 0,σ for every 0 < σ < 1 (Theorem 4.3). The proof is based on the Morrey space embedding, similar to the case of almost A-harmonic functions, as well as the case of almost minimizers with A = I, treated in [JP19]. We want to emphasize, however, that the results on almost Lipschitz and C 1,β regularity of almost minimizers (in the next section) do not require any symmetry condition that was imposed in [JP19].
We start with an auxiliary result on the solutions of the Signorini problem.
Proof. The difference of this proposition from [JP19, Proposition 3.2] is that h(y) is not assumed to be even symmetric in y n -variable. To circumvent that, we decompose h into the sum of even and odd functions in y n , i.e., It is easy to see that h * is a solution of the Signorini problem, even in y n -variable, and h is a harmonic function, odd in y n -variable.

Proposition 4.2 (cf. [JP19, Proposition 3.3]). Let U be an almost minimizer for the
Proof. Case 1. Suppose x 0 ∈ B 1 . Note that u x0 satisfies the Signorini property at 0 in B r with r = Λ −1/2 R. If h is the Signorini replacement of u x0 in B r (that is, h solves the Signorini problem in B r with thin obstacle 0 on Π and boundary values h = u x0 on ∂B r ), then h satisfies for any v ∈ K 0,ux 0 (B r , Π), which easily follows from the standard first variation argument. Plugging in v = u x0 , we obtain Br ∇h, ∇u x0 ≥ Br |∇h| 2 .
Then it follows that where in the last inequality we have used that h is the energy minimizer of the Dirichlet integral in K 0,ux 0 (B r , Π). Then, for ρ ≤ r, we have Now, we transform back from u x0 to U as we did in Proposition 3.4 to obtain (4.3) in this case.
Case 2. Now consider the case x 0 ∈ B + 1 . If ρ ≥ r/4, then we simply have Thus, we may assume ρ < r/4. Then, let d := dist(x 0 , B 1 ) > 0 and choose Since U is almost harmonic in B + 1 , we can apply Proposition 3.4 to obtain Thus, we may assume d ≤ r. Then we note that B d (x 0 ) ⊂ B + 1 and by a limiting argument from the previous estimate, we obtain To estimate B d (x0) |∇U | 2 in the right-hand side of the above inequality, we further consider the two subcases.
Case 2.2.2. It remains to consider the case ρ < d < r/4. Using Case 1 again, we have which also implies (4.3). This concludes the proof of the proposition.
As we have seen in [JP19], Proposition 4.2 implies the almost Lipschitz regularity of almost minimizers.

C 1,β regularity of almost minimizers
In this section we prove C 1,β regularity of the almost minimizers for the A-Signorini problem (Theorem 5.2). While we take advantage of the results available for the even symmetric almost minimizers with A = I in [JP19], removing the symmetry condition requires new additional steps, combined with "deskewing" arguments to generalize to the variable coefficient case.
We start again with an auxiliary result for the solutions of the Signorini problem.
Proposition 5.1. Let h be a solution of the Signorini problem in B r , 0 < r < 1. Define ∇h := ∇h(y , y n ), y n ≥ 0 ∇h(y , −y n ), y n < 0, the even extension of ∇h from B + r to B r . Then for 0 < α < 1, there are C 1 = C 1 (n, α), C 2 = C 2 (n, α) such that for all 0 < ρ ≤ s ≤ (3/4)r, Proof. This proposition differs from [JP19, Proposition 4.4] only by not requiring h(y) to be even in the y n -variable. As in the proof of Proposition 4.1 we split h into its even and odd parts Recall that h * is still a solution of the Signorini problem in B r , but now even in y n and h is a harmonic function in B r , odd in y n . Then, by [JP19, Proposition 4.4] we have Now we need a similar estimate for h . Since h is harmonic, by the standard interior estimates, we have Thus, taking the averages on B + ρ , we will therefore have which can be rewritten as Hence, combining (5.2)-(5.5), and using that both Br (h * ) 2 and Br (h ) 2 cannot exceed Br h 2 , we obtain the claimed estimate (5.1).
Theorem 5.2. Let U be an almost minimizer of the A-Signorini problem in B 1 . Then .
