Green Functions for Sub-Laplacian on Half Spaces of the Heisenberg Group

Published Online December 2011 in MECS

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Green functions play an important role in the study of partial differential equations. Itô [1] studied the Green type kernels on the half space in ⁿ . Wong [2] gave a formula of the Green function of a degenerate elliptic operator on ² related to Heisenberg group. The aim of this paper is to discuss the Green functions for sub-Laplacian on non-characteristic and characteristic half spaces of Heisenberg group, and deduce the integral formulas for the Dirichlet problems of the sub-Laplace equations.

We recall some notations and known results about the Heisenberg group ^{H n} . The group ^{H n} is a two step nilponent Lie group ^(D 2 n ⁺» with group law given by

⁵ °⁵ = ⁽ x , y , t И x , ^y , t ) = ( x + x , y + y , t + 1 + 2( x • y - x • y ))

in П² n ^{+ 1}, where 5 = ( z , t ) = ( x, y , t ), ^ = ( z , t ) = ( x , y , t ) e □ ^{2 n + 1}, x • y denote the usual inner product inD ⁿ .

The sub-Laplace operator in ^Hn is

* Corresponding author.

n

∆    = ∑ ( X ² + Y ²)

Hn  i=1 ii, where X = ∂/∂x +2y ⋅∂/∂t, Y = ∂/∂y -2x ⋅∂/∂t  (for i= 1,...,n), T= ∂/∂t. The generalized gradient on Hn is ∇=(X ,  ,X ,Y ,  ,Y ) . Hn allows to a family of dilations  {δ: λ> 0} defined by

δ ( ξ ) = ( λz , λ ² t ) , λ > 0. The Jacobi determinant of δ is λ^Q , where Q =2 n +2 is the homogeneous dimension of Hⁿ .

It is well known that the fundamental solution of sub-Laplace operator ^{-∆ Hn} at the origin is Γ( ξ )= c Q / d ( ξ ) ^{Q -2},                                                                                      (1)

where c _Q >  0 is a suitable constant, d ( ξ ) = ( z ⁴ + t ²)^{1 4}

(see [3]). If ξ ∈ Hⁿ , then ξ ^-1 =- ξ by the group law. Let d ( ξ , ξ ) = d ( ξ ^-1 ξ ) denote the distance between ξ and ξ in Hⁿ (see [4]). By translation and (1), the fundamental solution of -∆ with pole at ξ can be written as

Γ ( ξ , ξ ) =Γ ( ξ ^{- 1} ξ ) = c Q / d ( ξ ^{- 1} ξ ) ^{Q - 2}.                                                          (2)

It is easy to know that

∇ Γ ( ξ , ξ ) =- ( Q - 2) c Q d ( ξ ^{- 1} ξ )^{1 - Q} ⋅∇ d ( ξ ^{- 1} ξ ).                                                (3)

Ω ={ ξ ∈ Hⁿ d ( ξ )< r }

In general, we let ^r                       and

∂Ω = { ξ ∈ H ⁿ d ( ξ ) = r } denote the Heisenberg ball and sphere centered at the origin with radius r , respectively.

Note that H ⁿ is a non-abelian Lie group, whose stratification such that there exist characteristic points on the boundary of the smooth domain of Hⁿ . Thus it is difficult for the Dirichlet problem of sub-Laplace equation to have good regularity and we have to introduce non- characteristic domain in the following. Let φ( z , t ) be the boundary function in ω⊂ H ⁿ . If ∇ n Φ ( z , t ) ≠ 0, then ω is called the non-characteristic domain in Hⁿ . If there exist some points on the boundary such that ∇ n Φ ( z , t ) = 0 , then Ω is called the characteristic domain and these points are named characteristic points. In this paper, by simple calculation we know that

Π ₁ = { ξ = ( x ₁,   , x_n , y ₁,   , y_n , t ) ∈ Hⁿ x ₁ > 0}

(by taking Φ= x ) and

Π ₀ = ( ξ = ( x ₁,   , x_n , y ₁, , y_n , t ) ∈ Hⁿ t > 0)

(by taking Φ= t ) are non-characteristic half space and characteristic half space in H ⁿ , respectively.

