A new glance to the Alt-Caffarelli-Friedman monotonicity formula

In this paper we revisit the proof of the Alt-Caffarelli-Friedman monotonicity formula. Then, in the framework of the Heisenberg group, we discuss the existence of an analogous monotonicity formula introducing a necessary condition for its existence, recently proved in \cite{FeFo}.


Introduction
The Alt-Caffarelli-Friedman monotonicity formula was introduced in [1] as a fundamental tool for studying the main properties of the solutions of two-phase free boundary problems. Roughly saying, following [1], the result says that there exists r 0 > 0 such that for every non-negative u 1 , u 2 ∈ C(B 1 (0)) ∩ H 1 (B 1 (0)), if 0 ∈ F(u i ), ∆u i ≥ 0, i = 1, 2, u 1 (0) = u 2 (0) = 0 and u 1 u 2 = 0 in B 1 (0), where B 1 (0) is the Euclidean ball centered at 0 of radius 1 in R n , then in Ω − (u) := Int({x ∈ Ω : u(x) ≤ 0}), |∇u + | 2 − |∇u − | 2 = 1 on F(u) := ∂Ω + (u) ∩ Ω, see [1]. Thus, solutions of (3) satisfy, at least in a "weak" sense, the following property: for every P ∈ F(u) (u + ν (P )) 2 (u − ν (P )) 2 = lim r→0 + Φ(r) ≤ C, Date: January 20, 2020. F.F. and N.F. are partially supported by INDAM-GNAMPA-2019 projet: Proprietà di regolarità delle soluzioni viscose con applicazioni a problemi di frontiera libera. 1 where u + := sup{u, 0}, u − := sup{−u, 0}, ν is the unit vector, pointing inside Ω + (u) at P ∈ F(u) and inside Ω − (u) at P ∈ F(u) when this makes sense in the smooth case. See [10] for a more general viscosity meaning. Hence, if one of the two phases, let say u − , is sufficiently regular at P ∈ F(u), see [28], then by Hopf maximum principle it results u − ν (P ) > 0 so that, as a by-product, u + ν (P ) has to be bounded. In this way, the solutions of the free boundary problem are globally Lipschitz. After [1] many other important papers on this topic appeared. We remind some of them, without pretending of citing all the literature about this topic. In [8] it was proved that monotonicity formula holds for linear uniformly elliptic operators in divergence form with Hölder continuous coefficients, in [9] a formula for non-homogeneous free boundary problems was discovered, in [40] the Riemannian case was treated, while in [34] the non-divergence form case has been faced. Some very partial results have been obtained also in the nonlinear case in lower dimension: see [16] for the p−Laplace case. Moreover, this formula became popular and popular for other applications as well. Among them, there are further two-phase problems, see [7] for the elliptic homogeneous case, [2] and [20] for the parabolic homogeneous setting, and [15] for the elliptic linear non-homogeneous problems. In addition we also recall some segregation problems, see for instance: [35], [36], [39] and [38]. In this way, during the last decade, the Alt-Caffarelli-Friedman monotonicity formula has quickly increased its importance in literature.
The existence of such a tool for elliptic degenerate operators, for instance sublaplacians on groups, as far as we know, has not yet been understood. Anyhow, concerning other similar formulas about sublaplacians we find in literature some important contributions, see [27] and in particular [29], where the authors deal with the frequency function of Almgren in Carnot groups. Moreover, see [14] and [13] for further papers in non-commutative setting dealing with other free boundary problems, namely the obstacle problem. We also gently warn the reader about the existence of results about two-phase problems in the Heisenberg group, like [21] and [17] in particular, where the following parallel version of the Euler equations (3), of a two-phase problem in this non-commutative framework, has been achieved: In the Section 3 of this paper, we shall introduce the main notation that we need for working on this subject. Nevertheless, we ask to the reader that is not customary with this language to continue to follow this colloquial presentation having in mind that, in the Heisenberg group, there exists a natural translation of the classical Euclidean tools in terms of parallel intrinsic notions in the noncommutative structure H n . Hence, what we are going to discuss in a while in this introduction, it should be easily interpreted by all. We remark that, in this particular non-commutative context, the gradient jump |∇u + | 2 − |∇u − | 2 = 1 now is governed by the jump of the horizontal gradient ∇ H n of the solutions u of (20) in the Heisenberg group H n . As a first consequence, in this degenerate case associated with the sublaplacian ∆ H n , a new geometric problem, that in the Euclidean two-phase problem