Proof. Let K be a ball centered at 0. Fix a small r 0 = r 0 (n, α, M, K) > 0 to be determined later. In particular, we will ask r 1 := r Our goal is to show that for x 0 ∈ K, 0 < ρ < r < r 0 , Case 1. Suppose x 0 ∈ K ∩ B 1 . For given 0 < r < r 0 , we denote α := 1 − α 8n ∈ (0, 1), R := r 2n 2n+α . We then consider two cases: sup Let h be the Signorini replacement of u x0 in B R . If we define ∇u x0 (y , y n ) := ∇u x0 (y , y n ), y n ≥ 0 ∇u x0 (y , −y n ), y n < 0 and ∇h(y , y n ) := ∇h(y , y n ), y n ≥ 0 ∇h(y , −y n ), y n < 0, , then r < (3/4)R, thus by Proposition 5.1, the Signorini replacement h satisfies, for 0 < ρ < r, Combining the above three inequalities, we obtain Let us estimate the last term in the right-hand side of (5.10). Take δ = δ(n, α, M, K) > 0 such that δ < dist(K, ∂B 1 ) and is as in Proposition 4.2 and ε is as in Lemma 3.5. If r 0 ≤ Λ −1/2 δ 2n+α 2n , then Λ 1/2 R < δ, thus, by following the proof of Theorem 4.3 up to (4.4), we have It follows that Then by the change of variables (2.2), we have Now we can estimate the third term in the right-hand side of (5.10): To estimate the second term in the right-hand side of (5.10), we observe that Note that by (4.5), C 3 ≤ C(n, α, M, K) U W 1,2 (B1) . Thus, We now want to deduce (5.7) from (5.13). The complication here is that the mappinḡ T −1 x0 does not preserve the even symmetry with respect to the thin plane, since the conormal direction A(x 0 )e n might be different from the normal direction e n to Π at x 0 . To address this issue, by using the even symmetry of ∇u x0 , we rewrite (5.13) in terms of halfballs B + r = B r ∩ R n + (5.14) Repeating the argument that (3.4) implies (3.2) in the proof of Proposition 3.4, we have Then by the even symmetry of ∇U , (5.15) implies (5.7).
However, from U ≥ 0 on B 1 , the only possibility is If we let h again be the Signorini replacement of u x0 in B R , then the positivity of h = u x0 > 0 on ∂B R and superharmonicity of h in B R give that h > 0 in B R , and hence h is harmonic in B R . Thus, We next decompose h = h * + h in B R as in (4.2). Note that since both h and h are harmonic, h * must be harmonic as well. Then we have and similarly, Combining the above three inequalities, we have that for all 0 < ρ < r , then r ≤ R/2. By the harmonicity of both h * and h in B R , we have where the last inequality follows from (5.11). Also, note that ∇ y h = ∂ yn h * = 0 on B R/2 . Thus, for y = (y , y n ) ∈ B r , we have with C = (n, α, M, K). Hence, it follows that Combining (5.16) and (5.17), we obtain Note that (5.12) was induced in Case 1.1 without the use of the assumption sup ∂Er(x0) |U | ≤ C 3 Λ 1/2 R α , so it is also valid in this case. Finally, (5.8), (5.9), (5.12) and (5.18) give As we have seen in Case 1.1, this implies (5.7). This completes the proof of (5.7) when Case 2. The extension of (5.7) to general x 0 ∈ K follows from the combination of Case 1 and (3.5). The argument is the same as Case 2 in the proof of Theorem 4.6 in [JP19].
Thus, the estimate (5.7) holds in all possible cases.
To complete the proof of the theorem, we now apply Lemma 3.5 to the estimate (5.7) to obtain for 0 < ρ < r < r 0 Taking r r 0 = r 0 (n, α, M, K), we have with C = C(n, α, M, K). Then by the Campanato space embedding this readily implies that ∇U ∈ C 0,β (K) with , and combining with the bound in Theorem 4.3, we also deduce that we simply observe that the function U (y , −y n ) is also an almost minimizer of the Signorini problem with the appropriately modified coefficient matrix A.

Quasisymmetric almost minimizers
In the study of the free boundary in the Signorini problem, the even symmetry of the minimizer with respect to the thin space plays a crucial role. The even symmetry guarantees that the growth rate of the minimizer u over "thick" balls B r (x 0 ) ⊂ R n matches the growth rate over thin balls B r (x 0 ) ⊂ Π. This allows to use tools such as Almgren's monotonicity formula (see the next section) to classify the free boundary points. Without even symmetry, minimizers may have an odd component, vanishing on the thin space Π that may create a mismatch of growth rates on the thick and thin spaces.
In the case of minimizers of the Signorini problem (with A = I) or harmonic functions, it is easy to see that the even symmetrization is still a minimizer. Unfortunately, the even symmetrization may destroy the almost minimizing property, as well as the minimizing property with variable coefficients, as can be seen from the following simple example.
To overcome this difficulty, we need to impose the A-quasisymmetry condition on almost minimizers U , that we have already stated in Definition 1.2. In this section, we give more details on quasisymmetric almost minimizers.
