Báo cáo toán học: "Polar Coordinates on H-type Groups and Applications"

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Trong bài báo này chúng ta xây dựng các tọa độ cực vào các nhóm H-. Khi các ứng dụng, chúng tôi tính toán một cách rõ ràng khối lượng của quả bóng trong ý nghĩa của khoảng cách và liên tục trong các giải pháp cơ bản của p-sub-Laplacian vào nhóm H-loại. Ngoài ra, chúng tôi chứng minh một số kết quả không tồn tại của các giải pháp yếu cho một thoái hóa.

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Vietnam Journal of Mathematics 34:3 (2006) 307–316 9LHWQD P-RXUQDO RI 0$7+(0$7, &6 9$ 67 Polar Coordinates on H-type Groups and Applications* Junqiang Han and Pengcheng Niu Department of Applied Math., Northwestern Polytechnical University Xi’an, Shaanxi, 710072, China Received August 11, 2005 Revised November 14, 2005 Abstract. In this paper we construct polar coordinates on H-type groups. As ap- plications, we explicitly compute the volume of the ball in the sense of the distance and the constant in the fundamental solution of p-sub-Laplacian on the H-type group. Also, we prove some nonexistence results of weak solutions for a degenerate elliptic inequality on the H-type group. 2000 Mathematics Subject Classiﬁcation: 35R45, 35J70. Keywords: H-type group, polar coordinate, nonexistence, degenerate elliptic inequality. 1. Introduction The polar coordinates for the Heisenberg group H1 and Hn were deﬁned by Greiner [8] and D’Ambrosio [3], respectively. Using their introduction as in [3] we can explicitly compute the volume of the Heisenberg ball (see [6]) and the constant in the fundamental solution of Hn (see [4, 5]). In this paper we will construct polar coordinates on H-type groups. In [1], the polar coordinates were given in Carnot groups and groups of H-type, but the expression here is slightly diﬀerent. As an application, we will explicitly calculate the volume of the ball in the sense of the distance and the constant in the fundamental solution of ∗ The project was supported by National Natural Science Foundation of China, Grant No. 10371099.
308 Junqiang Han and Pengcheng Niu p-sub-Laplacian on the H-type groups. Nonexistence results of weak solutions for some degenerate and singular el- liptic, parabolic and hyperbolic inequalities on the Euclidean space Rn have been largely considered, see [13, 14] and their references. The singular sub-Laplace inequality and related evolution inequalities on the Heisenberg group Hn were studied in [3, 6]. In this paper we will discuss the nonexistence of weak solutions for some degenerate elliptic inequality on the H-type groups. We recall some known facts about the H-type group. H-type groups form an interesting class of Carnot groups of step two in connection with hypoellipticity questions. Such groups, which were introduced by Kaplan [9] in 1980, constitute a direct generalization of Heisenberg groups and are more complicated. There has been subsequently a considerable amount of work in the study of such groups. Let G be a Carnot group of step two whose Lie algebra g = V1 ⊕ V2 . Suppose that a scalar product < ·, · > is given on g for which V1 , V2 are orthogonal. With m = dimV1 , k = dimV2 , let X = {X1 , . . . , Xm } and Y = {Y1, . . . , Yk } be a basis of V1 and V2 , respectively. Assume that ξ1 and ξ2 are the projections of ξ ∈ g in V1 and V2 , respectively. The coordinate of ξ1 in the basis {X1 , . . . , Xm } is denoted by x = (x1, . . . , xm ) ∈ Rm ; the coordinate of ξ2 in the basis {Y1 , . . . , Yk } is denoted by y = (y1 , . . . , yk ) ∈ Rk . Deﬁne a linear map J : V2 → End(V1 ): < J (ξ2 )ξ1 , ξ1 >=< ξ2 , [ξ1, ξ1 ] >, ξ1 , ξ1 ∈ V1 , ξ2 ∈ V2 . A Carnot group of step two, G, is said of H-type if for every ξ2 ∈ V2 , with |ξ2| = 1, the map J (ξ2 ) : V1 → V1 is orthogonal (see [9]). As stated in [7], it has k ∂ 1 ∂ Xj = + [ξ, Xj ], Yi , j = 1, . . . , m. (1) ∂xj 2 ∂yi i=1 For a function u on G, we denote the horizontal gradient by Xu = (X1 u, . . . , Xm u) 1 m 2 2 . The sub-Laplacian on the group of H-type G and let |Xu| = j =1 |Xj u| is given by m 2 =− Xj . (2) G j =1 and the p-sub-Laplacian on G is m Xj |Xu|p−2Xj u ∆G,p u = − (3) j =1 for a function u on G. A family of non-isotropic dilations on G is δλ (x, y) = (λx, λ2y), λ > 0, (x, y) ∈ G. (4) The homogeneous dimension of G is Q = m + 2k.