The paper is organized as follows: In Section II, we give the Green function of sub-Laplacian on the general domain of H . In Section Ш and Section IV, we give the Green functions of sub-Laplacian on the non-characteristic half space and characteristic half space of H by the results in Section II, respectively. We also show the integral formulas for solutions of the Dirichlet problem of sub-Laplacian. By the discussion in this paper, we can see that the Green functions in these two half spaces are different in nature.

By [5], one has A_H n = div ( A V ), where

0   2 y

A = 0

1 ² y

I n   - 2 х

-2х  41 z2, and In denotes the identity matrix in □ n. The usual gradient in □ 2n+1

is denoted by V = ( d / dx , d / dy , d / dt ) ^T .

Let Q с H be a domain with smooth boundary. By [5], for v ^e C ⁽ H ),

I (u A n V - v A u ) d ^ =1* ( uA V v • n - vA V u • n ) dH ₂ ,(4) where n denotes the unit outward normal vector of

1 q     H          H       ~ JdQ dQ, dH2n denotes the 2n -dimensional Hausdorff measure in □ 2n+1. Assume that u e C” (Q) and let Qr (n) с Q be the Heisenberg ball centered at n with radius r. Letting v = 1 in (4), we have

£aH„ud^ ^ AVu• ndH2n .(5)

We substitute v with Г ( <' ¹ o ^ ) in Q\ Q_r ,

[ ГА ud ^ = r[ A V u • ndH + г[ A V u • ndH        -f uA VГ • ndH. -[   uA VГ • ndH. . (6

J Q \ Q r     H'    -      JdQ               2 n     JdQr               2 n              JdQ                2 n   JdQ r                2 nv

Note that n = V d / |V d | on dQ and n =-V d /|V d | on dQ_r . By (2.25) , (2.26) in [5], we get lim [   uA VГ • ndH. = - u ( n )

r ^0+JdQ r and lim Г[  AVu • ndH. = 0.

r ^0+ JdQr

Thus, let r ^ 0⁺ in (6) and then

u ( n ) = J ( uA Vr • n - Г A V u • n ) dH₂„

+£ГАЯ„ud^n eQ .(7)

In particular, if A _Hnu = 0 in Q , then we have the following basic integration formulas,

u(n) = [ (uAVГ • n - ГAVu • n)dH2n(8)

J dQ

Let h ( ^ , n ) belong to C ” ( Q ) with respect to ^ and satisfies -A _Hn h ( ^ ) = 0 in Q . For u ( ^ ) e C ” ( Q ), we take v = h in (4), then

J (uAVh • n - hAVu • n ) dH2„ + J hA^n ud^ = 0 .(9)

Let us add (8) and (9) and set

G (^,n) = Г(п-1 =)) + h (§,n)(10)

and G = 0on dQ , then h ( ^ , n ) = -Г( п ' °^ ), so

u(n) = J^u(^)AVG• ndH2n.(11)

v dQ

We say that G = G ( ^ , n ) is the Green function on Q .

• 3. Green function on non-characteristic half space

We consider the Dirichlet problem of sub -Laplace equation on the non-characteristic half space П in Hⁿ :

• -A„.u(k) = 0, k eRi;

u( ( k ) = ф(£, t ),   k е { X , = 0};

lim u ( k ) = 0,

I         k i R»

where k = (x,y,t) = (x,,---,Xn,y,,• • •,Уп,t) = (X,,k,t), k = (X2,   , Xn , X,"**, yn ).

Let      § , n еП,     , it is easy to know that the symmetric point of

П = ^{( x} , ^y , b =⁽ X , X ,---, x n , У ,—, ^y n , t ) =⁽^i, n ', ")

is n = (—- X i ,П, ")• Selecting h (k,n) = —F(( n *)^{—1 o} k ) in (10), the Green function on half space is

G ( k , n ) = Г( п⁴ °k )-F(( n *) "‘ o k ).

Because of n ^ П, h ( k , n ) satisfies —A _n h = 0 with respect to k еП . When k is over П,, we have G ( k , n ) = F( n° k ) — F(( n *)-‘ ° k ) = 0 .