Recall that for each x 0 ∈ B 1 , we defined a reflection matrix P x0 by From the ellipticity of A, we have a nn (x 0 ) ≥ λ, thus P x0 is well-defined. Note that P 2 x0 = I. Besides, P x0 Π = I Π and P x0 E r (x 0 ) = E r (x 0 ). We then define the "skewed" even/odd symmetrizations of the almost minimizer U in B 1 by Note that U * x0 and U x0 may not be defined in all of B 1 , but are defined in any ellipsoid E r (x 0 ) as long as it is contained in B 1 . Note also that U = U * x0 and U x0 = 0 on Π. Further, we note that transformed withT x0 , P x0 becomes an even reflection with respect to Π, i.e., the even/odd symmetrizations of u x0 about Π, we will have We also observe that the symmetries of u * x0 and u x0 imply the following decompositions which after a change of variables, can also be written as We now recall that by Definition 1.2, U ∈ W 1,2 (B 1 ) is called A-quasisymmetric if there is a constant Q > 0 such that (6.5) whenever E r (x 0 ) B 1 and x 0 ∈ B 1 . By the uniform ellipticity of A, (6.5) is equivalent to by changing Q to Q(Λ/λ), if necessary. Besides, using (6.4), (6.5) is also equivalent to (6.6) with some C = C(Q).
Lemma 6.2. Let U be an A-quasisymmetric almost minimizer for the A-Signorini problem in B 1 , with constant Q > 0. Then there are r 1 = r 1 (n, α, M, Q) > 0 and M 1 = M 1 (n, M, Q) > 0 such that . Remark 6.3. Since we are interested in local results, in what follows, we will assume without loss of generality that r 1 = 1 and M 1 = M .
Proof. Let V be the energy minimizer of Then v x0 = V •T −1 x0 is the energy minimizer of Br |∇v x0 | 2 on K 0,ux 0 (B r , Π).
Note that v * x0 is a solution of the Signorini problem, even in y n , with v * x0 = u * x0 on ∂B r . Similarly, v x0 is a harmonic function, odd in y n , with v x0 = u x0 on ∂B r . Thus, v * x0 is the energy minimizer of and so V * x0 is the energy minimizer of Thus, to show (6.7), it is enough to show To this end, we first observe that the quasisymmetry of U implies the quasisymmetry of u x0 : Using this, together with the symmetry of u * x0 , u x0 , v * x0 and v x0 , we have Therefore, for 0 < r < r 1 = (2CM ) −1/α , as desired.
We complete this section with a version of Signorini's complementarity condition that will play an important role in the analysis of the free boundary.
Proof. Let y 0 ∈ B R0 be such that u * x0 (y 0 ) > 0. Then we need to show that ∂ + yn u * x0 (y 0 ) = 0. Since u x0 = u * x0 on Π, we have u x0 (y 0 ) > 0 and by continuity u x0 > 0 in a small ball B δ (y 0 ). Then U > 0 in Ω =T −1 x0 (B δ (y 0 )). We claim now that U is almost A-harmonic in Ω. Indeed, if E r (y) Ω (not necessarily with y ∈ B 1 ) and V is A(y)-harmonic replacement of U on E r (y) (i.e. div(A(y)∇V ) = 0 in E r (y) with V = U on ∂E r (y)), then since V = U > 0 on ∂E r (y), by the minimum principle V > 0 on E r (y). This means that V ∈ K 0,U (E r (y), Π) and therefore we must have which also implies that U is an almost A-harmonic function in Ω. Hence, U ∈ C 1,α/2 (Ω) by Theorem 3.6, implying also that u x0 ∈ C 1,α/2 (B δ (y 0 )). Consequently, also u * x0 ∈ C 1,α/2 (B δ (y 0 )) and by even symmetry in the y n -variable, we therefore conclude that ∂ + yn u * x0 (y 0 ) = 0. The second part of the lemma now follows by the C 1,β regularity and the complementarity condition.

Weiss-and Almgren-type monotonicity formulas
In this section we introduce two technical tools: Weiss-and Almgren-type monotonicity formulas, that will play a fundamental role in the analysis of the free boundary. In fact, the proofs of these formulas follow immediately from the case A ≡ I, following the deskewing procedure.
