An extension of the estimation for solutions of certain Laplace equations

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R E S E A R C H

Open Access

An extension of the estimation for

solutions of certain Laplace equations

Bin Huang

1

_{, Jiaofeng Wang}

2

_{and Maria Hampshire Zylbersztejn}

3*

*_{Correspondence:}

m.h.zylbersztejn@gmail.com

3_{Departamento de Matemática,}

Universidade Federal do Rio de Janeiro, Caixa Postal 68530, Rio de Janeiro, RJ 21945-970, Brazil Full list of author information is available at the end of the article

Abstract

In this paper, by using a new type of Carleman formula with respect to a certain Laplace operator, we estimate the growth property for solutions of certain Laplace equations deﬁned in a smooth cone.

Keywords: growth property; certain Laplace equation; cone

1 Introduction and results

Let Rn(n≥) be then-dimensional Euclidean space. A point in Rnis denoted byV= (X,y), whereX= (x,x, . . . ,xn–). The boundary and the closure of a setEin Rnare denoted by∂EandE, respectively.

We introduce a system of spherical coordinates (l,),= (θ,θ, . . . ,θn–), in Rnwhich are related to cartesian coordinates (x,x, . . . ,xn–,y) byy=lcosθ.

The unit sphere in Rn_{is denoted by S}n–_{. For simplicity, a point (,}_{) on S}n–_{and the} set{; (,)∈}for a set,⊂Sn–_{are often identiﬁed with}_and_{, respectively. For} two sets⊂R+and⊂Sn–, the set

(l,)∈Rn;l∈, (,)∈

in Rn_{is simply denoted by}_×_.

We denote the set R+×in Rnwith the domainon Sn–byTn(). We call it a cone.

The setsI× andI×∂ with an interval on R are denoted byTn(;I) andSn(;I),

respectively. BySn(;l) we denoteTn()∩Sl. We denoteSn(; (, +∞)) bySn().

LetG(V,W) (P,Q∈Tn()) be the Green function inTn(). Then the ordinary Poisson

formula inTn() is deﬁned by

cnPI(V,W) =

∂G(V,W) ∂nW

,

where∂/∂nWdenotes the diﬀerentiation atQalong the inward normal intoTn(). Here, c=  andcn= (n– )wnwhenn≥, wherewnis the surface area of Sn–.

Let∗_nbe the spherical version of the Laplace operator andbe a domain on Sn–with smooth boundary∂. Consider the Dirichlet problem (see [])

∗_n+τψ=  on, (.)

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ψ=  on∂. (.)

We denote the least positive eigenvalue of (.) and (.) byτand the normalized positive eigenfunction corresponding toτ byψ(). In the sequel, for the sake of brevity, we shall writeχinstead ofℵ+_–_ℵ–_{, where}

ℵ±= –n+ ±(n– )_{+ }_τ_.

We use the standard notationsh+=max{h, }andh–= –min{h, }. All constants ap-pearing in the expressions in the following sections will be always writtenM, because we do not need to specify them. Throughout this paper, we will always assume thatη(t) and

ρ(t) are nondecreasing real-valued functions on an interval [, +∞) andρ(t) >ℵ+_{for any}

t∈[, +∞).

Recently, Li and Vetro (see [], Theorem ) obtained the lower bounds for functions harmonic in a smooth cone. Similar results for solutions of p-Laplace equations under Neumann boundary condition, we refer the reader to the papers by Guo and Gao (see []) and Rao and Pu (see []).

Theorem A Let K be a constant,h(V) (V= (R,))be harmonic on Tn()and continuous on Tn().If

h(V)≤KRρ(R), V= (R,)∈Tn

; (,∞)

and

h(V)≥–K, R≤, V= (R,)∈Tn(),

then

h(V)≥–KM +ρ(R)Rρ(R)ψ–n(),

where V∈Tn()and M is a constant independent of K,R,ψ(),and the function h(V).

In this paper, we shall extend Theorem A to solutions of a certain Laplace equation (see [] for the deﬁnition of this Laplace equation).

Theorem  Let h(V) (V= (R,))be solutions of certain Laplace equation deﬁned on Tn() and continuous on Tn().If

h(V)≤η(R)Rρ(R), V= (R,)∈Tn

; (,∞), (.)

and

h(V)≥–η(R), R≤, V= (R,)∈Tn(), (.)

then

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where V∈Tn(),c is a real number satisfying c≥and M is a constant independent of R,

ψ(),the functionsη(R)and h(V).

