Minimal surfaces and harmonic functions in the Heisenberg group

MINIMAL SURFACES AND HARMONIC FUNCTIONS IN THE HEISENBERG GROUP ROBERTO MONTI Abstract. We study the blow-up of H-perimeter minimizing sets in the Heisenberg group Hn , n ≥ 2. We show that the Lipschitz approximations rescaled by the square root of excess converge to a limit function. Assuming a stronger notion of local minimality, we prove that this limit function is harmonic for the Kohn Laplacian in a lower dimensional Heisenberg group. Contents 1. Introduction 2. Blow-up at the reduced boundary of minimizers 2.1. Small excess at the reduced boundary 2.2. Lipschitz approximation and intrinsic gradient 2.3. Blow-up of H-minimal boundaries 3. H-harmoncity of the limit function 3.1. First variation formula 3.2. H-harmonicity of ϕ References 1 5 5 6 8 12 12 13 19 1. Introduction One of the central facts in the regularity theory of minimal currents and of minimal boundaries in Rn is the existence of a harmonic function in the blow-up of the Lipschitz approximation of the current rescaled by excess. The heuristic idea behind this phenomenon is the fact that if a function f : D → R, with D ⊂ Rn−1 , is a local minimizer of the area functional Z p 1 + |∇f (x)|2 dx A(f ) = D 2010 Mathematics Subject Classification. 49Q05. Key words and phrases. Heisenberg group, regularity of H-minimal surfaces, H-harmonic functions. 1 2 R. MONTI and f is almost flat, i.e., |∇f (x)| is almost 0, then f is almost a minimizer of the Dirichlet functional Z 1 |∇f (x)|2 dx, D(f ) = 2 D that is the first order term in the Taylor development of the area functional. For this reason, a function f in the blow-up of a minimal boundary is harmonic x ∈ D ⊂ Rn−1 . ∆f (x) = 0, The L2 estimates on the derivatives of harmonic functions give the decay estimate of excess, that in turn implies the C 1,α regularity of the minimal boundary. In this paper, we investigate the existence of a similar phenomenon in the case of a nonelliptic perimeter, as the horizontal perimeter in the Heisenberg group. Our results are not satisfactory because they hold for sets that are H-perimeter minimizing in a stronger sense, that is not the natural one. However, they are the first example of “harmonic approximation” of minimal boundaries for a nonelliptic perimeter and they suggest an interesting research direction in the regularity theory. So far, the regularity theory for H-minimal surfaces always starts from some initial regularity (see [4], [5], [6], [7], [18]). See, however, the Lipschitz approximation [14] and the height estimate proved in [17]. The 2n + 1-dimensional Heisenberg group is the manifold Hn = Cn × R, n ∈ N, endowed with the group product
(z, t) ∗ (ζ, τ ) = z + ζ, t + τ + 2 Imhz, ζ̄i , (1.1) where t, τ ∈ R, z, ζ ∈ Cn and hz, ζ̄i = z1 ζ̄1 +. . .+zn ζ̄n . The Lie algebra of left-invariant vector fields in Hn is spanned by the vector fields Xj = ∂ ∂ + 2yj , ∂xj ∂t Yj = ∂ ∂ − 2xj , ∂yj ∂t and T = ∂ , ∂t (1.2) with zj = xj + iyj and j = 1, . . . , n. We denote by H the horizontal subbundle of T Hn . Namely, for any p = (z, t) ∈ Hn we let  Hp = span X1 (p), . . . , Xn (p), Y1 (p), . . . , Yn (p) . The H-perimeter of a L 2n+1 -measurable set E ⊂ Hn in an open set Ω ⊂ Hn is  Z 1 2n divH V dzdt : V ∈ Cc (Ω; R ), kV k∞ ≤ 1 , PH (E; Ω) = sup E where V : Ω → R2n is naturally identified with the horizontal vector field V = Pn j=1 Vj Xj + Vn+j Yj and the horizontal divergence of V is divH V = n X j=1 Xj Vj + Yj Vn+j . MINIMAL SURFACES AND HARMONIC FUNCTIONS 3 We use the notation µE (Ω) = PH (E; Ω). If µE (Ω) < ∞ we say that E has finite H-perimeter in Ω. If µE (A) < ∞ for any open set A ⊂⊂ Ω, we say that E has locally finite H-perimeter in Ω. In this case, the open sets mapping A 7→ µE (A) extends to a Radon measure µE on Ω that is called H-perimeter measure induced by E. Moreover, there exists a µE -measurable function νE : Ω → H such that |νE | = 1 µE -a.e. and the Gauss-Green integration by parts formula Z Z hV, νE i dµE = − divH V dzdt Ω Ω Cc1 (Ω; R2n ). holds for any V ∈ Here and hereafter, h·, ·i denotes the standard scalar 2n product in R . The vector νE is called horizontal inner normal of E in Ω. We consider a set E ⊂ Hn with 0 ∈ ∂ ∗ E, the H-reduced boundary of E, that is a local minimizer of H-perimeter in a neighborhood of 0 and we rescale E to a unitary scale to have infinitesimal excess. In this way, we have a sequence of sets Eh that are H-perimeter minimizing and have infinitesimal excess ηh , h ∈ N. In Section 2, we use the Lipschitz approximation