Polar Coordinates on H-type Groups and Applications 309 Let 1 d(x, y) = (|x|4 + 16|y|2) 4 . (5) Then d is a homogeneous norm on G. The open ball of radius R and centered at (0, 0) ∈ G is denoted by BR = {(x, y) ∈ G|d(x, y) < R}. 2 |x| Let ψ = , a direct computation shows d2 |Xd|2 = ψ. (6) As in [3], we need the following concepts. A function u : Ω ⊂ G → R is said to be cylindrical, if u(x, y) = u(|x|, |y|), and in particular, u is said to be radial, if u(x, y) = u(d(x, y)), that is u depends only on d. Let u ∈ C 2 (Ω). If u is radial, then it is easy to check that |Xu|2 = ψ|u |2 (7) and Q−1 Gu =ψ u + u . (8) d The following deﬁnitions are extensions of those introduced in [6]. Deﬁnition 1.1. For R > 0 and 1 < p < ∞ we deﬁne the volume of the ball BR as |Xd|p, |BR |p = (9) BR and the area of spherical surface ∂BR as d |∂BR |p = |BR |p. (10) dR We refer the following proposition to [2]. Proposition 1.1. Let 1 < p < Q and p−1 |x|pd2(p−2) Q−p −1 Cp,Q = (Q + 3p − 4) . (11) 3p+Q p−1 (1 + d4) G 4 The function p−Q Γp = Cp,Q d p−1 (12) is a fundamental solution of (3) with singularity of the identity element (0, 0) ∈ G. Here the integral in (11) is convergent, but it is not computed explicitly. We will give a description of polar coordinates on the H-type group G, and then compute explicitly |BR |p , |∂BR |p and Cp,Q in Sec. 2. In Sec. 3, we study
310 Junqiang Han and Pengcheng Niu some degenerate elliptic inequality on the H-type group. The main technique will be the so called test functions method introduced in [10, 11] and developed in [12]. Roughly speaking, this approach is based on the derivation of suitable a priori bounds of the weak solutions by carefully choosing special test functions and scaling argument. 2 In the sequel we shall use a function ϕ0 ∈ C0 (R) meeting the property 1, if |η| ≤ 1, 0 ≤ ϕ0 ≤ 1 and ϕ0 (η) = (13) 0, if |η| ≥ 2. The quantities |ϕ 0 ( η ) |q |ϕ 0 ( η ) |q dη or dη ϕ0 (η)q−1 ϕ0 (η)q−1 R R where q > 1, are said to be ﬁnite, if there exists a suitable ϕ0 with the property (13) such that the integrals are ﬁnite. Such a function ϕ0 satisfying above hypotheses is called an admissible function. q For q > 1, q = q−1 is the H¨lder exponent relative to q. o 2. Polar Coordinates and Applications Assume Ω = BR2 \B R1 , with 0 ≤ R1 < R2 ≤ +∞, u ∈ L1 (Ω) is a cylin- drical function. To compute Ω u, we consider the change of the variables (x1 , . . . , xm , y1, . . . , yk ) → (ρ, θ, θ1 , . . . , θm−1 , γ1 , . . . , γk−1) deﬁned by  1 1  x1 = ρ(sin θ) 2 cos θ1 ; x2 = ρ(sin θ) 2 sin θ1 cos θ2 ;     x = ρ(sin θ) 1 sin θ sin θ cos θ ; 3  2 1 2 3    ... ... ... ... ... ...     