Lp Polyharmonic Dirichlet Problems in the Upper Half Plane

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http://dx.doi.org/10.4236/apm.2015.514077

L

p

Polyharmonic Dirichlet Problems in the

Upper Half Plane

Kanda Pan

Department of Mathematics, Jinan University, Guangzhou, China

Received 11 November 2015; accepted 8 December 2015; published 11 December 2015

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

Abstract

In this article, a class of Dirichlet problem with Lp_{boundary data for polyharmonic function in the}

upper half plane is mainly investigated. By introducing a sequence of kernel functions called higher order Poisson kernels and a hierarchy of integral operators called higher order Pompeiu operators, we obtain a main result on integral representation solution as well as the uniqueness of the polyharmonic Dirichlet problem under a certain estimate.

Keywords

Dirichlet Problem, Polyharmonic Function, Higher Order Poisson Kernels, Higher Order Pompeiu Operators, Non-Tangential Maximal Function, Uniqueness

1. Introduction

Usually harmonic functions are defined by Laplace operator ∆ = ∂ ∂4 _z _z, where 1

2

z

x y

i

 _∂ _∂ 

∂ = _ + _

∂ ∂

  is the

Cauchy-Riemann operator and 1

2

z

x y

i

 _∂ _∂ 

∂ = _ − _

∂ ∂

  is the adjoint operator of C-R operator. By iterating the

Laplace operator, one can define the so-called polyharmonic functions by ∆n

(

n≥2

)

[1]. In [2], Goursat

obtained his decomposition formula, in [3], Vekua developed one method to construct an approximative solution

of the biharmonic Dirichlet problem in a simply connected domain. In recent years, the study of explicit solution

of BVPS (boundary value problems) has undergone a new phase of development [4]-[6]. There are Dirichlet,

Neumann and Robin boundary value problems in regular domain (in the disc [4]; and in the upper half plane [5])

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of them give a certain estimate about the uniqueness of the solution. Thus, the purpose of this article is devoted

to solving the unique solution of the following polyharmonic Dirichlet problems (for short, PHD) for Lp data

in the upper half plane, H, i.e.

( )

0 in H,

on ,

M

n

j j

P u u f

u L

∆ = 

∆ = 

 _∈



 

(1.1)

with

( )

_{( )} _{( )}

1

=0

M P P

n

j

L L

j

u C f

−

≤

∑

  , where ∆ is the Laplacian, and  is the real axis,

( )

P j

f ∈L  for some

suitable p>1, n∈, 0≤ <j n, M

( )

u is the non-tangential maximal function of u, which is defined by

( )( )

( )0

( )

0 0

M sup , for H,

X x

F x F X x

γ ∈Γ

= ∈∂

where Γγ

( )

x0 is the non-tangential approach region, viz.,

( )

x0

{

X H : X x0 dist X

(

, H

)

}

γ γ

Γ = ∈ − < ∂

where γ >1.

It is clear that all the boundary data in BVPs (1.1) are non-tangential.

2. Preliminary and Some Lemmas

Definition 2.1. If a real valued function f ∈C2n

( )

D satisfies the equation

(

∂ ∂z z

)

n f =0, in D, then f is called

an n-harmonic function in D, concisely, a polyharmonic function.

We use the notation Har Dn

( )

denoting the set of polyharmonic function of order n in D. Especially,

( )

n

Har D is the set of all harmonic functions in D.

Lemma 2.2. [7] Let D be a simply connected (bounded or unbounded) domain in the complex plane with

smooth boundary ∂D . If f∈Har Dn

( )

, then for any z0∈D , there exist functions 1,0

( )

j

j z

f ∈H D ,

0,1, , 1

j=  n− such that

( )

1

(

) ( )

0 0

2 , ,

n

j j j

f z z z f z z D

−

=

 

= ℜ_ − _ ∈



∑

 (2.1)

where ℜ denotes the real part. The above decomposition expression of f is unique in the sense of the equi-

valence relation ~, more precisely, ~j for fj.