proved in [14] to obtain a sequence of instrinsic Lipschitz functions (ϕh )h∈N whose graphs cover in measure a large part of the boundary of the rescaled sets Eh . By the Poincaré inequality recently proved in [8], we can show that there is subsequence of (ϕh /ηh )h∈N converging to a function ϕ in a suitable Sobolev space. To have this limit function, only the density estimates for minimal boundaries are in fact used and so the result extends to Λ-minima. The Poincaré inequality mentioned above is for functions in domains of R × Hn−1 and it holds only when n ≥ 2. This is one of the reasons why our discussion is limited to dimensions n ≥ 2. The area functional of an intrinsic Lipschitz function ϕ : D → R, where now D ⊂ R × Hn−1 , is of the form Z p A(ϕ) = 1 + |∇ϕ ϕ|2 dw, (1.3) D where dw is the Lebesgue measure on W = R × Hn−1 and ∇ϕ ϕ is a nonlinear gradient that is defined in the sense of distributions, known as “intrinsic gradient”, see Definition 2.3. The area formula (1.3) is obtained in [9] Theorem 6.5 part (vi) and in [2] Proposition 2.22. However, the Dirichlet functional Z 1 |∇ϕ ϕ|2 dw D(ϕ) = 2 D does not catch the correct regularity of the limit function, because in the blow-up there is a linearization of the nonlinear gradient ∇ϕ ϕ, see Theorem 2.5. After this linearization, the relevant Dirichlet functional turns out to be Z n n o 1 ∂ϕ 2 X DH (ϕ) = + (1.4) (Xj ϕ)2 + (Yj ϕ)2 dw, 2 D ∂y1 j=2 4 R. MONTI where y1 ∈ R is the variable of the factor R in the Cartesian product R × Hn−1 . The Dirichlet form (1.4) identifies the differentiability class where the limit of the (sub)sequence (ϕh /ηh )h∈N lies. In Section 3, we deduce from the minimality of E further properties of the limit function ϕ. We use the first order Taylor expansion of H-perimeter (3.41), that holds for any set with finite H-perimeter (these sets may be unrectifiable in the standard sense). We obtain two results. First, we prove that if E is a set that locally minimizes H-perimeter then the function ϕ : D ⊂ R × Hn−1 → R is independent of the variable y1 of the factor R, see the first claim of Theorem 3.2. This fact seems to have no counterpart in the classical theory. The second result holds for a stronger notion of minimality. The homogeneous cube centered at 0 ∈ Hn and with radius r > 0 is  Qr = (z, t) ∈ Hn : |xi | < r, |yi | < r, |t| < r2 , i = 1, . . . , n . (1.5) Definition 1.1. We say that a set E ⊂ Hn is H-perimeter minimizing in Qr if PH (E; Qr ) ≤ PH (F, Qr ) for any set F ⊂ Hn such that E∆F is a compact subset of Qr . Let E ⊂ Hn be a set with 0 ∈ ∂ ∗ E and νE (0) = X1 . Let J : H → H be the complex structure and consider Y1 = J(X1 ). We define the lateral closure of Qr relative to the positive direction Y1 as:  Q̄Yr 1 ,+ = (z, t) ∈ Hn : −r < y1 ≤ r, |x1 |, |xi |, |yi | < r, |t| < r2 , i = 2, . . . , n . We are adding to Qr the open face of the boundary where y1 = r. Definition 1.2. We say that a set E ⊂ Hn with 0 ∈ ∂ ∗ E and νE (0) = X1 is strongly H-perimeter minimizing in Qr if for any 0 < s ≤ r we have PH (E; Qs ) ≤ PH (F, Qs ) for any set F ⊂ Hn such that E∆F ∩ Q̄s is a compact subset of Q̄Ys 1 ,+ . Here and hereafter, E∆F = E \ F ∪ F \ E denotes the symmetric difference. In the second claim of Theorem 3.2, we show that if E is strongly H-perimeter minimizing then the function ϕ : D → R, where now D is a subset of Hn−1 , is H-harmonic, i.e., it solves the partial differential equation ∆H ϕ = 0, in D ⊂ Hn−1 , where ∆H is the Kohn Laplacian in the lower dimensional Heisenberg group Hn−1 ∆H = n X j=2 Xj2 + Yj2 . (1.6) MINIMAL SURFACES AND HARMONIC FUNCTIONS 5 It is not clear whether the strong H-perimeter minimality can be relaxed to the natural local minimality. The problem is related to the construction of suitable contact vector fields in Hn with compact support. This problem is explained in Section 3, along the proof of Theorem 3.2. The ideas presented in this paper are part of a joint research project with D. Vittone. 2. Blow-up at the reduced boundary of minimizers In this section, we show that in the blow-up of an H-perimeter minimizing set at a point of the reduced boundary there is a function belonging to a suitable Sobolev space. We use the box-norm kpk∞ = max{|z|, |t|1/2 } for p = (z, t) ∈ Hn , and the homogeneous balls Br = {p ∈ Hn : kpk∞ < r} and Br (p) = p ∗ Br , r > 0. The balls Br are equivalent to the cubes Qr . 