Therefore, by (10), for any n еП , n = ( X , n ',"), X >  0, it follows

u(n) = J o^(k,t)AVG• ndH’,, .(,3)

Next, we will calculate A V G • n . To do this, we note that for k , n еП_г , ^d(k,n) = ^d(n "‘ ° ^{k )}

nn

={[(x1— x )2+X( x—x) 4E( y—y.) ]2 i=2

nn

+(t— " + 2^ .X,. У — 2^ Xy )2} 1 /; z=,

^ d ^ ^ i ^ ) I X = 0 = ,/4 d (k, n )^—3{2[ ( x , — X , ) 2

+ E ^{( X} - ^{— X} - -^{)2 +} X ⁽ y ^— y ⁾² ] • ^{(2 X} , ^{— 2} y ) ⁺ z = 2                        z = ,

dx.

nn

2(t — t + 2XX^i — 2Xxy)(—2y)} x, = 0 z=1

nn

=— dx^nb—3{[x+Ё(x — X)1+Ё(y— y)2] • ^"i z=2

nn

+(t—"+2X 5ciy i—2X xiyi) -y,} i=1

• □ — M , / d ' ( k , n )³;

d d ( k , n )l         1 о/ ~ ^- nV -xl

---— n =--------- •* 2( t — t + 27 .x. y. — 2> xy. )

d t    ^{|x i} = 0     4 d ( k , n ) 3               ^ 1 '       ^{X        1 X i} = 0

= ^{1/2 d} '( ^{k ,} n ) ^{— 3(} t ^— "  + ² X ^{^t} i У i ^{— 2} X xy) г = 1                 z = 2

• □ M 2 / d ’ ( k , n )³;

d (k,n*) = d (( n *)^—1 o k )

nn

= <[ ( x , + x ) + j ( x i - xi ) + j ( у , ^{- y} ) ^]

i=2

+(t-1 -2.y, + i^Xyi -2 jj xiyi )2 }i-4, ^d^S’^Lll. =0 =     1     {2[(x, + X,)2

t,            d X i         4 d jj )³

j (xi - x) + j (y - y)2 ] ■ (2xi + 2x) + i=2

2t-1 - 2 x y, + 2 j xy - 2 j x y (-2 y, )}| x,=о i=2

= d'(jj*)-3{[ X2 + jj (Xi - XXi )2 + jj (y - Уi )2] ■ X - (1 - c - 2 -xi У, + 2 j xi-yi- - 21хуi) yi} i=2                     i=i                                                         i=2

□ n , / d 'jn *)³;

^{( d d}( Sjh / ^{d t} )| X , = 0

=i/4d(j, J)-3 ■2( 1 -t -2x,y, + 2 jxiy -2 jуу)|. =0 i=2

= I/2 d 'jj) "3( 1 -1 - 2 X у, + 2 jxi.y-2jj xy) □ N2/d jy . , i=2

where M , = ^{[ x + j ( x i - x ) ² + j ( y ^{- y} i ) ^{2 ]} ■ x ,

+(1 - c + 2 j xy t - 2 j x Ji) yi}, i=i

Ni =[-xi2 + jj(Xi - Xi)2 + jj(ji - .yi)2 ] ■ Xi i=2

• -(t-1 - 2 .xi y, + ijjxy - jj/J) J , i=2

M 2 = i/2( 1 -1 + 2jj;xiyi- 2]Гх,у), i=i

N2 = i/2( 1 -1 - 2 X y, + 2jj Х,у, - ijPxiyi), i=2

d' jj )and d '( j , j *) denotes d jj ) and d(j,n ) at . = 0, respectively.