To proceed, we fix a constant κ 0 > 0. We can take it as large as we want, however, some constants in what follows, will depend on κ 0 . Then for 0 < κ < κ 0 , we consider the Weiss-type energy functional introduced in [JP19]: (The formula in [JP19] corresponds to the case M = 1.) Based on that, we define an appropriate version of Weiss's functional for our problem. For a function V in E r (x 0 ), let for 0 < t < r, with a, b same as above, where the weight µ x0 is as in (2.4). Note that by the change of variables formulas (2.1)-(2.3), we have . Let now U be an A-quasisymmetric almost minimizer for the A-Signorini problem in B 1 and x 0 ∈ B 1/2 . By Lemma 6.2, U * x0 satisfies the almost A-Signorini property at x 0 in E (1/2)Λ −1/2 (x 0 ). Thus u * x0 also satisfies the almost Signorini property at 0 in B (1/2)Λ −1/2 . By using this observation, we then have the following Weiss-type monotonicity formulas for U * x0 and u * x0 . Theorem 7.1 (Weiss-type monotonicity formula). Let U be an A-quasisymmetric almost minimizer for the A-Signorini problem in B 1 . Suppose x 0 ∈ B 1/2 and U (x 0 ) = 0. Let 0 < κ < κ 0 with a fixed κ 0 > 0. Then, for 0 < t < t 0 = t 0 (n, α, κ 0 , M ), In particular, W κ (t, u * x0 , 0) and W A κ (t, U * x0 , x 0 ) are nondecreasing in t for 0 < t < t 0 . Proof. We note that the proof of [JP19, Theorem 5.1] for the monotonicity of W κ (t, v, x 0 ) requires the function v to be an almost minimizer for the Signorini problem with v(x 0 ) = 0 for the monotonicity of its energy. However, it is not hard to see that the almost minimizing property of v is used only when it is compared with the κ-homogeneous replacement w of v on balls centered at the given point x 0 to obtain Bt(x0) [JP19]. This means that the argument in the proof of [JP19, Theorem 5.1] also works in our case as long as u * x0 (0) = U (x 0 ) = 0 and implies the part of the theorem for u * x0 . We note that the constants a κ and b in our case will have an additional factor of M , as we work with ω(r) = M r α rather than ω(r) = r α in our case, but this change of the constants can be easily traced.
The part of the theorem for U * x0 follows by a change of variables.
The families of monotonicity formulas {W κ } 0<κ<κ0 and {W A κ } 0<κ<κ0 have an important feature that their intervals of monotonicity and the constant b can be taken the same for all 0 < κ < κ 0 . Because of that, their monotonicity indirectly implies that of another important quantity that we describe below. Namely, recall that for a function v in B r (x 0 ), Almgren's frequency of v at x 0 is defined as Note that this quantity is well-defined when v has an almost Signorini property at x 0 and x 0 ∈ Γ(v), since vanishing of ∂Bt(x0) v 2 for any t > 0, would imply vanishing of v in B t (x 0 ) by taking 0 as a competitor and consequently that x 0 / ∈ Γ(v). Next consider a modification of N , which we call the truncated frequency: where b is as in Weiss-type monotonicity formulas for κ < κ 0 . We next define the appropriate version of N , N κ0 in our setting. For a function V in E r (x 0 ), we define As observed in [JP19, Theorem 5.4], the Weiss-type monotonicity formula implies the following monotonicity of N A κ0 . Theorem 7.2 (Almgren-type monotonicity formula). Let U , κ 0 , and t 0 be as in Theorem 7.1, and x 0 ∈ B 1/2 a free boundary point. Then

Definition 7.3 (Almgren's frequency at free boundary point). For an
Remark 7.4. Note that even though the monotonicity of the truncated frequency is stated in Theorem 7.2 only for x 0 ∈ B 1/2 ∩ Γ(U ), by a simple recentering and a scaling argument, it will be monotone also at all x 0 ∈ Γ(U ), but for a possibly shorter interval of values 0 < t < t 0 (x 0 ) depending on x 0 . Thus, κ(x 0 ) exists at all x 0 ∈ Γ(U ).

Almgren rescalings and blowups
Our analysis of the free boundary is based on the analysis of blowups, which are the limits of rescalings of the solutions at free boundary points. In Signorini problem, there are a few types of rescalings that use different normalizations. In this section, we look at so-called Almgren rescalings and blowups that play well with the Almgren frequency formula.