x 1  m−1 = ρ(sin θ) 2 sin θ1 sin θ2 . . . sin θm−2 cos θm−1 ;     x = ρ(sin θ) 1 sin θ sin θ . . . sin θ m−2 sin θm−1 ; 2 m 1 2 (14)  y1 = ρ cos θ cos γ1 ; y2 = ρ cos θ sin γ1 cos γ2 ; 12 12   4 4    y3 = 1 ρ2 cos θ sin γ1 sin γ2 cos γ3 ;   4    ... ... ... ... ... ...      yk−1 = 1 ρ2 cos θ sin γ1 sin γ2 . . . sin γk−2 cos γk−1 ;    4  12 yk = 4 ρ cos θ sin γ1 sin γ2 . . . sin γk−2 sin γk−1 where R1 < ρ < R2, θ ∈ (0, π), θ1, . . . , θm−2 , γ1, . . . , γk−2 ∈ (0, π) and θm−1 , γk−1 ∈ (0, 2π). One easily sees that 1 1 s = |y| = ρ2 | cos θ|. r = |x| = ρ(sin θ) 2 , (15) 4 Using the ordinary spherical coordinates in Rm and Rk leads to dx = rm−1 drdωm , dy = sk−1dsdωk , (16) m−1 m k −1 in R where dωm and dωk denote the Lebesgue measures on S and S in Rk , respectively. From (14) and (15), we have
Polar Coordinates on H-type Groups and Applications 311 12 1 ρ (sin θ)− 2 dρ dθ, dr ds = (17) 4 and then 1 Q−1 m−2 (sin θ) 2 | cos θ|k−1dρ dθ dωm dωk . dx dy = ρ k 4 Therefore the following formula holds π R2 1 Q−1 m−2 1 (sin θ) 2 | cos θ|k−1u ρ(sin θ) 2 , u(r, s) = ωm ωk dθ ρ 4k Ω 0 R1 12 ρ cos θ dρ, 4 where π π π 2π ωm = dθ1 dθ2 . . . dθm−2 dθm−1 0 0 0 0 sinm−2 θ1 . . . sin2 θm−3 sin θm−2 , π π π 2π ωk = dγ1 dγ2 . . . dγk−2 dγk−1 0 0 0 0 sink−2 γ1 . . . sin2 γk−3 sin γk−2 are the Lebesgue measures of the unitary Euclidean spheres in Rm and Rk , respectively. Furthermore, if u is of the form u(x, y) = ψv(d), then π R2 ρ2 sin θ 1 Q−1 m−2 (sin θ) 2 | cos θ|k−1 ψv(d) = ωm ωk dθ ρ v(ρ)dρ 4k ρ2 Ω 0 R1 (19) R2 Q−1 = sm,k ρ v(ρ)dρ, R1 π m 1 (sin θ) 2 | cos θ|k−1dθ. where sm,k = ωω 4k m k 0 Theorem 2.1. We have the following formulae: m+k RQ π2 k p+m (1) |BR |p = B , ; (20) m k k −1 Q 4 2 4 Γ2Γ 2 m+k RQ−1 π2 k p+m (2) |∂BR |p = B , . (21) m k k −1 4 2 4 Γ2Γ 2 Proof. (1) By (9) and (14), |x |p |Xd|p = |BR |p = dp BR BR