By the expression

• _zd d     d d d d     d d    d d     d d    . ,2 d d_xT

AVd =(^ + 2 y^T ’^— 2 Х^Г,2 Уд— 2 x^ + 4 z ^7) and(3), we get dx      d 1  dy     d 1     dx     оyd

( A V G ) ■ n | x = 0 = [ A Vr ( j , j ) - A Vr ( j , j *)] ■ n | . = 0 = -( Q - 2) c_Q ( d ' ( j n )^i- Q A V d ( j , j )

• - d '( j , J - Q A V d j j J П = 0

= -( Q -2) с_е [ d'(j, j )^i- Q ( ^dj^- + 2 y , ^ d ^ j ’ ^ ))   - d '( j , n?" Q ( ^{5 d} ( f ^{, n} ) + 2 У , ^8d ( j^J ) )]| x , = 0

dx,             d 1                          dx              81

= - ( Q - 2 ) c_Q [ d '(jj ) ^i- Q ( - M ⁱ ₃ + 2 y,    M ² ₃ ) - d '( j , n *)¹ " Q (--- N 4— + 2 yN^ )]

^V      ' Q ^{V V}     d '( j , j )³         d '( j , j )³                      d'(j’j )³        d '( j ’ j *)³

= ( Q - 2) Cq [ d X7 - 2 - ^Q ( M i — 2 У 1 M 2 )

+ d X^nT ^{- Q} ( N + 2 У 1 N 2 )].

Substituting it into (13), yields

Theorem 1. For any 7 еЦ , the integral formula of solutions for Dirichlet problem (12) is

u ( 7 ) = ( Q - 2 ) Cq • J   ^ ( § ', t ) [ d §7 ) ^{- 2 - Q} ( M 1 — 2 У 1 M 2 ) + d X§7T" ^Q ( N + 2 y 1 N 2 )] d § dt , where   M 1 , M ₂, N 1 , N 2    are

X 1 = 0

defined as above.

• 4.    Green function on characteristic half space

Consider the Dirichlet problem on П_о in Hⁿ :

-A Hu ( § ) = 0,   § е^П ₀;

u ( § ) = ^ ( § '),   § е { t = 0};

lim u ( § ) = 0, II § 1

where § = ( x_x ,•••, x_n , y ,•••, y_n , t ) = ( § ,t ). The symmetric point of 7 = ( x,y ,^f) = (7,t) еП₀ is 7 7* = ( 7 ', -t) . Let us take h ( § , 7 7 ) = -Г(( 7 *)^-1 о § ) in (10), the Green function on half space is given by

G ( § , 7 ) = Г( 7 -¹ o § )-Г(( 7 *)^-1 o § ).

Because 7 7 * £ П_о, h ( § , 7 7 ) is sub-harmonic with respect to § е П_о. When § is over the hyper-plane t = 0, we have

G ( § , 7 ) = Г( 7 o § )-Г(( 7 *)^-1 o § ) = 0.

Hence, by (10), for any 7 = ( 7 ', t) е П_о, t > 0 , we get

u ( 7 ) = J ₀ ^ ( § ') AVG • ndH 2 n .                                                                     (15)

1 = 0

We will calculate AVG • n . For § , 7 е П_о, we have

d(§,7) = d(7"‘ °§) = n    n       nn

{Ё(xi - xi)2 +Z(y - yi )2]2 + (t- f + 2Z Xyii - 2Z Xyi)2}V4, dd(§,7)/ dXi = i=1                         i=1                                               i=1

nn

1/4 d ( § , 7 )-³{2 j ( x i. - x , )² + £ ( y - y )² • 2( x -;? , )}1 1 = 0

i=1

nn

= d '( § , 7 ) ^- 3 {[ X ( X - ^x?)² + Z ( y - y i )²] • ( X -^ X i )} □ P ( X - X?)/d '( § , 7 )³, i = 1,.-., n ;

i=1

dd (§,7)/ Syi = nn

———{2[ ^ ( x - X, )² + £ ( y i - y i )²]-2( y i -j y i )}| , = ₀

4d (§,7)      i=1

nn

= d '( § , 7 ) -³{[ £ ( X i - X)² + z ( y i - y i -)²] • ( y i - y i )} □ P ( y - yd/d '( § , 7 )³, i = 1,-, n ;

i=1

dd (§,7) 1   _ dt      !'=0

4d (§,7)                 i=1

= 1/2 d '^,n ) ^{- 3}( - t + 2 У Xy - 2 ^y xy) i = 1                  ' = 1

• □    M ' / d '( £л )³;

d(§,^) = d ( П Т* ° £ ) =

- d 4,nT Q A^ d ( £л *)] • A, =0

= -( Q - 2) C q { d '(Ы" Q •

[f (2y 8d(^> -2x 8d<^) + 4 | z p 8d t=о

E       dx,       '   dy,d nn

{[E( x- A +E( y- y.)]2 i=1

+(t + t + itxy-2Ёх.Л)!}". i=1

Therefore dd (§s^^/dx =

—J - . {2lE(X-x)2 + Z(у-y-)2]•2(X-x)}t=о = d'•-У > А1У(X -X)2 + E(у -y'-)2]•(у -y'» 4d(£>n )       '=1                   '=1                                                                   '=1

• □    P ( x - 5c, )/ d '( ^ , n *)³,   ' = 1,-, n ;

ed ( 5 ^{, n ,)/} 8y, =

¹ -v ^CE (( X ' - -X ' )² + ( y. - y a²)] • 2( y - : У ' »1 t =о

4d(9,n )

nn

= d '( ^ , 7 ) - ^{[ E ( x ' - x) ⁺ Z ( y ^-У ) ^]• ( y ^{- y} ) }

'=1

• □    P ( y - y, )/ d '( ^ , n *)³, '■ = 1,-, n ;

  ^{d d(} § , n) d t

  
  

  4 d (^)³

  
  

  n

  ^{2( t + 1 + ² E xj ,

  i = 1

  
  

  - 2 ^ 5 - У)}| t = о

  
  

  = 1 /^{2 d ,(} ^ ’ n * ) ^{- 3 (} t ^{+ 2} E x,y , ^{- 2} E X^y i )

  
  

• □    N ' / d '( ^ , n *)³.

where M' = l^/2( ^- 1 + 2 У xc,y ^- 2 У x^, ), i = 1                ' = 2

N ' = 1/2( t + 2] T ( 5 ' У ' - 5 ' У ' )), i = 1

^P = E ⁽⁽ x ^{- 5} - ^{)2 +} ( y ^{- y} - )²);

d'(^,n),d'(§,n*) denotes d(^,n),d(§,л) at t = 0, respectively. By (3),

( A V G ) • n | t =о = [ A VF( ^ , n ) - A VF( ^ , ₇ ’)] • n 1 1 =о

= -( Q - 2) C q [ d '( ^ , n ) 1- Q A V d (^, n)

+ ( Q - 2) c Q { d ′ ( ξ , η ^*)^{1 - Q} ⋅ [ ∑ ⁿ (2 y i ^{∂ d ( ξ , η *)} - 2 x i ^{∂ d ( ξ , η * )} ) i = 1            ∂ x_i                ∂ y_i

_{+ 4| z} 2 _| ∂ d ( ξη ,^* )_{] t } = 0}

=-(Q-2)cQ[d′(ξ,η)-2-Q⋅(P∑n (2xiyˆi-2yixˆi)+4z2M′)]+ i=1

( Q - 2) c Q [ d ′ ( ξ , η ^*) ^{- 2 - Q} ⋅ ( P ∑ ⁿ (2 x i y ˆ i - 2 y i x ˆ i ) + 4 z ² N ′ )].

i = 1

Using (14), we have

Theorem 2. For any η ∈ Π , the integral formula for Dirichlet problem (14) on Π is

u ( η ) =- ( Q - 2) c_Q ⋅

( P ∑ ⁿ (2 x i y ˆ i - 2 y i x ˆ i ) + 4 z ² M ′ )  ( P ∑ ⁿ (2 x i y ˆ i - 2 y i x ˆ i ) + 4 z ² N ′ )

ϕ ξ ′ (   ^{i = 1} -    ^{i = 1}) d ξ ' where M ′ N ′ P are defined as above.

^∫ t = 0 ^{( )}             d ′ ( ξ , η )^{2 + Q}d ′ ( ξ , η ^*)^{2 + Q ,              ,,}

• 5.    Conclusion

We obtain several kinds of Green functions in different domains of the Heisenberg group, which extend some results in the Euclidean space to the more general case. Theses results are new and very important in the study of partial differential equations.