Let V ∈ W 1,2 (B 1 ) and x 0 ∈ B 1/2 be a free boundary point. For small r > 0 define the Almgren rescaling of V at x 0 by The Almgren rescalings have the following normalization and scaling properties Here N A(x0) denotes Almgren's frequency for a constant matrix A(x 0 ). Thus, we also have N A (r, V, x 0 ) = N A(x0) (r, V, x 0 ). Note that when A = I, then is same as the Almgren rescaling in [JP19], and satisfies V I x0,r L 2 (∂B1) = 1 N (ρ, V I x0,r , 0) = N (ρr, V, x 0 ). We will call the limits of V A x0,r over any subsequence r = r j → 0+ Almgren blowups of V at x 0 and denote them by V A x0,0 . By using a change of variables, we can express Almgren rescalings of V in terms of those of v x0 = V •T −1 x0 and vice versa. Namely, we have (v x0 ) I r (y) = (det a x0 ) 1/2 V A x0,r (ā x0 y), wherever they are defined. Applied to the particular case V = U * x0 , we have (u * x0 ) I r (y) = (det a x0 ) 1/2 (U * x0 ) A x0,r (ā x0 y). Proposition 8.1 (Existence of Almgren blowups). Let U be an A-quasisymmetric almost minimizer for the A-Signorini problem in B 1 , and x 0 ∈ B 1/2 ∩ Γ(U ) be such that κ(x 0 ) < κ 0 . Then, every sequence of Almgren rescalings (U * x0 ) A x0,tj , with t j → 0+, contains a subsequence, sill denoted t j such that for a function in C 1 loc (a x0 (B ± 1 ∪ B 1 )). Moreover, (U * x0 ) A x0,0 extends to a nonzero solution of the A(x 0 )-Signorini problem in R n , (U * x0 ) A x0,0 (x) = (U * x0 ) A x0,0 (P x0 x), and it is homogeneous of degree κ(x 0 ) in R n . Similarly, every sequence of Almgren rescalings (u * x0 ) I tj , with t j → 0+ contains a subsequence, sill denoted t j such that for a function (u * x0 . Moreover, (u * x0 ) I 0 extends to a nonzero solution of the Signorini problem in R n , even in y n , and it is homogeneous of degree κ(x 0 ) in R n . Proof.
Step 2. For 0 < t < 1 and x ∈ B 1/(2t) (x 0 ), let Then by a simple scaling argument, we have that U x0,t is an almost minimizer of the A x0,t -Signorini problem in B 1/(2t) (x 0 ) with a gauge function µ t (r) = (tr) α ≤ r α . In particular, for any R > 0, we will have that U x0, . Then, arguing as in the proof of Theorem 6.5, by using the quasisymmetry of U , we obtain that Next, observing that (u * x0 ) I t is a positive constant multiple of (U x0,t ) * Taking R = 1, combined with the boundedness of (u * x0 ) I t in W 1,2 (B 1 ) for small t > 0, it follows that up to a subsequence, Step 3. Next, we claim that the blowup (u * x0 ) I 0 is a solution of the Signorini problem in B 1 . Indeed, fix 0 < R < 1, and for each t j let h tj be the Signorini replacement of (u * x0 ) I tj in B R . Then a first variation argument gives (see [JP19,(3.2)]) Since (u * x0 ) I tj has an almost Signorini property at 0 with a gauge function r → C(t j r) α , it follows that This implies that h tj → (u * x0 ) I 0 weakly in W 1,2 (B R ). On the other hand, by the boundedness of the sequence h tj in W 1,2 (B R ), we have also boundedness in C 1,1/2 norm locally in (B ± R ∪ B R ) and hence, over a subsequence, . By this convergence, we then conclude that (u * x0 hence, by letting R → 1, (u * x0 ) I 0 itself solves the Signorini problem in B 1 .
Step 4. Recall now that the blowup (u * x0 ) I 0 is nonzero in B 1 . In particular, ∂Br ((u * x0 ) I 0 ) 2 > 0 for any 0 < r < 1, otherwise we would have that (u * x0 ) I 0 is identically zero on ∂B r and consequently also on B r . Using this fact, combined with C 1 loc convergence in B ± 1 ∪ B 1 , we have that for any 0 < r < 1 Thus, Almgren's frequency of (u * x0 ) I 0 is constant κ(x 0 ) on 0 < r < 1 which is possible only if (u * x0 ) I 0 is a κ(x 0 )-homogeneous solution of the Signorini problem in B 1 , see [PSU12,Theorem 9.4]. Finally, by using the homogeneity, we readily extend (u * x0 ) I 0 to a solution of the Signorini problem in all of R n . This completes the proof for (u * x0 ) I 0 . The corresponding result for (U * x0 ) A x0,tj follows now by a change of variables. With Proposition 8.1 at hand, we can repeat the argument in the proof of Lemma 6.2 in [JP19] with u * x0 to obtain the following, which is possible since u * x0 satisfies the complementarity condition and an Almgren-type monotonicity formula with a blowup as a nonzero solution of the Signorini problem.