Corollary 2.3. If the sequence of functions

{ }

fn defined in D satisfy (1) f1∈Har D1

( )

;

(2)

(

∂ ∂z z

)

fn= fn−1 in D for n>1.

Then fn∈Har Dn

( )

for n>1, and

1

, 1, 1, 1 1

zfn j j fn j j n

− − −

∂ = ≤ ≤ − (2.2)

where fn j, is the analytic jth decomposition component of the n-harmonic function fn. It must be noted that

(2.2) holds in the sense of the equivalence relation ~.

Definition 2.4. A sequence of real-valued functions of two variables

{

( )

}

1

,

n _n

G z t ∞₌ defined on H× is

called a sequence of higher order Poisson kernels, more precisely, G_n

( )

z t, is called the nth order Poisson kernel, if they satisfy the following conditions.

(1) For all n∈,G_n

( )

z t, ∈C

(

H×

)

; Gn

( )

.,t ∈C2n

( )

H with any fixed t∈; and

( )

,. , 1

p n

G z ∈L p> ,

with any fixed z∈H, and the non-tangential boundary value

( )

,

lim n , n ,

z s z s

G z t G s t

→ ∈ ∈

=

 H

(3)

(2) 1

( )

1 1 1

, 2

G z t

i t z t z

 

= _ − _

− −

  and G i tn

( )

, =0,n≥2 and t∈, and for any n∈

( )

,

n

M G z t

t z

≤ ′ −

uniformly on Dc× ∈

{

t :t >T

}

whenever z′∈Dc, where Dc is any compact set in H, M, T are positive constants depending only on Dc and n;

(3)

(

∂ ∂z z

) ( )

G z t1 , =0 and ∆Gn

( )

z t, =Gn−1

( )

z t, for n>1;

(4) 1

( ) ( )

( )

,

1

lim , d

π

z s z G z t γ t t γ s

+∞ −∞

→ ∈H

∫

= , a.e., for any

( )

, 1

p

L P

γ ∈  ≥ ;

(5)

( ) ( )

,

1

lim , d 0

π n

z s z G z t γ t t

+∞ −∞

→ ∈H

∫

= , for any

( )

, 1, 2

P

L P n

γ∈ _ ≥ ≥ ,

where all limits are non-tangential.

Definition 2.5. Let D be a simply connected (bounded or unbounded) domain in the plane with smooth boundary ∂D, and H D

( )

denote the set of all analytic functions in D. If f is a continuous function defined on

D× ∂D satisfying f

( )

.,t ∈H D

( )

for any fixed t∈∂D, and f z

( )

,. ∈Lp

( )

∂D , P≥1, for any fixed z∈D, then f is called H×LP on D× ∂D and this is noted by f ∈

(

H×LP

)

(

D× ∂D

)

.

Lemma 2.6. [8] If Gn

( )

z t, n₁

∞

= is a sequence of higher order Poisson kernels defined on H×, i.e.,

( )

, 1

n n

G z t ∞₌ fulfills the aforementioned properties 1 - 5 in Definition 2.4, then, for n>1, there exist functions

( )

,0 , , ,1 , , , , 1 ,

n n n n

G z t G z t  G ₋ z t defined on H× such that

( )

1

(

)

( )

, 0

, 2 , , , ,

n

j

n n j

j

G z t z i G z t z t

−

=

 

= ℜ +  ∈ ∈



∑

 H  (2.3)

with

( )

1

( )

, , 1, 1 ,

n j n j

zG z t j G− − − z t

∂ = (2.4)

for 1≤ ≤ −j n 1

( )

, , 0

k zGn j i t

∂ = (2.5)

for 1≤ ≤ −k j 1 with respect to t∈ and

( )

1

(

)

( )

,0 ,

1

, ,

n

j

n n j

j

G z t z i G z t −

=

= −

∑

+ (2.6)

Moreover,

( )

1

1 1 1

, 2

G z t

i t z t z

 

= _ − _

− −

  (2.7)

is the classical Poisson kernel for the upper half plane. All of the above ,

(

)

(

)

p n j

G ∈ H×L H× , the non-

tangential boundary value

( )

, ,

,

lim n j , n j ,

z s z s

G z t G s t

→ ∈ ∈

=

 H

(2.8)

exists on , except t∈ and ,

( )

,.

