An application of the inequality for modified Poisson kernel

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R E S E A R C H

Open Access

An application of the inequality for

modiﬁed Poisson kernel

Gaixian Xue

1

_{and Junfei Wang}

2*

*_{Correspondence:} [email protected]

2_{Foundation Department, Wuhai}

Vocational and Technical College, Wuhai, 016000, China

Full list of author information is available at the end of the article

Abstract

As an application of an inequality for modiﬁed Poisson kernel obtained by Qiao and Deng (Bull. Malays. Math. Sci. Soc. (2) 36(2):511-523, 2013), we give the generalized solution of the Dirichlet problem with arbitrary growth data.

Keywords: growth property; Dirichlet problem; modiﬁed Poisson kernel

1 Introduction and results

Let Rn₍_n_≥_{) be the}_n_{-dimensional Euclidean space. The boundary and the closure of a}

setEin Rn_{are denoted by}_∂_E_and_E_{, respectively. The Euclidean distance of two points}_P

andQin Rnis denoted by|P–Q|. Especially,|P|denotes the distance of two pointsPand

Oin Rn, whereOis the origin in Rn.

We introduce a system of spherical coordinates (r,),= (θ,θ, . . . ,θn–), in Rnwhich are related to Cartesian coordinates (x,x, . . . ,xn–,xn) byxn=rcosθ.

LetB(P,r) denote the open ball with center atPand radiusr(>) in Rn_{. The unit sphere}

and the upper half unit sphere in Rn_{are denoted by S}n–_{and S}n–

+ , respectively. The sur-face area πn/{(n/)}–of Sn–is denotedwn. Let⊂Sn–, a point (,) and the set

{; (,)∈}are denotedand, respectively. For two sets⊂R+ and⊂Sn–, we denote×={(r,)∈Rn_;_r_∈_{, (,}₎_∈_}_{, where R}

+is the set of all positive real numbers.

For the set⊂Sn–, we denote the set R+×in RnbyCn(), which is called a cone.

For the setI⊂R, the setsI×andI×∂are denotedCn(;I) andSn(;I), respectively,

where R is the set of all real numbers. Especially, the setSn(; R+) is denotedSn().

Given a continuous function f onSn(), we say thathis a solution of the Dirichlet

problem inCn() withf, ifhis a harmonic function inCn() and

lim P→Q∈Sn(),P∈Cn()

h(P) =f(Q).

Let⊂Sn–_and ∗_{be a Laplace-Beltrami on the unit sphere. Consider the Dirichlet} problem (see,e.g.[], p.)

∗_ϕ₍_{) +}_λϕ₍_{) = } _in_,

ϕ() =  in∂.

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We denote the non-decreasing sequence of positive eigenvalues of it, repeating accord-ingly to their multiplicities, and the corresponding eigenfunctions are denoted, respec-tively, by{λi}∞i=and{ϕi()}∞i=. Especially, we denote the least positive eigenvalue of itλ and the normalized positive eigenfunction toλϕ(). In the sequel, for the sake of brevity, we shall writeλandϕinstead ofλandϕ, respectively.

The set of sequential eigenfunctions corresponding to the same value of{λi}∞i= in the sequence{ϕi()}∞i=makes an orthonormal basis for the eigenspace of the eigenvalueλi.

Hence for each⊂Sn–_{there is a sequence}_{_k

j}of positive integers such thatk= ,λkj<

λkj+,λkj=λkj+=λkj+=· · ·=λkj+–and{ϕkj,ϕkj+, . . . ,ϕkj+–}is an orthonormal basis for

the eigenspace of the eigenvalue{λkj}

∞

j=. ByI(km) we denote the set of all positive integers

less than{km}∞m=. In spite of the fact

I(k) =∅,

the summation overI(k) of a functionS(k) of a variablekwill be used by promising

k∈I(k)

S(k) = .

If we denote the solutions of the equation

t+ (n– )t–λi=  (i= , , , . . .)

byℵ+

i andℵ–i, then the functions

rℵ±i_ϕ_i₍₎ ₍_i_{= , , , . . .)}

are harmonic functions inCn() and vanish onSn().

LetG(P,Q) be the Green function ofCn() for anyP= (r,)∈Cn() and anyQ=

(t,)∈Cn(). Then the Poisson kernel inCn() can be deﬁned by

PI(P,Q) = 

cn

∂ ∂nQ

G(P,Q),

whereP∈Cn(),Q∈Sn(),∂/∂nQdenotes the diﬀerentiation atQalong the inward

nor-mal intoCn() and

cn= ⎧ ⎨ ⎩

π ifn= ,

(n– )wn ifn≥.