Lemma 8.2 (Minimal frequency). Let U be an A-quasisymmetric almost minimizer for the A-Signorini problem in B 1 . If Consequently, we also have

Growth estimates
The first result in this section (Lemma 9.1) provides growth estimates for the quasisymmetric almost minimizers near free boundary points x 0 with κ(x 0 ) ≥ κ. Such estimates were obtained in [JP19, Lemmas 7.1] in the case A ≡ I as a consequence of Weiss-type monotonicity formulas. However, they contain an unwanted logarithmic term that creates difficulties in the blowup analysis of the problem.
The results in this section are stated in terms of both u * x0 and U * x0 , as we need both forms in the subsequent arguments. We note that the estimates for u * x0 follow directly from [JP19, Lemmas 7.1, 7.3, 7.4] and the ones for U * x0 are obtained by using the deskewing procedure and therefore we skip all proofs in this section.
In the estimates below, as well in the rest of the paper, we use the notation which is the radius of the largest ball B R0 , where u * x0 is guaranteed to exists for any x 0 ∈ B 1/2 for an almost minimizer U in B 1 .

3/2-almost homogeneous rescalings and blowups
In this section we study another kind of rescalings and blowups that will play a fundamental role in the analysis of regular free boundary points where κ(x 0 ) = 3/2 (see the next section), namely 3/2-almost homogeneous blowups. The main result that we prove in this section is the uniqueness and Hölder continuous dependence of such blowups at a free boundary point x 0 (Lemma 10.3).
For a function v in B 1 and x 0 ∈ B 1/2 , we define the 3/2-almost homogeneous with b as in the Weiss-type monotonicity formulas W A 3/2 and W 3/2 . When x 0 = 0, we simply write v φ 0,t = v φ t . The name is explained by the fact that lim t→0 φ(t) t 3/2 = 1, and the reason to look at such rescalings instead of 3/2-homogeneous rescalings (that would correspond to φ(t) = t 3/2 ) is how they play well with the Weiss-type monotonicity formulas W A 3/2 and W 3/2 . Now, if U is an A-quasisymmetric almost minimizer and x 0 ∈ B 1/2 ∩ Γ(U ), then for any fixed R > 1, if t = t j > 0 is small, then by Lemma 9.3, with C = C(n, α, M ), R 0 = (1/2)Λ −1/2 . Hence, (u * x0 ) φ tj is a bounded sequence in W 1,2 (B R ). Next, arguing as in the proof of Proposition 8.1, we will have that with C = C(n, α, M, R, K) for K B R . Thus, by letting R → ∞ and using Cantor's diagonal argument, we can conclude that over a subsequence t = t j → 0+, We call such (u * x0 ) φ 0 a 3/2-homogeneous blowup of u * x0 at 0. (We may skip the "almost" modifier here as the limit is the same as for 3/2-homogeneous rescalings.) Furthermore, from the relation (u * x0 ) φ t (y) = (U * x0 ) φ x0,t (ā x0 y), we also conclude that for any sequence t j → 0+, there is a subsequence, still denoted by t j , such that Apriori, the blowups (u * x0 ) φ 0 and (U * x0 ) φ x0,0 may depend on the sequence t j → 0+. However, this does not happen in the case of 3/2-homogeneous blowups. We start with what we call a rotation estimate for rescalings.
Lemma 10.1 (Rotation estimate). Let U be an A-quasisymmetric almost minimizer for the A-Signorini problem in B 1 , x 0 ∈ B 1/2 a free boundary point, and δ as in Lemma 9.2. Then, for s < t < t 0 = t 0 (n, α, M ) and C = C(n, α, M ).
Proof. This is an analogue of Lemma 8.2 in [JP19], which follows from the computation done in the proof of [JP19, Lemma 7.1], the growth estimate for W 3/2 in [JP19, Lemma 7.3] and a dyadic argument. The analogues of those results in our case are stated in Lemma 9.1 and 9.2. This proves the lemma for u * x0 . The estimate for (U * x0 ) φ x0,t then follows from the equality The uniqueness of 3/2-homogeneous blowup now follows.
Proof. If (u * x0 ) φ 0 is the limit of (u * x0 ) φ tj for t j → 0, then the first part of the lemma follows immediately from Lemma 10.1, by taking s = t j → 0 and passing to the limit.
To see the uniqueness of blowups, we observe that (u * x0 ) φ 0 is a solution of the Signorini problem in B 1 , by arguing as in the proof of Proposition 8.1 for Almgren blowups. Now, if v 0 is another blowup, over a possibly different sequence t j → 0, then passing to the limit in the first part of the lemma we will have implying that both v 0 and (u * x0 ) φ 0 are solutions of the Signorini problem in B 1 with the same boundary values on ∂B 1 . By the uniqueness of such solutions, we have v 0 = (u * x0 ) φ 0 in B 1 . The equality propagates to all of R n by the unique continuation of harmonic functions in R n ± . This completes the proof for u * x0 . An analogous argument holds for U * x0 using the equalities ,0 (ā x0 y), y ∈ R n . The rotation estimate for rescalings implies not only the uniqueness of blowups and the convergence rate to blowups, but also the continuous dependence of blowups on a free boundary point.