did not exist, now appears. In fact, since classical smooth free boundaries of (20), in principle, might have characteristic points, then the jump 2 of the horizontal gradient of u on F(u) could be no satisfied pointwise, because the horizontal gradient vanishes on characteristic points, see Section 3. It is also worthwhile to recall that it has been already proved that a minimum u of the functional Ω ⊂ H n , see Section 3 in [21], is endowed by a locally bounded horizontal gradient ∇ H n u and moreover that every minimum u satisfies ∆ H n u = 0 in Ω + (u), as well as ∆ H n u = 0 in Ω − (u), even if no word has been spent about the behavior of the free boundary of these local minima. Indeed, an alternatively way of proving that a local minimum of the functional E H n is intrinsically Lipschitz, instead of using the monotonicity formula, has been shown in [21]. The proof of the monotonicity formula in the Euclidean framework is quite long and based on many highly non-trivial results. Thus, we like to revisit it in Section 2, by commenting the key points of the proof and then focusing our attention to the parallel steps that we would need to prove in the Heisenberg group, including the statement of our following main result proved in [19] as well. In order to reach our goal, let us introduce the following family of functionals depending on a real number β > 0 : Following the main steps of the Euclidean proof, in [19] we proved the following result as a corollary of an estimation of the first eigenvalue of an operator defined on the boundary of the Koranyi ball of radius one. In fact, as people who work with Heisenberg group stuff well know, this set takes the place of the boundary of the classical Euclidean ball of radius one, when we need to work with the fundamental solution of the sublaplacian ∆ H 1 , see [18].
We stated this result in the first Heisenberg group only, because we did not prove a monotonicity formula for all the Heisenberg groups, but simply we have proved that if this formula holds in the non-commutative framework given by H 1 , then the right exponent β has to be smaller or equal than 4. The proof in higher Heisenberg groups requires more computations, but it may be obtained with some further efforts, that we do not discuss here, following the same ideas. On the other hand, the breakthrough that we would need for concluding that, at least in H 1 , the sharp exponent β is exactly 4, depends on a long standing open question. In fact the best profile of the set that realizes the equality in the isoperimetric inequality in the Heisenberg group (and as a byproduct the descendant Polya-Szëgo inequality on the surface of the Koranyi ball of radius one) is still open, see [12] for an introduction to this problem. So that, considering previous arguments, we have decided to state our result only in H 1 . We shall discuss this part in Section 5. In the remaining Section 4, we describe the main tools we need for obtaining the key estimate on the Rayleigh quotient in H 1 , see [19] for the details.

The Euclidean setting
In this section, following the original paper [1], and [10], we try to focus on the main steps we need to achieve for proving the Alt-Caffarelli-Friedman monotonicity formula in the Euclidean setting. After a straightforward differentiation, it results where for i = 1, 2 : By a rescaling argument the problem may be reduced to Precisely, we have where here y denotes the coordinates on ∂B 1 (0). Thus, we get which implies, if we define . As a consequence, if we write y = x and (u i ) r = u i the last equality gives , and so (6) becomes (7).
Since the gradient may split in two orthogonal parts involving the radial part and the tangential part, respectively denoted by ∇ ρ u i and ∇ θ u i , it results Then, we can rewrite J i as At this point, we estimate the numerator and denominator of (8) separately. As regards the numerator, we define first and λ(Γ i ), i = 1, 2, is the Rayleigh quotient. By the definition of λ(Γ i ), we thus obtain, for every β i ∈ (0, 1), hence, by Cauchy inequality, we havê Concerning the denominator, instead, we compute since u i ∆u i ≥ 0 by the assumptions on u i . Consequently, we achieve the following estimate: In fact, previous inequality follows after an integration by parts, using the facts that |x| 2−n is, up to a multiplicative constant, the fundamental solution of ∆ and 0 ∈ F(u i ), i = 1, 2, and by Hölder inequality because: Now, putting together (9) and (10), we get, in view of (8), and The last equality easily follows by elementary arguments. Now, if it were possible to choose β i ∈ (0, 1) in such a way that