( )

p n j

G s ∈L  for any fixed s∈. We can further show that Gn j,

( )

.,t

can be continuously extended to H\

{ }

t for any fixed t∈, and

( )

,

1 ,

n j

G z t M t z

≤

′

− (2.9)

uniformly on D_c× ∈

{

t :t >T

}

whenever z′∈Dc which is any compact set in H, where M, T are positive

constants depending only on Dc.

(4)

( )

(

)

( )

, ,

lim n j , and lim n j , 0

z s z s

G z s z s G z s

→ →

∈ ∈ ∈ ∈

= +∞ − =

 

H H

(2.10)

for any s∈ and n≥2.

Remark 2.7. Lemma 2.6 provides a algorithm to obtain all explicit expressions of higher order Poisson kernels appeared in [8].

3. Homogeneous PHD Problem in the Upper Half Plane

In order to solve the homogeneous PHD problems (1.1) and get the uniqueness of its solution, we need the following lemmas.

Lemma 3.1. [8] Let D be a simply connected unbounded domain in the plane with smooth boundless

boundary ∂D. If f ∈

(

H×L1

)

(

D× ∂D

)

and there exists p

( )

, 1

g∈L ∂D P≥ , such that

( )

0

, g t

f z t M t z ≤

− (3.1)

uniformly on Dc× ∈∂

{

t D t: >T

}

whenever z0∈Dc which is any compact set in D, where M, T, are positive

constants depending only on Dc. Then

( )

(

, d

)

( )

, d

D D

z z

f f z t t z t t

∂ ∂

=

∂

∫

∂ .

Lemma 3.2. [8] Let

{

( )

}

1

,

n _n

G z t ∞₌ be the sequence of higher order Poisson kernels defined on H×, then

for any n>1 and p

( )

, 1

L P

γ∈ _ ≥ ,

( ) ( )

(

)

( ) ( )

2

1

, d , d

n n

z z

G z t γ t t G z t γ t t

+∞ +∞

−

−∞ −∞

∂ ₌

∂ ∂

∫

. (3.3)

Lemma 3.3. [9] Let f∈LP

( )

 , 1≤ ≤ ∞p , and u z

( )

be the Poisson integral of f (in our notations,

( )

1

( ) ( )

1

, d

π

u z =

∫

_−∞+∞G z t f t t, z∈H), then

[ ]( )

( )0

( )

0 sup 0

z x

M u x u z C f x

α

α ∈Γ

= ≤ M (3.4)

where Γ_α

( )

x₀ is the cone in H with the vertex at

(

x₀, 0

)

and the aperture α, x₀∈, α>0; C_α is a

positive constant depending only on α, M u

[ ]

is the non-tangential maximal function, and Mf is the

standard Hardy-Littlewood maximal function defined by

( )

0

( )

0

0 0

1

sup d

2

x x

f x _ρρ f x x

ρ ρ

+ − >

=

_∫

M (3.5)

Lemma 3.4. (Hardy-Littlewood maximal theorem, see [10]) Let f∈LP

( )

 , 1≤ ≤ ∞P , then Mf is finite

almost everywhere on . Moreover,

(1) If f ∈L1

( )

 , then Mf is in L1,∞

( )

 , more precisely

( )

{

}

1

( )

2

: _L .

x f x λ f

λ

∈_ M > ≤ _ (3.6)

(2) If f ∈LP

( )

 , 1< ≤ ∞P , then

( ) ( ),

P p P

L L

f _ ≤A f _

M (3.7)

where Ap is a constant depending only on p.