LetF() be a function deﬁned in. We denoteNi(F) by

F()ϕi()d,

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For any two pointsP= (r,) andQ= (t,) inCn() andSn(), respectively, we deﬁne

Km(P,Q) = ⎧ ⎨ ⎩

 if  <t< ,

Km

(P,Q) if ≤t<∞,

wheremis a non-negative integer and

Km(P,Q) =

i∈I_km_+

ℵ+i+n–_N_i_PI_(,_{), (,}₎ _rℵ+i_t–ℵi+–n+_ϕ_i₍_).

To obtain the solution of the Dirichlet problem in a cone, as in [, , ], we use the modiﬁed Poisson kernel deﬁned by

PIm(P,Q) =PI(P,Q) –Km(P,Q),

whereP∈Cn() andQ∈Sn(), which has the following estimates (see []):

PI(P,Q) –Km(P,Q)≤M(r)

ℵ+

km+_t–ℵ +

km+–n+ _()

for anyP= (r,)∈Cn() and anyQ= (t,)∈Sn() satisfying  <r_t < _, whereMis a

constant independent ofP,Q, andm. For the construction and applications of a modiﬁed Green function in a half space, we refer the reader to the paper by Qiao (see []).

Write

Um[f](P) =

Sn()

PIm(P,Q)f(Q)dσQ,

wheref(Q) is a continuous function on∂Cn() anddσQthe (n– )-dimensional volume

elements induced by the Euclidean metric on∂Cn().

Recently, Qiao and Deng (cf.[]) gave the solution of the Dirichlet problem in a cone. Applications of modiﬁed Poisson kernel with respect to the Schrödinger operator, we refer the reader to the papers by Huang and Ychussie (see []) and Li and Ychussie (see []).

Theorem A If+ℵ+_{–  > ,}_–_n_{+ }_{≤ ℵ}+

km+<–n+ and f(Q) (Q= (t,))is a continuous function on∂Cn()satisfying

Sn()

|f(Q)|

 +tdσQ<∞, ()

then the function Um

[f](P)is a solution of the Dirichlet problem in Cn()with f and

lim r→∞,P=(r,)∈Cn()

rn––ϕn–()Um[f](P) = .

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Theorem B Let <p<∞,γ > (–ℵ+_–_n_{+ )}_p₊_n_{– }_and

γ –n+ 

p <ℵ

+ km+<

γ –n+ 

p + .

If f(Q) (Q= (t,))is a continuous function on Sn()satisfying

Sn()

|f(Q)|p

 +tγ dσQ<∞, ()

then the function Um

[f](P)satisﬁes

lim r→∞,P=(r,)∈Cn()r

n–γ–

p _ϕn–₍₎_Um

[f](P) = .

It is natural to ask if the continuous functionusatisfying () and () can be replaced by arbitrary continuous function? In this paper, we shall give an affirmative answer to this question. To do this, we first construct a modified Poisson kernel. Letφ(l) be a positive function ofl≥ satisfying

ℵ+φ() = .

Denote the set

l≥; –ℵ+_k_ilog =logln–φ(l)

by π(φ,i). Then ∈π(φ,i). When there is an integer N such thatπ(φ,N)=and

π(φ,N+ ) =, denote

J(φ) ={i; ≤i≤N}

of integers. Otherwise, denote the set of all positive integers byJ(φ). Letl(i) =l(φ,i+ ) be the minimum elementsl inπ(φ,i) for eachi∈J(φ). In the former case, we put

l(N+ ) =∞. Thenl() = . The kernel functionKφ(P,Q) is deﬁned by

Kφ(P,Q) = ⎧ ⎨ ⎩

 if  <t< ,

Ki(P,Q) ifl(i)≤t<l(i+ ) andi∈J(φ),

whereP∈Cn() andQ= (t,)∈Sn().

The generalized Poisson kernelPφ(P,Q) is deﬁned by

PIφ(P,Q) =PI(P,Q) –K φ (P,Q),

whereP∈Cn() andQ∈Sn().

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Theorem Let g(Q)be a continuous function on Sn().Then there is a positive continuous

functionφg(l)of l≥depending on g such that

Hφg(P) =

Sn()

PIφg(P,Q)g(Q)dσQ

is a solution of the Dirichlet problem in Cn()with g.