we would realize, by denoting α i := (β i λ(Γ i )) 1 2 , that previous equation is satisfied if and only if α 2 i + (n − 2)α i − λ(Γ i ) = 0. On the other hand, since a function u = ρ α g(θ), θ := (θ 1 , . . . , θ n−1 ), is harmonic in a cone determined by a domain Γ whenever ρ α−2 ((α(α − 1) + α(n − 1))g(θ) + ∆ θ g) = 0, we deduce that there exists α i such that By the structure of the equation, it immediately comes out that there always exists a strictly positive solution α i = α i (Γ i ), which is called the characteristic constant of Γ i . Therefore, we have to prove the existence of β i ∈ (0, 1) such that Specifically, (12) is equivalent to solve Since the continuous positive function defined in [0, +∞) as (n−2) 2 +z 2 = 1, we conclude that for every λ(Γ i ) > 0, there exists β i such that (12) holds. In particular, we get Hence, with previous choice of β i , if we denote which is also the exponent corresponding to the eigenvalue given by the Rayleigh quotient λ(Γ i ), we conclude that, whenever So, for completing this proof, we would need to know that (13) holds. To this end, by [37] we know that Here H n−1 denotes the (n − 1)-dimensional Hausdorff measure on ∂B 1 (0). Precisely, [37] shows that if u ∈ C ∞ (∂B 1 (0), R), then , where u * is the symmetrized function of u, depending only on the latitude of the argument. Moreover, we also have that , that is the pushforward measures of u and u * coincide in the Borel sets of R, which entails for any function φ : R → R µ * -measurable, where µ * is the outer measure defined on the power set P(R) of R as and F ∈ P(R). Hence, choosing φ = x 2 in (15), we obtain which gives, together with (14), λ(Γ i ) ≥ λ(Γ * i ), and thus, using the expression of derives from a property of u * which says that On the other hand, from [25] we achieve that H n−1 (∂B 1 (0)) and ψ(s), s ∈ (0, 1), is convex and decreasing. In particular, ψ(s) is defined as Precisely, the proof of α i (Γ * i ) ≥ ψ(s i ) is organized in some steps. First of all, we denote α(E) = α(s, n), where α(E) is the characteristic constant of the spherical cap E ⊂ ∂B 1 (0), s = H n−1 (E) H n−1 (∂B 1 (0)) , and n is the dimension. At this point, Theorem 2 in [25] tells us that α(s, n) is a monotone decreasing function of n for fixed s, so (17) α(s, ∞) = lim n→∞ α(s, n) is well defined and satisfies α(s, ∞) ≤ α(s, n) for every n. It is thus sufficient to show that α(s, ∞) ≥ ψ(s) defined in (16). To this end, Theorem F in [25], which is taken by [31], says that α(s) ≥ ψ(s), where where Γ is the Euler gamma function. In particular, H α (x) is the Hermite's function of order α. Now, Theorem 3 in [25] shows that α(s, ∞) defined in (17) is equal to α(s) of Theorem F, since s of α(s, n) converges to s of α(s) as n goes to ∞, i.e.
Hence, being α(s, n) ≥ α(s, ∞) for all n, we finally have that α(s, n) ≥ ψ(s) for every n and for all s ∈ (0, 1). As a consequence, recalling that s i = (16) is convex and decreasing, we get which finally gives (13). An alternative proof of this result is given in [10], where, using [3] and [6], the two authors directly show that α(s 1 ) + α(s 2 ) ≥ 2, exploiting the properties of α(s) of Theorem F in [25], which is the first Dirichlet eigenvalue on [h, ∞) associated to the Hermite operator

The main notation in the Heisenberg group
We denote by H n the set R 2n+1 , n ∈ N, n ≥ 1, endowed with the non-commutative inner law in such a way that for every P ≡ (x 1 , y 1 , t 1 ) ∈ R 2n+1 , M ≡ (x 2 , y 2 , t 2 ) ∈ R 2n+1 , x i ∈ R n , y i ∈ R n , i = 1, 2 : where ·, · denotes the usual inner product in R n . Let X i = (e i , 0, 2y i ) and Y i = (0, e i , −2x i ), i = 1, . . . , n, where {e i } 1≤i≤n is the canonical basis for R n . We use the same symbol to denote the vector fields associated with the previous vectors, so that for i = 1, . . . , n,
In particular, we get a norm associated with the metric on the space span{X 1 , . . . , X n , Y 1 , . . . , Y n }, which is For example, the norm of the intrinsic gradient of a smooth function u in P is If ∇ H n u(P ) = 0, instead, we say that the point P is characteristic for the smooth surface {u = u(P )}. Hence, for every point M ∈ {u = u(P )}, which is not characteristic, it is well defined the intrinsic normal to the surface {u = u(P )} as follows: At this point, we introduce in the Heisenberg group H n the following gauge norm: In particular, for every positive number r, the gauge ball of radius r centered in 0 is B H n r (0) := {P ∈ H n : |P | H n < r}. In the Heisenberg group, a dilation semigroup is defined as follows: for every r > 0 and for every P = (x, y, t) ∈ H n , let δ r (P ) := (rx, ry, r 2 t).