Corollary 3.5. M u

[ ]

LP_{( )} ≤Cp,α f LP_{( )} for any

( )

P

f ∈L  with 1< ≤ ∞p , where Cp,α is a constant

depending only on p,α . Moreover,

[ ]( )

{

}

1_{( )}

2

: C _L

x M u x λ α f

λ

(5)

for any f∈L1

( )

 , and for any f∈LP

( )

 , 1≤ ≤ ∞P , M u

[ ]

is finite almost everywhere on , C_α is a

positive constant depending only on α.

Theorem 1. Let

{

( )

}

1

,

n _n

G z t ∞₌ be the sequence of higher order Poisson kernels defined on H×, then for any n≥1,

( )

1

( )

1

( )

1 4 , d π j n j j j

u z G z t f t t

− _+∞

− −∞

=

∑

_∫

(3.9)

is the unique solution of PHD problem (1.1)

Proof. Since the higher order Poisson kernels possess the inductive property as stated in Definition 2.4. Act

on the two sides of (3.9) with the polyharmonic operator ∆l, 1≤ ≤ −l n 1. We have

( )

1

( )

1

( )

1 4 , d π j l n l

j l j

j l

u z G z t f t t

− + _+∞

− −

−∞ = +

∆ =

∑

_∫

(3.10)

since the Laplace operator is

2

4

z z

∂ ∆ =

∂ ∂ . Thus, for 0

l h

u

∆ = on ,

( )

, , 0 1,

l

u s f s s l n

∆ = ∈ ≤ ≤ − (3.11)

follow from Lemma 2.6 and the nice property of G, i.e.,

( ) ( )

( )

1 H,

1

lim , d

π

z s z s

G z t γ t t γ s

+∞ −∞ → ∈ ∈ =

∫

 (3.12)

for any γ∈Lp

( )

 ,p≥1.

Similarly, letting the polyharmonic operator ∆n act on the two sides of (3.9), we have ∆nu z

( )

=0 for any

z∈H. Thus (3.9) is a solution of the PHD problem (1.1).

Next we turn to the estimate and uniqueness of the solution. By Definition 2.4 and Corollary 3.5, we have

( )

( ) ( ) 1 1 1 1 1 1 1 0 4 , d π 4 , d π P P P P j n j j L j L j n n

j j _{j L}

j j

L

Mu M G z t f t t

M G z t f t t C f − _+∞ − −∞ = − _+∞ − − −∞ = =   = _ _     ≤ _ _ ≤  

∑

∫

∑

∫

∑

   

As discussed above, the uniqueness of solution follows.

4. Inhomogeneous PHD Problem in the Upper Plane

Due to the limited knowledge of the author, at this section, we only consider the bounded domain D for in-

homogeneous PHD problem in the upper half-plane, i.e.

in ,

on ,

n

j j u g D u f D

∆ = 

∆ = ∂

 (4.1)

where g∈L1loc

( )

 , such that, for some δ >0,

( )

(

)

m n

g z =O z− − −δ , as z→ ∞ and fj∈Lp

( )

∂D for some

suitable p>1. In order to solve the inhomogeneous PHD problem (4.1), we need the higher order Pompeiu

operators which are higher order analogues of the classical Pompeiu operators.

Definition 4.1. [11] Let kernels

( )

( ) ( )

(

)

( ) ( )

(

)

(

) (

)

1 1 1 1 , 1 1 2 1 1 1 1 ! 1 , 0; 1 !π ! 1 , 0; 1 !π

1 1 1

log , , 1

1 ! 1 !π

m m n n m n m n m n m n k l m

z z m n

n

K z z z n

m

z z z m n

m n k l

(6)

where m and n are integer, with m+ ≥n 0 but

(

m n,

) ( )

≠ 0, 0 . Then, we formally define operators Tm n D, , ,

acting on suitable complex valued function w defined in D, a domain in the plane, according to

( )

(

) ( )

, , , d d

m n D _D m n

T w z =

∫∫

K z−ζ w ζ ξ η (4.3)

The following properties of Tm n D, , are needed in the sequel. They are partial results from [11].