2.1. Small excess at the reduced boundary. Let E ⊂ Hn be a set with locally finite H-perimeter in Hn . We say that 0 ∈ Hn is a point of the H-reduced boundary of E, 0 ∈ ∂ ∗ E, if the following three conditions hold: µE (Br ) > 0 for all r > 0, we have Z 1 νE dµE = νE (0), lim r→0 µE (Br ) B r and |νE (0)| = 1. This definition is introduced and studied in [9]. The horizontal excess of E in Br , r > 0, is Z 1 |νE (p) − ν|2 dµE . Exc(E, Br ) = min ν∈S2n r Q−1 Br We refer the reader to [13] for an account on excess in the Euclidean setting. Notice that r1−Q C1 ≤ µE (Br ) ≤ r1−Q C2 for constants 0 < C1 < C2 < ∞ and Q = 2n + 2. For minimizers, the constants are independent of the point in the reduced boundary. Thus, if 0 ∈ ∂ ∗ E is a point in the H-reduced boundary of E then there exists a sequence rh → 0+ such that Z 1 1 |νE − νE (0)|2 dµE < , µE (Brh ) Brh h and so we have Exc(E, Brh ) < 1/h. We consider the anisotropic dilations (z, t) 7→ (λz, λ2 t) = δλ (z, t), λ > 0. The rescaled sets Eh = δ1/rh E, h ∈ N, satisfy suph∈N PH (Eh ; B1 ) < ∞. Moreover, we have: 6 R. MONTI i) If E is H-perimeter minimizing near 0 ∈ ∂E ∗ , then each set Eh is H-perimeter minimizing in B1 ; ii) Since excess is scale invariant, there holds Exc(Eh , B1 ) < 1/h; iii) 0 ∈ ∂ ∗ Eh . Rotating each set Eh by an isometry fixing the t-axis, we may assume that Z 1 |νEh − ν|2 dµEh < , Exc(Eh , B1 ) = h B1 (2.7) where ν ∈ S2n is a vector independent of h. In fact, we may assume that ν = νE (0). Possibly taking a subsequence, by the compactness theorem for sets with finite Hperimeter, there exists a set F ⊂ Hn such that lim χEh = χF , h→∞ in L1 (B1 ). Moreover, by the lower semicontinuity of excess we have Exc(F, B1 ) = 0. Since 0 ∈ ∂F , when n ≥ 2 this implies that F ∩ B1 = {(z, t) ∈ B1 : hz, νi ≥ 0}, see [9]. When n = 1, this fact does no longer hold, i.e., ∂F needs not be flat in any neighborhood of 0, see [14]. 2.2. Lipschitz approximation and intrinsic gradient. We identify the vertical  hyperplane W = R × Hn−1 = (z, t) ∈ Hn : x1 = 0 with R2n via the coordinates w = (x2 , . . . , xn , y1 , . . . , yn , t). The line flow of the vector field X1 starting from the point (z, t) ∈ W is exp(sX1 )(z, t) = (z + se1 , t + 2y1 s), s ∈ R, where e1 = (1, 0, . . . , 0) ∈ R2n and z = (x, y) ∈ Cn = R2n , with x = (x1 , . . . , xn ), x1 = 0, and y = (y1 , . . . , yn ). Let D ⊂ W be a set and let ϕ : D → R be a function. The set  Eϕ = exp(sX1 )(w) ∈ Hn : s > ϕ(w), w ∈ D (2.8) is called intrinsic epigraph of ϕ along X1 . The set  gr(ϕ) = exp(ϕ(w)X1 )(w) ∈ Hn : w ∈ D (2.9) is called intrinsic graph of ϕ along X1 . We identify ν = (1, 0, . . . , 0) ∈ R2n with (ν, 0) ∈ Hn . For any p ∈ Hn , we let ν(p) = hp, νiν ∈ Hn and we define ν ⊥ (p) ∈ W ⊂ Hn as the unique point such that p = ν ⊥ (p) ∗ ν(p). (2.10) The (open) cone with vertex 0 ∈ Hn , axis ν ∈ R2n , |ν| = 1, and aperture α ∈ (0, ∞] is the set  (2.11) C(0, ν, α) = p ∈ Hn : kν ⊥ (p)k∞ < αkν(p)k∞ . MINIMAL SURFACES AND HARMONIC FUNCTIONS 7 The cone with vertex p ∈ Hn , axis ν ∈ R2n , and aperture α ∈ (0, ∞] is the set C(p, ν, α) = p ∗ C(0, ν, α). Definition 2.1 (Intrinsic Lipschitz graphs). Let D ⊂ W be a set and let ϕ : D → R be a function. The function ϕ is L-intrinsic Lipschitz with 0 < L < ∞ if for any p ∈ gr(ϕ) there holds gr(ϕ) ∩ C(p, ν, 1/L) = ∅. (2.12) The starting point of our argument is the following result of [14] on the Lipschitz approximation of H-minimal boundaries. We denote by S Q−1 the (2n + 1)-dimensional spherical Hausdorff measure constructed using any homogeneous left invariant metric on Hn . We shall use freely the identity S Q−1 ∂ ∗ E = µE . (2.13) Recall that for an H-perimeter minimizing set E, the reduced boundary ∂ ∗ E coincides with the essential boundary, that is denoted by ∂E. Theorem 2.2. Let n ≥ 2. For any L > 0 there are constants k > 1 and c(L, n) > 0 such that for any H-perimeter minimizing set E in Bkr , with 0 ∈ ∂E and νE (0) = ν = X1 , there exists an L-intrinsic Lipschitz function ϕ : W → R such that
(2.14) S Q−1 (gr(ϕ)∆∂E) ∩ Br ≤ c(L, n)(kr)Q−1 Exc(E, Bkr ), r > 0. Theorem 2.2 holds also for n = 1. In this case, the Lipschitz constant L has to be suitably large. We introduce a nonlinear gradient for functions ϕ : D → R with D ⊂ W open set. The Burgers’ operator B : Liploc (D) → L∞ loc (D) is Bϕ = ∂ϕ ∂ϕ − 4ϕ . ∂y1 ∂t (2.15) When ϕ ∈ C(D) is only continuous, we say that Bϕ exists in the sense of distributions and is represented by a locally bounded function, if there exists a function ϑ ∈ L∞ loc (D) 1 such that for all ψ ∈ Cc (D) there holds Z n Z ∂ψ ∂ψ o ϕ dw. (2.16) ϑψ dw = − − 2ϕ2 ∂y1 ∂t D D In this case, we let Bϕ = ϑ. Next, notice that the vector fields X2 , . . . , Xn , Y2 , . . . , Yn can be naturally restricted to W and that they are self-adjoint. Definition 2.3 (Intrinsic gradient). Let D ⊂ W = R2n be an open set and let ϕ ∈ C(D) be a continuous function. We say that the intrinsic gradient ∇ϕ ϕ ∈ 2n−1 L∞ ) exists in the sense of distributions if the distributional derivatives loc (D; R 8 R. MONTI Xi ϕ, Bϕ, Yi ϕ, i = 2, . . . , n, are represented by locally bounded functions in D. In this case, we let ∇ϕ ϕ = X2 ϕ, . . . , Xn ϕ, Bϕ, Y2 ϕ, . . . , Yn ϕ), (2.17) and we call ∇ϕ ϕ the intrinsic gradient of ϕ. When n = 1, the intrinsic gradient reduces to ∇ϕ ϕ = Bϕ. Theorem 2.4. Let D ⊂ W be an open set and ϕ : D → R be a continuous function. The following statements are equivalent: 2n−1 A) We have ∇ϕ ϕ ∈ L∞ ). loc (D; R B) For any D′ ⊂⊂ D, the function ϕ : D′ → R is intrinsic Lipschitz. Moreover, if A) or B) holds then the intrinsic epigraph Eϕ ⊂ Hn has locally finite H-perimeter in the cylinder D ∗ R = {w ∗ (se1 ) ∈ Hn : w ∈ D, s ∈ R}, for L 2n a.e. w ∈ D the inner horizontal normal to ∂Eϕ is   −∇ϕ ϕ(w) 1 , (2.18) ,p νEϕ (w ∗ ϕ(w)) = p 1 + |∇ϕ ϕ(w)|2 1 + |∇ϕ ϕ(w)|2 and, for any D′ ⊂ D, we have the area formula Z p ′ 1 + |∇ϕ ϕ|2 dw. PH (Eϕ ; D ∗ R) = (2.19) D′ The equivalence between A) and B) is a deep result that is proved in [3], Theorem 1.1. Formula (2.18) for the normal and the area formula (2.20) are proved in [8] Corollary 4.2 and Corollary 4.3, respectively. Part of these results is the fact that k∇ϕ ϕk∞ is equivalent to the Lipschitz constant. The area formula (2.19) can be improved in the following way Z Z p g(w ∗ ϕ(w)) 1 + |∇ϕ ϕ(w)|2 dw, g(p) dµEϕ = (2.20) ∂Eϕ ∩(D ′ ∗R) D′ where g : ∂Eϕ → R is a Borel function. A result related to Theorem 2.4 can be found in [16], where it is proved that if E ⊂ n H is a set with finite H-perimeter having controlled normal νE , say hνE , e1 i ≥ k > 0 µE -a.e., then the reduced boundary ∂ ∗ E is an intrinsic Lipschitz graph along X1 . 2.3. Blow-up of H-minimal boundaries. Let E ⊂ Hn be an H-perimeter minimizing set in a neighborhood of 0 ∈ Hn , with 0 ∈ ∂E and νE (0) = X1 . Let Eh be the rescaled sets of E introduced before equation (2.7). The square root of excess p (2.21) ηh = Exc(Eh , B1 ) is infinitesimal, and we may assume that ηh > 0. Let σ > 0 be a small number, e.g., 0 < σ ≤ 1/k where k > 1 is the geometric constant given by Theorem 2.2, and let 0 < L ≤ 1 be a Lipschitz constant. Since MINIMAL SURFACES AND HARMONIC FUNCTIONS 9 each set Eh is H-perimeter minimizing in the ball B1 , by Theorem 2.2 there exist L-intrinsic Lipschitz functions ϕh : W → R such that
(2.22) S Q−1 (gr(ϕh )∆∂Eh ) ∩ Bσ ≤ c(L, n, σ)Exc(Eh , B1 ) = c0 ηh2 , where c0 = c(L, n, σ). In this section we prove the following theorem. Recall that the Sobolev space WH1,2 (D) is the set of all ϕ ∈ L2 (D) such that the distributional derivatives X2 ϕ, . . . , Xn ϕ, ∂ϕ , Y2 ϕ, . . . , Yn ϕ ∈ L2 (D) ∂y1 are squared integrable. In this case, we let   ∂ϕ ∇H ϕ = X2 ϕ, . . . , Xn ϕ, , Y2 ϕ, . . . , Yn ϕ . ∂y1 Theorem 2.5. Let n ≥ 2. Under the assumptions made at the beginning of this section, there exist an open neighborhood D ⊂ W of 0 ∈ W , constants ϕ̄h ∈ R, a function ϕ ∈ WH1,2 (D), and a selection of indices k 7→ hk such that, for k → ∞ we have ϕhk − ϕ̄hk ⇀ϕ ηh k ∇ ϕ h k ϕ hk ⇀ ∇H ϕ ηhk weakly in L2 (D), weakly in L2 (D; R2n−1 ). In the proof of Theorem 2.5, we use the Poincaré inequality of [8]. As in Section 2.1 of [8] (but with our normalization (1.2) of the vector fields), for w = (z, t) ∈ W and ϕ : W → R we let dϕ (w, 0) =   1 1 max |z|, |t + 4ϕ(w)y1 |1/2 + max |z|, |t + 4ϕ(0)y1 |1/2 , 2 2 (2.23) and, for r > 0,  Uϕ (r) = w ∈ W : dϕ (w, 0) < r . (2.24) Theorem 2.6. Let n ≥ 2 and let ϕ : W → R be an L-intrinsic Lipschitz function. There exist constants C1 , C2 > 0 depending on L and n such that Z Z 2 2 |∇ϕ ϕ(w)|2 dw, r > 0, (2.25) |ϕ(w) − ϕUϕ (r) | dw ≤ C1 r Uϕ (C2 r) Uϕ (r) where ϕUϕ (r) See Corollary 4.5 in [8]. 1 = 2n L (Uϕ (r)) Z ϕ(w)dw. Uϕ (r) (2.26) 10 R. MONTI Proof of Theorem 2.5. By the lower density estimate PH (Eh ; Bσ/2 ) ≥ Cσ Q−1 with a constant C > 0 independent of h and, from (2.22), we deduce that gr(ϕh ) ∩ Bσ/2 6= ∅ for all h ∈ N large enough. It follows that (details are omitted) there exists ε1 > 0 such that gr(ϕh ) ∩ {w ∈ W : |w| < ε1 } ∗ R ⊂ Bσ . (2.27) Without loss of generality we can assume that kϕh k∞ ≤ 1 for all h ∈ N. Thus, from (2.23) and (2.24), it follows that there exist ε0 > 0 and r > 0 such that   D := w ∈ W : |w| < ε0 ⊂ Uϕh (r) ⊂ Uϕh (C2 r) ⊂ w ∈ W : |w| < ε1 =: D′ . (2.28) Then, by (2.22), we deduce the estimate