Next note that we can incorporate the weight µ x0 /det a x0 with µ x0 as in (2.4) in the integral on the left hand side of (10.4) because of the bounds Then, by using Lemma 10.2, we have ≤ 2Cr δ/2 + I r + II r + III r ≤ Cd τ δ/2 + I r + II r + III r , Step 2. By the definition of the almost homogeneous rescalings, we have This gives |∇U * x0 |dzds.
Notice that the last integral is taken over where the third inequality follows from Lemma 9.3.
Step 6. To prove (10.5), we first observe that from (10.4), On the other hand, where the last inequality follows from (10.8). (It is easy to see that we can enlarge the domain in (10.8).) Therefore, combining the preceding two estimates, we conclude that Step 7. Finally, (10.5) implies (10.6), by arguing precisely as in [GPS16, Proposition 7.4].

Regularity of the regular set
In this section we combine the uniqueness and Hölder continuous dependence of 3/2-homogeneous blowups of the symmetrized almost minimizers (U * x0 ) φ x0,0 (Lemma 10.3) with a classification of such blowups at so-called regular points (Proposition 11.3) to prove one of the main results of this paper, the C 1,γ regularity of the regular set (Theorem 11.7). While some arguments follow directly from those in the case A ≡ I by a coordinate transformationT x0 , the dependence of these transformations on x 0 creates an additional difficulty. We start by defining the regular set.
Definition 11.1 (Regular points). For an A-quasisymmetric almost minimizer U for the A-Signorini problem in B 1 , we say that a free boundary point x 0 of U is regular if κ(x 0 ) = 3/2. We denote the set of all regular points of U by R(U ) and call it the regular set.
The proofs of the following two results (Lemma 11.2 and Proposition 11.3) are established precisely as in [JP19, Lemma 9.2, Proposition 9.3] for the transformed functions u * x0 . The equivalent statements for U * x0 are obtained by changing back to the original variables.
Corollary 11.5. The regular set R(U ) is a relatively open subset of the free boundary.
The combination of Proposition 11.3 and Lemma 10.3 implies the following lemma.
Lemma 11.6. Let U and x 0 be as in Lemma 11.2. Then there exists ρ > 0, Proof. The proof follows by repeating the argument in Lemma 7.5 in [GPS16] with Now we are ready to prove the main result on the regularity of the regular set.
Proof. The proof of the theorem is similar to those of in [JP19, Theorem 9.7] and [GPS16, Theorem 1.2]. However, we provide full details since there are technical differences.
Step 1. By relative openness of R(U ) in Γ(U ), for small ρ > 0 we have B 2ρ (x 0 ) ∩ Γ(U ) ⊂ R(U ). We then claim that for any ε > 0, there is r ε > 0 such that for x ∈ B ρ (x 0 ) ∩ Γ(U ), r < r ε , we have that Assuming the contrary, there is a sequence of pointsx j ∈ B ρ (x 0 ) ∩ Γ(U ) and radii Taking a subsequence if necessary, we may assumex j →x 0 ∈ B ρ (x 0 )∩Γ(U ). Using estimates (10.1)-(10.3), we can see that ∇(u * xj ) φ rj are uniformly bounded in C 0,β (B ± 2 ∪ B 2 ). Since (u * xj ) φ rj (0) = 0, we also have that (u * xj ) φ rj is uniformly bounded in C 1,β (B ± 2 ∪ B 2 ). Thus, we may assume that for some w (u * xj ) φ rj → w in C 1 (B ± 1 ). By arguing as in the proof of Proposition 8.1, we see that the limit w is a solution of the Signorini problem in B 1 . Further, by Lemma 10.2, we have On the other hand, by Lemma 11.6, we have Since both w and (u * x0 ) φ 0 are solutions of the Signorini problem, they must coincide also in B 1 . Therefore 1 ) → 0, which contradicts our assumption.

Singular points
In this section we study another type of free boundary points for almost minimizers, the so-called singular set Σ(U ). Because of the machinery developed in the earlier sections, we are able to prove a stratification type result for Σ(U ) (Theorem 12.8), following a similar approach for the minimizers and almost minimizers with A = I.