Lemma 4.2. Assume m+ ≥n 1, and let w be a complex valued function in L1loc

( )

 such that for some

0 δ > ,

( )

(

m n

)

, as .

w z =O z− − −δ z→ ∞

Then, the integral T_{m n}_, w z

( )

converges absolutely for almost all z in  and, provides that p satisfies conditions,

( )

1 2, when 1,

1 , when 2, 0,

1 , when 3,

a p m n

b p m n mn

c p m n mn

d p m n

 ≤ < + =

 _{≤ ≤ ∞} _{+ =} _≤



 _{≤ < ∞} _{+ =} _>



 _{≤ ≤ ∞} _{+ ≥}



( )

,

p m n loc T w∈L  .

Proof. See Corollary 4.6 in [11].

Lemma 4.3. Assume m+ ≥n 1, and let w be a measurazble complex valued function in  such that for some δ >0,

( )

(

m n

)

, as .

w z =O z− − −δ z→ ∞ (4.4)

(a) If m+ ≥n 2 and w∈L1loc

( )

 , then in the sense of Sobolev derivatives in the entire plane ,

, 1, ,

z m nT w Tm−nw

∂ = (4.5)

, , 1

zTm nw Tm n−w

∂ = (4.6)

(b) If m+ =n 1 and w∈Llocp

( )

 for some p>1, then (4.5) and (4.6) again hold in the sense of Sobolev

derivatives in ; moreover, the formulas

1,0 0,1

zT w zT w w

∂ = ∂ = (4.7)

are valid in  even in the case of p=1.

Proof. See Corollary 5.4 in [11].

Theorem 2. The problem of (4.1) is solvable and its unique solution is

( )

, ,

( )

1

( )

1

( )

1 , 1 ,

( )

1

4

, d

π

j n

n n j j n j n j

j

u z T g z G z t f t T g t t

− _+∞

− + − + − −∞

=

 

= _ +

∑

_∫

_ − _ _ (4.8)

where z∈H and T_{l l}_{, ,}_

(

1≤ ≤l n

)

are the higher order Pompeiu operators, G_j

( )

z t, and

(

1≤ ≤j n

)

are the

former n higher order Poisson kernel functions.

Proof. Through Lemma 4.2 and Lemma 4.3, we get

( )

, , , , , 0

k l

z zTm n g z Tm k n l− − g z k l m n

∂ ∂ _ = _ ≤ + ≤ + (4.9)

in the Sobolev sense. Moreover,

( )

, , as 0

p m k n l loc

T − − g z ∈L  ≤ + ≤ +k l m n. (4.10)

Noting (4.9) we know that u z

( )

=Tn n, ,g z

( )

is a week solution of the inhomogeneous equation

(

)

( )

1

( )

, , ,

n n

z z u z g z z g z Lloc

∂ ∂ = ∈ ∈ 

(7)

( )

(

)

0, g z O z m n δ , asz .

δ _> ₌ − − − _{→ ∞}

(4.11)

By the aforementioned, the problem (4.1) is equivalent to the PHD problem of simplified form

(

)

( ) ( )

(

)

(

)

( )

1 , ,

, , , ,

0, , , for some 0, as

, , 0 , 1.

m n n

n n loc

j p

z z n n j n j n j j

u T g g L g z O z z

u T g f T g f L D j n p

δ _δ

− − −

− −

∆ − = ∈ = > → ∞

 

 ∂ ∂ _ − _= − ∈ ∂ ≤ < >





 



So, through Theorem 1 as well as the estimate of the solution, we complete the proof of Theorem 2.

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