(2.29) S Q−1 (gr(ϕh ) \ ∂Eh ) ∩ D′ ∗ R ≤ c0 ηh2 . Let Dh ⊂ D′ be the set of the points w ∈ D′ such that   −∇ϕh ϕh (w) 1 νEϕh (w ∗ ϕh (w)) = p , ,p 1 + |∇ϕh ϕh (w)|2 1 + |∇ϕh ϕh (w)|2 (2.30) and νEϕh (w ∗ ϕh (w)) = νEh (w ∗ ϕh (w)). (2.31) By Theorem 2.4, see formula (2.18), identity (2.30) holds for L 2n -a.e. w ∈ D′ . By the locality of H-perimeter (see Corollary 2.5 in [1]) and by the area formula (2.19), identity (2.31) holds for L 2n -a.e. w ∈ π(gr(ϕh ) ∩ ∂Eh ), where π : Hn → W is the projection along X1 . Since each function ϕh is L-intrinsic Lipschitz with 0 < L ≤ 1, we can assume k∇ϕh ϕh k∞ ≤ 1. Then for any point w ∈ Dh we have: 1 |νEh (w ∗ ϕh (w)) − ν|2 = |νEϕh (w) − ν|2 ≥ |∇ϕh ϕh (w)|2 , 2 where ν = (1, 0, . . . , 0) ∈ S2n . By the area formula (2.20) for intrinsic Lipschitz functions and by (2.7), we obtain the estimate Z Z ϕh 2 |∇ ϕh (w)| dw ≤ 2 |νEh − ν|2 dµEh ≤ 2ηh2 . (2.32) Dh B1 Again by k∇ϕh ϕh k∞ ≤ 1, by the area formula, and by (2.22), we obtain Z |∇ϕh ϕh (w)|2 dw ≤ L 2n (D′ \ Dh ) D ′ \Dh ≤S Q−1 ((gr(ϕh ) \ ∂Eh ) ∩ Bσ ) ≤ (2.33) c0 ηh2 . It follows that the sequence of functions |∇ϕh ϕh |/ηh , h ∈ N, is uniformly bounded in L2 (D′ ). Then there exists a function Φ ∈ L2 (D′ ; R2n−1 ) such that, possibly taking MINIMAL SURFACES AND HARMONIC FUNCTIONS 11 a subsequence, we have as h → ∞ ∇ϕh ϕ h ⇀ Φ weakly in L2 (D′ ; R2n−1 ). ηh (2.34) After a relabeling, we assume here and hereafter that the full sequence is converging. We denote by ϕ̄h the mean of ϕh defined in (2.26), namely, 1 ϕ̄h = 2n L (Uϕh (r)) Z ϕh (w)dw, (2.35) Uϕh (r) where r > 0 is such that the inclusions in (2.28) hold. By the Poincaré inequality (2.25), by the inclusions in (2.28), (2.32), and (2.33) we have Z 2 |ϕh (w) − ϕ̄h | dw ≤ D Z |ϕh (w) − ϕ̄h |2 dw Uϕh (r) ≤ C1 r 2 ≤ C1 r 2 Z Uϕh (C2 r) Z D′ |∇ϕh ϕh (w)|2 dw |∇ϕh ϕh (w)|2 dw ≤ C1 r2 (2 + c0 )ηh2 . Then, the sequence (ϕh − ϕ̄h )/ηh is uniformly bounded in L2 (D). It follows that we have ϕh − ϕ̄h → 0 in L2 (D). As the sequence of sets (Eh )h∈N is converging to a half-plane inside the ball B1 , we deduce that ϕ̄h → 0 as h → ∞. Finally, by weak compactness there exists a function ϕ ∈ L2 (D) such that, possibly taking a further subsequence, we have ϕh − ϕ̄h ⇀ ϕ weakly in L2 (D). ηh (2.36) 1,2 (D) and that We claim that ϕ ∈ WH   ∂ϕ , Y2 ϕ, . . . , Yn ϕ , Φ = ∇H ϕ = X2 ϕ, . . . , Xn ϕ, ∂y1 (2.37) in the sense of weak derivatives in L2 (D). Notice that the nonlinear derivative Bϕh /ηh is converging to the linear derivative ∂y1 ϕ. By (2.36), for any test function ψ ∈ Cc1 (D) we have lim h→∞ Z D ϕh − ϕ̄h ψdw = ηh Z ϕ ψ dw. D (2.38) 12 R. MONTI On the other hand, by the distributional definition (2.16) of the derivative Bϕh we have Z Z  1 1 ϕh ψy1 − 2ϕ2h ψt dw ψ Bϕh dw = − ηh D ηh D Z  1 (ϕh − ϕ̄h )ψy1 − 2(ϕ2h − ϕ̄2h )ψt dw =− ηh D Z n o ϕh − ϕ̄h ϕh − ϕ̄h =− ψ y1 − 2 (ϕh + ϕ̄h )ψt dw. ηh ηh D Since ϕh + ϕ̄h is converging to zero strongly in L2 (D) and (ϕh − ϕ̄h )/ηh is uniformly bounded in L2 (D), we obtain Z Z 1 lim ψ Bϕh dw = − ϕ ψy1 dw. h→∞ ηh D D A similar argument shows that for any Z ∈ {X, Y } and j = 2, . . . , n we have Z Z 1 ϕ Zj ψdw. ψ Zj ϕh dw = − lim h→∞ ηh D D This finishes the proof of (2.37).  3. H-harmoncity of the limit function In this section, we prove that the limit function ϕ given by Theorem 2.5 is independent of the variable y1 dual in the complex sense to the graph direction x1 . If the set E is a strong minimizer in the sense of Definition 1.2, we show that the function ϕ is H-harmonic in Hn−1 , the lower dimensional Heisenberg group. 3.1. First variation formula. We recall the first variation formula for H-perimeter of sets in Hn that are deformed along a contact flow. A diffeomorphism Ψ : Ω → Ψ(Ω), with Ω ⊂ Hn open set, is a contact map if for any p ∈ Ω the differential Ψ∗ maps the horizontal space Hp into HΨ(p) . A one-parameter flow (Ψs )s∈R of diffeomorphisms in Hn is a contact flow if each Ψs is a contact map. Contact flows are generated by contact vector fields (see [12]). A contact vector field in Hn is a vector field of the form n X Vψ = (Yj ψ)Xj − (Xj ψ)Yj − 4ψT, (3.39) j=1 where ψ ∈ C ∞ (Hn ) is the generating function. For any compact set K ⊂ Hn we have the flow Ψ : [−δ, δ] × K → Hn that is defined by Ψ̇(s, p) = Vψ (Ψ(s, p)) and Ψ(0, p) = p for any s ∈ [−δ, δ] and p ∈ K, for some δ = δ(ψ, K) > 0. We call Ψ the flow generated by ψ. We also let Ψs = Ψ(s, ·). MINIMAL SURFACES AND HARMONIC FUNCTIONS 13 Related to the generating function ψ, we have, at any point p ∈ Hn , the real quadratic form Qψ : Hp → R Qψ n X  xj Xj +yj Yj = j=1 n X xi xj Xj Yi ψ +xj yi (Yi Yj ψ −Xj Xi ψ)−yi yj Yj Xi ψ, (3.40) i,j=1 where xj , yj ∈ R, and ψ with its derivatives are evaluated at p. The quadratic form Qψ appears in the first variation of H-perimeter along the flow generated by ψ. In the following, we identify Hp with R2n by declaring X1 , . . . , Xn , Yn , . . . , Yn an orthonormal basis. Theorem 3.1. Let Ω ⊂ Hn be a bounded open set and let Ψ : [−δ, δ] × Ω → Hn be the flow generated by ψ ∈ C ∞ (Hn ). Then there exists C = C(ψ, Ω) > 0 such that for any set E ⊂ Hn with finite H-perimeter in Ω we have Z  4(n + 1)T ψ + Qψ (νE ) dµE ≤ C PH (E, Ω) s2 PH (Ψs (E), Ψs (Ω)) − PH (E, Ω) + s Ω (3.41) for any s ∈ [−δ, δ]. The proof of Theorem 3.1 when ∂E ∩ Ω is a C ∞ -smooth hypersurface can be found in [15]. The proof for a set with finite H-perimeter will appear elsewhere. 3.2. H-harmonicity of ϕ. Let E ⊂ Hn be a set with locally finite H-perimeter in Hn . Assume that 0 ∈ Hn is a point of the H-reduced boundary of E, 0 ∈ ∂ ∗ E, with νE (0) = (1, 0, . . . , 0) ∈ R2n , and that E is H-perimeter minimizing in a neighborhood of 0, in the sense of Definition 1.1. Let (Eh )h∈N be the sequence of rescaled sets introduced in Section 2.1. We can assume that each set Eh is H-perimeter minimizing in the cube QR = {(z, t) ∈ Hn : |xi |, |yi |, |t|2 < R, i = 1, . . . , n}, for some large R > 0. Let (ϕh )h∈N be the sequence of L-intrinsic Lipschitz functions satisfying (2.22), with 0 < L ≤ 1. We can assume that each ϕh is defined on D1 = {(z, t) ∈ Q1 : x1 = 0}. Finally, let ϕ ∈ WH1,2 (D1 ) be the limit function of a subsequence of (ϕh )h∈N , as in Theorem 2.5. Without loss of generality, we can assume that ϕ is defined on the whole D1 . Let D1/4 = {(z, t) ∈ Q1/4 : x1 = 0}. Theorem 3.2. Let n ≥ 2 and let E be a set with locally finite H-perimeter, as above. Then: i) If E is H-perimeter minimizing in a neighborhood of 0 ∈ Hn , then the function ϕ : D1/4 ⊂ R × Hn−1 → R is independent of the variable y1 of the factor R. 14 R. MONTI ii) If E is strongly H-perimeter minimizing in a neighborhood of 0 ∈ Hn , then the function ϕ is H-harmonic, i.e., it is of class C ∞ and it solves the partial differential equation ∆H ϕ = 0 D1/4 ∩ {y1 = 0}, in (3.42) where ∆H is the Kohn Laplacian (1.6) in Hn−1 . Proof. Let ψ ∈ C ∞ (Hn ) be the generating function of a contact vector field Vψ . We assume that ψ has the following structure. First we assume that we have 1 ψ = α + x1 β + x21 γ, 2 where α, β, γ ∈ C ∞ (Hn ) are smooth functions such that X1 α = X1 β = X1 γ = 0 in the stripe {(z, t) ∈ Hn : |x1 | < 1/4}. (3.43) After a Taylor development in the variable x1 along the flow of X1 , the function ψ has this structure plus a remainder. The functions β, γ are always assumed to satisfy β, γ ∈ Cc∞ (Q1/2 ). (3.44) As far as the function α is concerned, we distinguish two cases, according to the claims i) and ii): i) In this case, we assume also that α ∈ Cc∞ (Q1/2 ). (3.45) ii) In this case, we let α(x1 , y1 , z2 , . . . , zn , t) = Z y1 ϑ(x1 , s, z2 , . . . , zn , t)ds, x1 ∈ R, (3.46) 0 where ϑ ∈ Cc∞ (Q1/2 ) is an arbitrary smooth compactly supported function such that X1 ϑ = 0 in {|x1 | < 1/4}. We consider the sets Eh′ = Φsh (Eh ), where sh > 0 are small numbers that will be fixed later. We can assume that ∂Eh ⊂ {|x1 | < 1/4} for all h ∈ N. In the stripe (3.43), the vector field Vψ has the form Vψ = (Y1 ψ)X1 − (β + x1 γ)Y1 + n X (Yj ψ)Xj − (Xj ψ)Yj − 4ψT. (3.47) j=2 It follows that PH (Ψsh (Eh ), Ψ(Q1 )) = PH (Eh′ , Q1 ) for all large h ∈ N. In case i), each Eh is H-perimeter minimizing in the cube Q1 ; in fact we have ′ Eh ∆Eh ⊂⊂ Q1 . In case ii), each Eh is strongly H-perimeter minimizing in the cube MINIMAL SURFACES AND HARMONIC FUNCTIONS 15 Q1 ; in fact, we have Eh′ ∆Eh ∩ Q̄1 ⊂ Q̄Y1 1 ,+ . In both cases, by Theorem 3.1 the minimality condition PH (Eh , Q1 ) ≤ PH (Eh′ , Q1 ) gives Z n o ′ 0 ≤ PH (Eh , Q1 ) − PH (Eh , Q1 ) = −sh 4(n + 1)T ψ + Qψ (νEh ) dµEh + O(s2h ), Q1 O(s2h )/s2h where is bounded by a constant independent of h. We fix sh > 0 such that sh = o(ηh ) as h → ∞, where ηh > 0 is the excess (2.21), and we obtain Z n o 1 4(n + 1)T ψ + Qψ (νEh ) dµEh + o(1), 0≤− ηh Q where o(1) is infinitesimal as h → ∞. Replacing ψ with −ψ and using the identity Q−ψ (νEh ) = −Qψ (νEh ), we also have the opposite inequality. We therefore deduce that Z n o 1 lim 4(n + 1)T ψ + Qψ (νEh ) dµEh = 0. (3.48) h→∞ ηh Q 1 Notice that the excess ηh in (2.21) can be equivalently defined using homogeneous cubes in place of balls. From now on, we let D = D1 . Let Eϕh ⊂ Hn be the intrinsic epigraph of ϕh , as in (2.8). Let gr(ϕh ) be the intrinsic graph of ϕh over D, as in (2.9). With a slightly abuse of notation, for any h ∈ N let Dh ⊂ D be the set of points w ∈ D such that (2.30) and (2.31) hold. By (2.22), (2.31), and (2.20) we have Z n Z n o o 4(n + 1)T ψ + Qψ (νEh ) dµEh + O(ηh2 ) 4(n + 1)T ψ+Qψ (νEh ) dµEh = Q1 Q1 ∩gr(ϕh ) = Z Q1 ∩gr(ϕh ) = Z n Dh n o 4(n + 1)T ψ + Qψ (νEϕh ) dµEϕh + O(ηh2 ) op 4(n + 1)T ψ + Qψ (νEϕh ) 1 + |∇ϕh ϕh (w)|2 dw + O(ηh2 ), where νEϕh is the vector in (2.30) and the bracket {. . .