Let P := (ξ, η, σ) ∈ H n and O = (0, 0, 0), then we define For every P, T ∈ H n is well defined that is a distance d K on the Heisenberg group H n , known as the Koranyi distance. This distance is left invariant, that is for every P, T, R ∈ H n d K (R • P, R • T ) = d K (P, T ).
As a consequence, we may perform our computation supposing of dealing with d K (P, O) = |P | H n , where O = (0, 0, 0), simply by multiplying the left hand side by T −1 .
In particular, for every i = 1, . . . , n we obtain: Moreover, for every i = 1, . . . , n : As a consequence, and Thus, for every i = 1, . . . , n, denoting by Q := 2n + 2 homogeneous dimension we get: In conclusion, |P | 2−Q H n is, up to a constant, the fundamental solution of the sublaplacian ∆ H n in the Heisenberg group, with the pole in the origin, and Γ(P, The definition of H n −subharmonic function, as well as the one of H n −superharmonic function in a set Ω ⊂ H n , can be stated, as usual, in the classical way, requiring respectively that ∆ H n u(P ) ≥ 0 for every P ∈ Ω, for the H n −subharmonicity, and that ∆ H n u(P ) ≤ 0 for every P ∈ Ω for having H n −superharmonicity. We refer to [4] for further details. Concerning the natural Sobolev spaces to consider in the Heisenberg group H n , we refer to the literature, see for instance [26]. Here, we simply recall that: is a Hilbert space with respect to the norm Now, if E ⊂ H n is a measurable set, a notion of H n -perimeter measure |∂E| H n has been introduced in [26] in a more general setting, even if here we recall some results in the framework of the Heisenberg group, the simplest non-trivial example of Carnot group. We refer to [26], [22], [24], [23] for a detailed presentation. For our applications, we restrict ourselves to remind that, if E has locally finite H nperimeter (is a H n -Caccioppoli set), then |∂E| H n is a Radon measure in H n , invariant under group translations and H n -homogeneous of degree Q − 1. Moreover, the following representation theorem holds (see [11]).
where n E = n E (x) is the Euclidean unit outward normal to ∂E.
We also have: If E is a regular bounded open set with Euclidean C 1 boundary and φ is a horizontal vector field, continuously differentiable on Ω, then where ν H n (x) is the intrinsic horizontal unit outward normal to ∂E, given by the (normalized) projection of n E (x) on the fiber HH n x of the horizontal fiber bundle HH n . Remark 3.3. The definition of ν H n is well done, since HH n x is transversal to the tangent space of E at x, for P E H n (∂E)-a.e. x ∈ ∂E (see [33]). Now, adapting the approach described in [1] and recalled in Section 2 to the Heisenberg case, we conclude, by applying the definition of solution in the sense of the domain variation to the functional Ω ⊂ H n , that the parallel two-phase problem to (3) is, see [17]: Thus, it seems natural to consider, as a candidate for an Alt-Caffarelli-Friedman monotonicity formula in the Heisenberg group, the following function: where β > 0 is a suitable fixed exponent and u + := sup{u, 0} and u − := sup{−u, 0}, being 0 ∈ F(u).
On the other hand, the first eigenvalue λ ϕ 0 ,ϕ 1 is determined by the Rayleigh quotient given, in this case, by λ ϕ 0 ,ϕ 1 := inf f ∈H 1 0 (ϕ 0 ,ϕ 1 ) . 16 Thus, it is fundamental to know if the result by [25], that is the cap on ∂B 1 (0) having the same H n−1 measure of some sets Σ on ∂B 1 (0) has the smallest Rayleigh quotient, is true even in the Heisenberg case.
Let say that we would like to know if there exists a set Γ * ⊂ ∂B H 1 1 (0) such that for every Γ ⊂ ∂B H 1 1 (0), The existence in the Heisenberg group of the properties of the characteristic number associated with the set Γ, as far as we know, is still unknown. This part corresponds to the topic discussed in [37] in the Euclidean setting. In fact, just for having an idea about the difficulty in solving the problem, we remark that where Γ = T ({1} × Ω). At this point, we may decide to symmetrize the set Ω in many ways. For instance, for every ϕ, we might define Ω * ϕ in such a way that H 1 (Ω * ϕ ) = 2θ ϕ = H 1 (Ω ϕ ), and consider Ω * := ∪ ϕ∈Π 2 (Ω) Ω * ϕ , being Π 2 (Ω) := {ϕ : Ω ϕ = ∅}. Unfortunately, the lack of an isoperimetric result does not permit to conclude anything.