Definition 12.1 (Singular points). Let U be an A-quasisymmetric almost minimizer for the A-Signorini problem in B 1 . We say that a free boundary point x 0 is singular if the coincidence set Λ(U ) = {U (·, 0) = 0} ⊂ B 1 has zero H n−1 -density at x 0 , i.e., We denote the set of all singular points by Σ(U ) and call it the singular set.
Recall now that for x 0 ∈ Γ(u), u x0 (y) = U (ā x0 y + x 0 ) and note thatā x0 B r + x 0 = E r (x 0 ). Thus, . Now, by (12.2) and (12.1), In terms of Almgren rescalings (u * x0 ) I r , we can rewrite the condition above as lim We then have the following characterization of singular points.
Proposition 12.2 (Characterization of singular points). Let U be an A-quasisymmetric almost minimizer for the A-Signorini problem in B 1 , and x 0 ∈ B 1/2 ∩ Γ(U ) be such that κ(x 0 ) = κ < κ 0 . Then the following statements are equivalent.
(iii) any Almgren blowup (U * x0 ) A x0,0 of U * x0 at x 0 is a nonzero polynomial from the class Proof. This is the analogue of [JP19, Proposition 10.2] in the case A ≡ I.
Clearly, (ii) and (iii) are equivalent. By Proposition 8.1, any Almgren blowup (u * x0 ) I 0 of u * x0 at 0 is a nonzero global solution of the Signorini problem, homogeneous of degree κ. Moreover, (u * x0 ) I 0 is a C 1 loc limit of Almgren rescalings (u * x0 ) I tj in R n ± ∪ R n−1 . Since u * x0 also satisfies the complementarity condition in Lemma 6.6, the equivalence among (i), (ii) and (iv) follows by repeating the arguments in [JP19, Proposition 10.2].
In order to proceed with the blowup analysis at singular points, we need to remove the logarithmic term from the growth estimates in Lemma 9.1. This was achieved in [JP19, Lemma 10.8] in the case A ≡ I by using a bootstrapping argument [JP19, Lemmas 10.4-10.6, Corollary 10.7], based on the log-epiperimetric inequality of [CSV20]. All the arguments above work directly for u * x0 (and then for U * x0 , by deskewing) and we obtain the following optimal growth estimate.
The growth estimates in Lemma 12.3 enable us to consider κ-homogeneous blowups , for t = t j → 0+, similar to 3/2-homogeneous blowups in Section 10.
Furthermore, the arguments in [JP19, Proposition 10.10] also go through for u * x0 (and then for U * x0 , by deskewing), and we obtain the following rotation estimate for almost homogeneous rescalings.
Proposition 12.4 (Rotation estimate). For U and x 0 as in Lemma 12.3, there exist C > 0 and t 0 > 0 such that for 0 < s < t < t 0 . In particular, the blowups (u * x0 ) φ 0 and (U * x0 ) φ x0,0 are unique. We next show that the rotation estimate as above holds uniformly for u * x0 replaced with its Almgren rescalings (u * x0 ) I r , 0 < r < 1. (Note that the objects (u * x0 ) I r φ t in the proposition below are κ-almost homogeneous rescalings of Almgren rescalings.) Proposition 12.5. For U and x 0 as in Lemma 12.3 and 0 < r < 1, there are C > 0 and t 0 > 0, independent of r such that for 0 < s < t < t 0 . In particular, the κ-homogeneous blowup (u * x0 ) I r φ 0 is unique. Proof. We first observe that since u * x0 has the almost Signorini property at 0, (u * x0 ) I r also has the almost Signorini property at 0. This implies that W κ (ρ, (u * x0 ) I r , 0) and N κ0 (ρ, (u * x0 ) I r , 0) are monotone nondecreasing on ρ. Thus .
Those are all we need to proceed all the arguments with (u * x0 ) I r as in Lemmas 10.4-10.6, Corollary 10.7, Lemma 10.8, and Proposition 10.10 in [JP19]. This completes the proof.
Once we have Proposition 12.5, we can argue as in [JP19, Lemma 10.11] to obtain the nondegeneracy for u * x0 , and also for U * x0 .
Note that the last equality follows from the implication (iv) ⇒ (i) in Proposition 12.2.
Lemma 12.7. The set Σ κ (U ) is of topological type F σ ; i.e., it is a countable union of closed sets.
Proof. For j ∈ N, j ≥ 2, let F j := x 0 ∈ Σ κ (U ) ∩ B 1−1/j : Note that if x j → x 0 , then by the local uniform continuity of U and A, Using this, together with Lemma 12.3, Lemma 12.6 and Lemma 9.1, we can argue as in [JP19, Lemma 10.12] to prove that Σ κ (U ) = ∪ ∞ j=2 F j and each F j is closed.