} in the last line is evaluated at w ∗ ϕh (w). By (2.22), we have L 2n (D \ Dh ) = O(ηh2 ), and so we deduce that Z n o 1 4(n + 1)T ψ(w ∗ ϕh (w)) + Qψ (νEϕh (w ∗ ϕh (w))) dw = 0. (3.49) lim h→∞ ηh D We compute the limit in (3.49). We start from the integral of T ψ(w ∗ ϕh (w)). The sequence (ϕh )h∈N is converging to 0 uniformly. We omit details of the proof of this fact. Then we can assume that kϕh k∞ < 1/4 and thus, by (3.43), we have X1 T α = T X1 α = 0. This implies that T α(w ∗ ϕh (w)) = T α(w) = αt (w), where we are using the notation αt = ∂α/∂t. The same holds for β and γ. Thus we have, for any w ∈ D, 1 T ψ(w ∗ ϕh (w)) = αt + ϕh βt + ϕ2h γt , 2 where the right hand-side is evaluated at w. With abuse of notation, here and in the following we denote by ψ, α, β, γ, ϑ also the restriction of the functions to {x1 = 0}. 16 R. MONTI Since we have  supp(α), supp(β), supp(γ) ⊂ (z, t) ∈ Hn : |t|2 < 1/2 , (3.50) then there holds Z αt dw = D Z βt dw = D Z γt dw = 0. D Let ϕ̄h ∈ R be the numbers given by Theorem 2.5. By (2.38), we have Z Z Z 1 ϕh − ϕ̄h lim ϕh βt dw = lim βt dw = ϕ βt dw, h→∞ ηh D h→∞ D ηh D (3.51) and 1 lim h→∞ ηh Z D ϕ2h γt dw = lim h→∞ Z D ϕh − ϕ̄h (ϕh + ϕ̄h )γt dw = 0, ηh (3.52) because ϕh + ϕ̄h is converging to 0 strongly in L2 . From (3.51) and (3.52), we deduce that Z Z 1 lim ϕ βt dw. (3.53) 4(n + 1)T ψ(w ∗ ϕh (w))dw = 4(n + 1) h→∞ ηh D D We compute the limit of the integral of Qψ (νEϕh ) in (3.49). Letting νEϕh = (νX1 , . . . , νXn , νY1 , . . . , νYn ) ∈ S2n , we isolate in (3.40) the terms containing νX1 . Namely, we have Qψ (νEϕh ) = 2 (X1 Y1 ψ)νX 1 + n X (Xj Y1 ψ + X1 Yj ψ)νX1 νXj n X (Yj Y1 ψ − X1 Xj ψ)νX1 νYj j=2 + (3.54) j=1 + Eψ (νEϕh ), where Eψ (νEϕh ) is a quadratic form that does not contain νX1 . Inserting into formula (3.54) the derivatives X1 Y1 ψ = Y1 X1 ψ − 4T ψ 1 2  = Y1 β + x1 Y1 γ − 4 αt + x1 βt + x1 γt , 2 1 2 Xj Y1 ψ = Y1 Xj α + x1 Y1 Xj β + x1 Y1 Xj γ, j ≥ 2, 2 X1 Yj ψ = Yj β + x1 Yj γ, j ≥ 2,  1 Yj Y1 ψ = Yj Y1 α + x1 Yj Y1 β + x21 Yj Y1 γ, 2 j ≥ 1, (3.55) MINIMAL SURFACES AND HARMONIC FUNCTIONS 17 we obtain 1 2 o 2 Qψ (νEϕh ) = Y1 β + x1 Y1 γ − 4 αt + x1 βt + x1 γt νX1 2 n n o X 1 + Y1 Xj α + x1 Y1 Xj β + x21 Y1 Xj γ + Yj β + x1 Yj γ νX1 νXj 2 j=2 n + n n X j=1  (3.56) o 1 Yj Y1 α + x1 Yj Y1 β + x21 Yj Y1 γ − Xj β − x1 Xj γ νX1 νYj 2 + Eψ (νEϕh ), where, by (2.18) and (2.17), we have Bϕh 1 , νY1 = − p , ν X1 = p ϕ 2 1 + |∇ h ϕh | 1 + |∇ϕh ϕh |2 Zj ϕ h νZj = − p , Z ∈ {X, Y }, j ≥ 2. 1 + |∇ϕh ϕh |2 (3.57) Above, Bϕh is the Burgers’ operator. In particular, since each ϕh is intrinsic LLipschitz with 0 < L ≤ 1 we can assume that suph∈N k∇ϕh ϕh k∞ < ∞ and thus there exists an absolute constant C > 0 such that |Eψ (νEϕh )| ≤ C|∇ϕh ϕh |2 . (3.58) So, from (2.32) and (2.33) we have 1 lim h→∞ ηh Z D |Eψ (νEϕh (w ∗ ϕh (w))|dw = 0. In other words, the limit (3.49) of the integral of Eψ in (3.49) vanishes. We compute the limit of the integral of the first three lines in (3.56), separately. By (3.43), we have X1 Y1 β = Y1 X1 β − 4T β = −4T β and thus Y1 β(w ∗ ϕh (w)) = βy1 (w) − 4ϕh (w)βt (w). Similarly, there holds Y1 γ(w ∗ ϕh (w)) = γy1 (w) − 4ϕh (w)γt (w). The limit of the integral of terms in the first line of (3.56) containing x21 vanishes, by a computation analogous to (3.52). Moreover, by (2.32), (2.33), and (3.57) the 2 function νX may be replaced by 1. Thus, the limit of the integral of the first line in 1 18 R. MONTI (3.56) is 1 lim h→∞ ηh 1 2 o 2 βy1 − 4ϕh βt + ϕh (γy1 − 4ϕh γt ) − 4 αt + ϕh βt + ϕh γt νX1 dw = 2 B Z ϕh − ϕ̄h dw (γy1 − 8βt ) = lim h→∞ D ηh Z = (γy1 − 8βt )ϕ dw. Z n  D (3.59) We used Theorem 2.5. We compute the limit of the integral of the second line in (3.56). In this case, the limit of the integral of terms containing x1 or x21 vanishes. So we have: Z X n n o 1 2 1 Y1 Xj α + ϕh Y1 Xj β + ϕh Y1 Xj γ + Yj β + ϕh Yj γ νX1 νXj dw = lim h→∞ ηh D 2 j=2 Z X n X j ϕh (Y1 Xj α + Yj β) = − lim dw h→∞ D ηh j=2 Z X n   ∂ Xj α + Yj β Xj ϕ dw. =− D j=2 ∂y1 (3.60) We used Theorem 2.5. Finally, we compute the limit of the integral of the third line in (3.56): Z X n n o 1 1 lim Yj Y1 α + ϕh Yj Y1 β + ϕ2h Yj Y1 γ − Xj β − ϕh Xj γ νX1 νYj dw = h→∞ ηh D 2 j=1 Z n n X  Bϕh 2 Yj ϕ h o dw Y α+ Yj Y1 α − X j β = − lim (3.61) h→∞ D ηh 1 ηh j=2 Z n n   o X 2 ∂y1 ϕ Y1 α + =− Yj Y1 α − Xj β Yj ϕ dw. D j=2 We used Theorem 2.5. Putting together (3.53), (3.59), (3.60), and (3.61), we obtain: Z n
