Received 11 November 2015; accepted 8 December 2015; published 11 December 2015
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1. Introduction
Usually harmonic functions are defined by Laplace operator
, where
is the Cauchy-Riemann operator and
is the adjoint operator of C-R operator. By iterating the
Laplace operator, one can define the so-called polyharmonic functions by
[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] ) and in irregular domains (Lipschitz domains [6] ). Although, there are many marked works about the BVPS, few 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
data in the upper half plane, H, i.e.
(1.1)
with
, where
is the Laplacian, and
is the real axis,
for some
suitable
,
,
,
is the non-tangential maximal function of u, which is defined by
![](//html.scirp.org/file/3-5301013x20.png)
where
is the non-tangential approach region, viz.,
![]()
where
.
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
satisfies the equation
, in D, then f is called an n-harmonic function in D, concisely, a polyharmonic function.
We use the notation
denoting the set of polyharmonic function of order n in D. Especially,
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
. If
, then for any
, there exist functions
,
such that
(2.1)
where
denotes the real part. The above decomposition expression of f is unique in the sense of the equi- valence relation
, more precisely,
for
.
Corollary 2.3. If the sequence of functions
defined in D satisfy
(1)
;
(2)
in D for
.
Then
for
, and
(2.2)
where
is the analytic jth decomposition component of the n-harmonic function
. 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
defined on
is called a sequence of higher order Poisson kernels, more precisely,
is called the nth order Poisson kernel, if they satisfy the following conditions.
(1) For all
;
with any fixed
; and
, with any fixed
, and the non-tangential boundary value
![]()
exists for all t and
;
can be continuously extended to
for any fixed
;
(2)
and
and
, and for any ![]()
![]()
uniformly on
whenever
, where
is any compact set in
, M, T are positive constants depending only on
and n;
(3)
and
for
;
(4)
, a.e., for any
;
(5)
, for any
,
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
, and
denote the set of all analytic functions in D. If f is a continuous function defined on
satisfying
for any fixed
, and
,
, for any fixed
, then f is called
on
and this is noted by
.
Lemma 2.6. [8] If
is a sequence of higher order Poisson kernels defined on
, i.e.,
fulfills the aforementioned properties 1 - 5 in Definition 2.4, then, for
, there exist functions
defined on
such that
(2.3)
with
(2.4)
for ![]()
(2.5)
for
with respect to
and
(2.6)
Moreover,
(2.7)
is the classical Poisson kernel for the upper half plane. All of the above
, the non- tangential boundary value
(2.8)
exists on
, except
and
for any fixed
. We can further show that ![]()
can be continuously extended to
for any fixed
, and
(2.9)
uniformly on
whenever
which is any compact set in
, where M, T are positive constants depending only on
.
Moreover,
(2.10)
for any
and
.
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
. If
and there exists
, such that
(3.1)
uniformly on
whenever
which is any compact set in D, where M, T, are positive constants depending only on
. Then
.
Lemma 3.2. [8] Let
be the sequence of higher order Poisson kernels defined on
, then
for any
and
,
. (3.3)
Lemma 3.3. [9] Let
,
, and
be the Poisson integral of f (in our notations,
,
), then
(3.4)
where
is the cone in
with the vertex at
and the aperture
,
,
;
is a positive constant depending only on
,
is the non-tangential maximal function, and
is the standard Hardy-Littlewood maximal function defined by
(3.5)
Lemma 3.4. (Hardy-Littlewood maximal theorem, see [10] ) Let
,
, then
is finite almost everywhere on
. Moreover,
(1) If
, then
is in
, more precisely
(3.6)
(2) If
,
, then
(3.7)
where
is a constant depending only on p.
Corollary 3.5.
for any
with
, where
is a constant depending only on
. Moreover,
(3.8)
for any
, and for any
,
,
is finite almost everywhere on
,
is a positive constant depending only on
.
Theorem 1. Let
be the sequence of higher order Poisson kernels defined on
, then for any
,
(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
,
. We have
(3.10)
since the Laplace operator is
. Thus, for
on
,
(3.11)
follow from Lemma 2.6 and the nice property of G, i.e.,
(3.12)
for any
.
Similarly, letting the polyharmonic operator
act on the two sides of (3.9), we have
for any
. 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
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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.
(4.1)
where
, such that, for some
,
, as
and
for some suitable
. 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
(4.2)
where m and n are integer, with
but
. Then, we formally define operators
, acting on suitable complex valued function w defined in D, a domain in the plane, according to
(4.3)
The following properties of
are needed in the sequel. They are partial results from [11] .
Lemma 4.2. Assume
, and let w be a complex valued function in
such that for some
,
![]()
Then, the integral
converges absolutely for almost all z in
and, provides that p satisfies con- ditions,
![]()
.
Proof. See Corollary 4.6 in [11] .
Lemma 4.3. Assume
, and let w be a measurazble complex valued function in
such that for some
,
(4.4)
(a) If
and
, then in the sense of Sobolev derivatives in the entire plane
,
(4.5)
(4.6)
(b) If
and
for some
, then (4.5) and (4.6) again hold in the sense of Sobolev derivatives in
; moreover, the formulas
(4.7)
are valid in
even in the case of
.
Proof. See Corollary 5.4 in [11] .
Theorem 2. The problem of (4.1) is solvable and its unique solution is
(4.8)
where
and
are the higher order Pompeiu operators,
and
are the former n higher order Poisson kernel functions.
Proof. Through Lemma 4.2 and Lemma 4.3, we get
(4.9)
in the Sobolev sense. Moreover,
. (4.10)
Noting (4.9) we know that
is a week solution of the inhomogeneous equation
![]()
and for some
(4.11)
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.