_{1}

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In this article, a class of Dirichlet problem with
L^{p} boundary data for poly-harmonic 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.

Usually harmonic functions are defined by Laplace operator

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

with

suitable

where

where

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

Definition 2.1. If a real valued function

We use the notation

Lemma 2.2. [

where

Corollary 2.3. If the sequence of functions

(1)

(2)

Then

where

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

(1) For all

exists for all t and

(2)

uniformly on

(3)

(4)

(5)

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

Lemma 2.6. [

with

for

for

Moreover,

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

exists on

can be continuously extended to

uniformly on

Moreover,

for any

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

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. [

uniformly on

Lemma 3.2. [

for any

Lemma 3.3. [

where

Lemma 3.4. (Hardy-Littlewood maximal theorem, see [

(1) If

(2) If

where

Corollary 3.5.

for any

Theorem 1. Let

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

since the Laplace operator is

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

for any

Similarly, letting the polyharmonic operator

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

As discussed above, the uniqueness of solution follows.

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.

where

Definition 4.1. [

where m and n are integer, with

The following properties of

Lemma 4.2. Assume

Then, the integral

Proof. See Corollary 4.6 in [

Lemma 4.3. Assume

(a) If

(b) If

are valid in

Proof. See Corollary 5.4 in [

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

where

Proof. Through Lemma 4.2 and Lemma 4.3, we get

in the Sobolev sense. Moreover,

Noting (4.9) we know that

and for some

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

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

Kanda Pan, (2015) L^{p} Polyharmonic Dirichlet Problems in the Upper Half Plane. Advances in Pure Mathematics,05,828-834. doi: 10.4236/apm.2015.514077