4(n + 1)βt + γy1 − 8βt ϕ − ∂y1 ϕY12 α− D n  (3.62)    o X ∂ Xj α + Yj β Xj ϕ − Yj Y1 α − Xj β Yj ϕ dw = 0. − ∂y1 j=2 When α = β = 0, this equation reads Z Z γϕy1 dw, γy1 ϕ dw = − 0= D D MINIMAL SURFACES AND HARMONIC FUNCTIONS 19 for any test function γ ∈ Cc∞ (D1/2 ). This implies that ϕ is independent of y1 . This proves claim i) of the theorem. When α = γ = 0, equation (3.62) reads Z n n o X 0= 4(n − 1)βt ϕ + Xj βYj ϕ − Yj βXj ϕ dw D = j=2 Z n 4(n − 1)βt − D n X o Yj Xj β − Xj Yj β ϕ dw, j=2 for any β ∈ Cc∞ (D1/2 ). This information is empty. In fact, the equation is satisfied for any test function because Yj Xj − Xj Yj = [Yj , Xj ] = 4T . When β = γ = 0, by Y1 ϕ = 0 and (3.46) equation (3.62) reads Z n n o X 2 0= Y1 αY1 ϕ + Y1 Xj αXj ϕ + Yj Y1 αYj ϕ dw D j=2 =− Z D =− Z D n ∂α X 2 (X ϕ + Yj2 ϕ)dw ∂y1 j=2 j ϑ ∆H ϕ dw, for any test function ϑ ∈ Cc∞ (D1/2 ). Then the function ϕ ∈ WH1,2 (D) solves the partial differential equation ∆H ϕ = 0 in the weak sense in D1/4 ∩ {y1 = 0}. It follows that ϕ is smooth, by hypoellipticity, and ϕ is a classical solution. This proves claim ii).  References [1] L. Ambrosio, M. Scienza, Locality of the perimeter in Carnot groups and chain rule. Ann. Mat. Pura Appl. (4) 189 (2010), no. 4, 661–678. [2] L. Ambrosio, F. Serra Cassano & D. Vittone, Intrinsic regular hypersurfaces in Heisenberg groups. J. Geom. Anal., 16 (2006), no. 2, 187–232. [3] F. Bigolin, L. Caravenna & F. Serra Cassano, Intrinsic Lipschitz graphs in the Heisenberg group and continuous solutions of a balance equation. Preprint 2013. [4] L. Capogna, G. Citti, M. Manfredini, Regularity of non-characteristic minimal graphs in the Heisenberg group H1 . Indiana Univ. Math. J. 58 (2009), no. 5, 2115–2160. , Smoothness of Lipschitz minimal intrinsic graphs in Heisenberg groups Hn , n > 1. J. [5] Reine Angew. Math. 648 (2010), 75–110. [6] J.-H. Cheng, J.-F. Hwang, P. Yang, Regularity of C 1 smooth surfaces with prescribed p-mean curvature in the Heisenberg group. Math. Ann. 344 (2009), no. 1, 1–35. [7] J.-H. Cheng, J.-F. Hwang, A. Malchiodi, P. Yang, Minimal surfaces in pseudohermitian geometry. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4 (2005), 129–177. [8] G. Citti, M. Manfredini, A. Pinamonti, F. Serra Cassano, Poincaré type inequality for intrinsic Lipschitz continuous vector fields in the Heisenberg group, preprint 2013. 20 R. MONTI [9] B. Franchi, R. Serapioni, F. Serra Cassano, Rectifiability and perimeter in the Heisenberg group. Math. Ann. 321 (2001), 479–531. [10] B. Franchi, R. Serapioni & F. Serra Cassano, Regular submanifolds, graphs and area formula in Heisenberg groups, Adv. Math. 211 (2007), no. 1, 152–203. [11] N. Garofalo, D.-M. Nhieu, Isoperimetric and Sobolev inequalities for Carnot–Carathéodory spaces and the existence of minimal surfaces. Comm. Pure Appl. Math. 49 (1996), 1081–1144. [12] A. Korányi, H. M. Reimann. Foundations for the theory of quasiconformal mappings on the Heisenberg group. Adv. Math. 111 (1995), no. 1, 1–87. [13] F. Maggi, Sets of finite perimeter and geometric variational problems. An introduction to geometric measure theory. Cambridge Studies in Advanced Mathematics, 135. Cambridge University Press, Cambridge, 2012. xx+454 pp. [14] R. Monti, Lipschitz approximation of H-perimeter minimizing boundaries, Calc. Var. Partial Differential Equations 50 (2014), no. 1-2, 171–198. [15] R. Monti, Isoperimetric problem and minimal surfaces in the Heisenberg group, Lecture notes of the ERC School Geometric Measure Theory and Real Analysis, 57–130, Edizioni SNS Pisa, 2014-2015. [16] R. Monti, D. Vittone, Sets with finite H-perimeter and controlled normal. Math. Z. 270 (2012), no. 1-2, 351–367. [17] R. Monti, D. Vittone, Height estimate and slicing formulas in the Heisenberg group. Preprint 2014. [18] F. Serra Cassano, D. Vittone, Graphs of bounded variation, existence and local boundedness of non-parametric minimal surfaces in the Heisenberg group, Adv. Calc. Var. 7 (2014), no. 4, 409–492. E-mail address: monti@math.unipd.it (Monti) Università di Padova, Dipartimento di Matematica, via Trieste 63, 35121 Padova, Italy