2 Lemmas

Lemma  Letφ(l)be a positive continuous function of l≥satisfying φ() = –ℵ+.

Then

PI(P,Q) –K φ

(P,Q)≤Mφ(l)

for any P= (r,)∈Cn()and any Q= (t,)∈Sn()satisfying

t>max{, r}. ()

Proof We can choose two pointsP= (r,)∈Cn() andQ= (t,)∈Sn(), which satisﬁes

(). Moreover, we also can choose an integeri=i(P,Q)∈J(φ) such that

l(i– )≤t<l(i). ()

Then

Kφ(P,Q) =Ki–(P,Q).

Hence we have from (), (), and () that

PI(P,Q) –Kφ(P,Q)≤M –ℵ+

ki≤_M_φ₍_l_),

which is the conclusion.

Lemma (See []) Let g(Q)be a continuous function on∂Cn()and V(P,Q)be a locally

integrable function on∂Cn()for any ﬁxed P∈Cn(),where Q∈∂Cn().Deﬁne

W(P,Q) =PI(P,Q) –V(P,Q)

for any P∈Cn()and any Q∈∂Cn().

Suppose that the following two conditions are satisﬁed:

(I) For anyQ∈∂Cn() and any> , there exists a neighborhoodB(Q) ofQsuch that

Sn(;[R,∞))

W(P,Q)u(Q)dσQ< ()

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(II) For anyQ∈∂Cn(), we have

lim sup P→Q,P∈Cn()

Sn(;(,R))

V(P,Q)u(Q)dσQ=  ()

for any positive real numberR. Then

lim sup P→Q,P∈Cn()

Sn()

W(P,Q)uQ dσQ≤u(Q)

for anyQ∈∂Cn().

3 Proof of Theorem

Take a positive continuous functionφ(l) (l≥) such that

φ()ℵ+=  ()

and

φ(l)

∂

g(l,)dσ≤

L ln

forl> , where

L= –ℵ+

∂

g(,)dσ.

For any ﬁxedP= (r,)∈Cn(), we can choose a numberRsatisfyingR>max{, r}.

Then we see from Lemma  that

Sn(;(R,∞))

PIφg(P,Q)g(Q)dσQ

≤M

∞

R

∂

g(,)dσ

φ(l)ln–dl

≤ ML

∞

R

l–dl

<∞. ()

Obviously, we have

Sn(;(,R)) _PIφg

(P,Q)g(Q)dσQ<∞,

which gives

Sn()

PIφg(P,Q)g(Q)dσQ<∞.

To see thatHφg(P) is a harmonic function inCn(), we remark thatH φg

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Finally we shall show that

lim P∈Cn(),P→QH

φg

(P) =g

Q

for anyQ= (t,)∈∂Cn(). Set

V(P,Q) =Kφg(P,Q)

in Lemma , which is locally integrable onSn() for any ﬁxedP∈Cn(). Then we apply

Lemma  tog(Q) and –g(Q).

For any>  and a positive numberδ, by () we can choose a numberR(>max{, (t+

δ)}) such that () holds, whereP∈Cn()∩B(Q,δ).

Since

lim

→ϕi() =  (i= , ,  . . .)

asP= (r,)→Q= (t,)∈Sn(), we have

lim P∈Cn(),P→Q

Kφg(P,Q) = ,

whereQ∈Sn() andQ∈Sn(). Then () holds.

Thus we complete the proof of Theorem.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript and read and approved the ﬁnal manuscript.

Author details

1_{School of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou, 450002, China.} 2_{Foundation Department, Wuhai Vocational and Technical College, Wuhai, 016000, China.}

Acknowledgements

The authors are very thankful to the anonymous referees for their valuable comments and constructive suggestions.

Received: 14 September 2015 Accepted: 4 January 2016

References

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3. Qiao, L: Integral representations for harmonic functions of inﬁnite order in a cone. Results Math.61(1-2), 63-74 (2012) 4. Qiao, L: Growth of certain harmonic functions in ann-dimensional cone. Front. Math. China8(4), 891-905 (2013) 5. Qiao, L: Modiﬁed Poisson integral and Green potential on a half-space. Abstr. Appl. Anal.,2012, Article ID 765965

(2012)

6. Huang, J, Ychussie, B: The modification of Poisson-Sch integral on cones and its applications. Filomat (to appear) 7. Li, Z, Ychussie, B: Sharp geometrical properties of a-rarefied sets via fixed point index for the Schrödinger operator