312 Junqiang Han and Pengcheng Niu 1 [ρ(sin θ) 2 ]p 1 Q−1 m−2 (sin θ) 2 |cos θ |k−1dρdθdωm dωk = · kρ ρp 4 BR π 11 p+m−2 = ωm ωk · k RQ |cos θ |k−1dθ (sin θ) 2 4Q 0 m k RQ 2π 2π 2 2 = · ·k· Γm Γk 4Q 2 2 π π 2 2 p+m−2 p+m−2 k −1 (sin θ)k−1 dθ (sin θ) (cos θ) dθ + (cos θ) 2 2 0 0 m+k p+ m Γ p+ m Γ k ΓkΓ RQ π2 2 4 4 2 = + p+ m 2Γ k + p+m ΓmΓ k 2Γ k + k −1Q 4 2 2 2 4 2 4 m+k m+k p+ m k RQ RQ Γ Γ π2 π2 k p+m 2 4 = = B , . p+ m ΓmΓ k k ΓmΓ k k −1Q k −1 Q 4 4 2 4 Γ + 2 2 2 4 2 2 (2) From (1.10), the conclusion is obvious. Remark 1. On Heisenberg groups we can analogously obtain |BR |p = 1 2π n+ 2 RQ Γ( n + p ) 2 4 by using the polar coordinates introduced in [3]. QΓ(n)Γ( 1 + n + p ) 2 2 4 −1 Next we compute explicitly Cp,Q in Prpposition 1.1. Theorem 2.2. We have m+k p−1 k m+ p π2 B 2, 4 Q−p −1 Cp,Q = . (22) m k 4k−1 Γ p−1 2Γ2 Proof. By (14), it follows that |x|pd2(p−2) 3p+Q (1 + d4) G 4 p π +∞ p 2(p−2) 1 Q−1 k −1 ρ (sin θ ) 2 · ρ m−2 = ωm ωk dθ ρ (sin θ) 2 |cos θ | dρ 3p+Q k 4 (1 + ρ4 ) 4 0 0 π +∞ ρQ+3p−5 1 m+p−2 |cos θ |k−1 dθ = ωm ωk k (sin θ) dρ 2 3p+Q 4 (1 + ρ4 ) 0 0 4 m+ p m k k 1Γ Γ 2π 2 2π 2 1 2 4 = m· ·k · k 4Γ −4 + 3p + Q Γ2 Γ2 2 k + m+ p 4
Polar Coordinates on H-type Groups and Applications 313 m+k k m+ p π2 B 2, 4 1 = , m Γk Q + 3p − 4 4k−1 Γ 2 2 and so p−1 |x|pd2(p−2) Q−p −1 Cp,Q = (Q + 3p − 4) 3p+Q p−1 (1 + d4) Rm+k 4 m+k p−1 k m+ p π2 B 2, 4 Q−p = . m k 4k−1 Γ p−1 2Γ2 Remark 2. In [15] the fundamental solution of p-sub-Laplacian on the Heisen- p−1 p−Q |z |p d2(p−2) Q−p −1 berg group is Cp,Q d p−1 , where Cp,Q = (Q+3p−4)· dzdt. Hn 3p+Q p−1 (1+d4 ) 4 p−1 Q−p −1 One deduces easily by using the polar coordinates in [3] Cp,Q = · p−1 1 2π n + 2 Γ 2n4 p + () . Especially when p = 2, the constant appears in the fundamental 2+2n+p Γ(n)Γ( )4 solution of the sub-Laplacian in [4]. 3. A Degenerate Elliptic Inequality The target of this section is to deal with the inequality d2 ∆G (au) ≥ |u|q on G\{(0, 0)}, − (23) ψ where a ∈ L∞ (G). Deﬁnition 3.1. Let q ≥ 1. A function u is called a weak solution of (23), if u ∈ Lq (G\{(0, 0)}) and loc |u |q au∆G (d2−Qϕ) dxdy ψϕ dxdy ≤ − (24) dQ G G 2 for any nonnegative ϕ ∈ C0 (G\{(0, 0)}). Theorem 3.1. For any q > 1, (23) has no nontrivial weak solutions. 2 Proof. Let u be a nontrivial weak solution of (23) and ϕ ∈ C0 (G\{(0, 0)}), ϕ ≥ 0. We set F = 2(2 − Q)d < Xd, Xϕ > +d2∆G ϕ. Using (6) and (8), we have 1 F ∆G (d2−Q ϕ) = [2(2 − Q)d < Xd, Xϕ > +d2∆G ϕ] = Q . (25) Q d d By (24), (25) and H¨lder’s inequality, we get o
314 Junqiang Han and Pengcheng Niu |u |q au∆G (d2−Qϕ) dxdy ψϕ dxdy ≤ − dQ G G auF |u| |F | =− dxdy ≤ a dxdy ∞ dQ dQ G G 1 1 |u |q |F |q q q ≤a ψϕ dxdy dxdy , ∞ dQ dQ ψq −1 ϕq −1 G G and therefore |u |q |F |q q q ψϕ dxdy ≤ a dxdy = a ∞ I1 , (26) ∞ dQ dQ ψq −1 ϕq −1 G G q where I1 = G dQ ψq|F |1 ϕq −1 dxdy. − We select the function ϕ by letting ϕ = ϕ(d). Clearly, F becomes Q−1 F = 2(2 − Q)dϕ (d)ψ + d2ψ ϕ (d) + ϕ (d) d = ψ[d2 ϕ (d) + (3 − Q)dϕ (d)]. Hence, we have from (19) |d2ϕ (d) + (3 − Q)dϕ (d)|q I1 = ψ dxdy dQ ϕq −1 G +∞ |ρ2 ϕ (ρ) + (3 − Q)ρϕ (ρ)|q = sm,k dρ. ρϕq −1 0 Letting s = ln ρ and ϕ(s) = ϕ(ρ), leads to +∞ |ϕ (s) + (2 − Q)ϕ (s)|q I1 = sm,k ds. ϕ(s)q −1 −∞ s We perform our choice of ϕ by taking ϕ(s) = ϕ0 ( R ) with ϕ0 as in (13) and obtain 1 s 1 s | R2 ϕ0 ( R ) + (2 − Q) R ϕ0 ( R )|q I1 = sm,k ds s q −1 ϕ0 ( R ) R≤|s|≤2R ϕ0 ( τ ) + (2 − Q)ϕ0 (τ )|q 1−q | R = sm,k R dτ ϕ0 (τ )q −1 1≤|τ |≤2 = sm,k R1−q I2 , (27) where ϕ0 ( τ ) + (2 − Q)ϕ0 (τ )|q | R I2 = dτ. ϕ0(τ )q −1 1≤|τ |≤2 Let ϕ0 be an admissible function. For R > 1, it follows that
Polar Coordinates on H-type Groups and Applications 315 q |ϕ0 ( τ ) | + (Q − 2)|ϕ0(τ )| R I2 ≤ dτ ϕ0 (τ )q −1 1≤|τ |≤2 q |ϕ0 ( τ ) | 2q −1 + ((Q − 2)|ϕ0(τ )|)q R ≤ dτ ϕ0 (τ )q −1 1≤|τ |≤2 2q −1 |ϕ 0 ( τ ) |q |ϕ 0 ( τ ) |q dτ + 2q −1 (Q − 2)q = dτ Rq ϕ0 (τ )q −1 ϕ0 (τ )q −1 1≤|τ |≤2 1≤|τ |≤2 ≤ M < +∞, with M independent of R. Merging (26) into (27) and considering ϕ(x, y) = ϕ(ln d) = ϕ0 ( ln d ), we have R |u |q ψ dxdy ≤ M a q sm,k R1−q = CR1−q . ∞ Q −R ≤d≤eR d e Letting R → +∞, it induces u = 0. This contradiction completes the proof. Remark 3. Arguing as in [3], we can treat the evolution inequalities d2 ≥ |u | q on G\{(0, 0)} × (0, +∞), ut − ∆G (au) ψ on G\{(0, 0)}, u(x, y, 0) = u0 (x, y) where a ∈ R, and  d2 ≥ |u | q  utt − on G\{(0, 0)} × (0, +∞), ∆G (au)  ψ on G\{(0, 0)}, u(x, y, 0) = u0(x, y)   on G\{(0, 0)}, ut(x, y, 0) = u1 (x, y) where a ∈ L∞ (G × [0, +∞)), in the setting of the H-type group. References 1. Z. M. Balogh and J. T. Tyson, Polar coordinates in Carnot groups, Math. Z. 241 (2002) 697–730. 2. L. Capogna, D. Danielli, and N. Garofalo, Capacitary estimates and the local behavior of solutions of nonlinear subelliptic equations, Amer. J. Math. 118 (1997) 1153–1196. 3. L. D’Ambrosio, Critical degenerate inequalities on the Heisenberg group, Manus- cripta Math. 106 (2001) 519–536. 4. G. B. Folland, A fundamental solution for a subelliptic operator, Bull. Amer. Math. Soc. 79 (1973) 373–376. ¯ 5. G. B. Folland and E. M. Stein, Estimates for the ∂b complex and analysis on the Heisenberg group, Comm. Pure. Appl. Math. 27 (1974) 429–522.
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