Remark In the casec≡ andη(R)≡K, whereK is a constant, Theorem  reduces to Theorem A.

2 Lemmas

In order to prove our result, we ﬁrst introduce a new type of Carleman formula for func-tions harmonic in a cone (see []). For the Carleman formula for harmonic funcfunc-tions and its application, we refer the reader to the paper by Yang and Ren (see [], Lemma ). Re-cently, it has been extended to Schrödinger subharmonic functions in a cone (see [], Lemma ). For applications, we also refer the reader to the paper by Wanget al.(see [], Theorem ).

Lemma  Let h be harmonic on a domain containing Tn(; (,R)),where R> .Then

χ

Sn(;R)

hψRℵ––dSR+

Sn(;(,R))

htℵ––tℵ+R–χ∂ψ/∂n dσW+d+dR–χ= ,

where dSRdenotes the(n– )-dimensional volume elements induced by the Euclidean met-ric on SR,∂/∂n denotes diﬀerentiation along the interior normal,

d=

Sn(;)

ℵ–_h_ψ_–_ψ₍_∂_h_/_∂_n₎_dS 

and

d=

Sn(;)

ψ(∂h/∂n) –ℵ+_h_ψ_dS .

Lemma (See [], Lemma ) We have

PI(V,W)≤Mrℵ

–

tℵ+–ψ()∂ψ()

∂n

for any V = (r,)∈Tn()and any W= (t,)∈Sn()satisfying <t_r≤_.

PI(V,W)≤M ψ()

tn– ∂ψ()

∂n

+M rψ() |P–Q|n

∂ψ()

∂n

for any V = (r,)∈Tn()and any W= (t,)∈Sn(; (_r,_r)). Let G,R(V,W)be the Green function of Tn(, (,R)).Then

∂G,R(V,W)

∂R ≤Mr

ℵ+

Rℵ––ψ()ψ(),

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3 Proof of Theorem 1

We ﬁrst apply Lemma  toh=h+_–_h–_{and obtain}

χ

Sn(;R)

h+Rℵ––ψdSR+

Sn(;(,R))

h+tℵ––tℵ+R–χ∂ψ/∂n dσW+d+dR–χ

=χ

Sn(;R)

h–Rℵ––ψdSR+

Sn(;(,R))

h–tℵ––tℵ+R–χ∂ψ/∂n dσW, (.)

It is easy to see that

χ

Sn(;R)

h+Rℵ––ψdSR≤Mη(R)Rρ(cR)–ℵ

+

(.)

and

Sn(;(,R))

h+tℵ––tℵ+R–χ∂ψ/∂n dσW≤Mη(R)Rρ(cR)–ℵ

+

(.)

from (.). We remark that

d+dR–χ≤Mη(R)Rρ(cR)–ℵ

+

. (.)

We have

χ

Sn(;R)

h–Rℵ––ψdSR≤Mη(R)Rρ(cR)–ℵ

+

(.)

and

Sn(;(,R))

h–tℵ––tℵ+R–χ∂ψ/∂n dσW≤Mη(R)Rρ(cR)–ℵ

+

(.)

from (.), (.), (.), and (.). It follows from (.) that

Sn(;(,R))

h–tℵ–∂ψ

∂n dσW≤Mη(R)

(ρ(cR) + )χ (ρ(cR) + )χ_{– (}_ρ₍_cR₎₎χ

ρ(cR) + 

ρ(cR) R

ρ(ρ_ρ(cR₍_cR)+₎ R)–ℵ+

,

which shows that

Sn(;(,R))

h–tℵ–∂ψ/∂n dσW≤Mη(R)ρ(cR)Rρ(cR)–ℵ

+

. (.)

By the Riesz decomposition theorem (see []), we have

–h(V) =

Sn(;(,R))

PI(V,W) –h(W)dσW

+

Sn(;R)

∂G,R(V,W)

∂R –h(W)dSR, (.)

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We next distinguish three cases.

Case.V= (l,)∈Tn(; (/,∞)) andR= l/.

Since –h(V)≤h–₍_V_{), we have}

–h(V) = 

i=

Ui(V) (.)

from (.), where

U(V) =

Sn(;(,])

PI(V,W) –h(W)dσW,

U(V) =

Sn(;(,l/])

U(V) =

Sn(;(l/,R))

and

U(V) =

Sn(;R)

PI(V,W) –h(W)dσW.

We have the following estimates:

U(V)≤Mη(R)ψ() (.)

and

U(V)≤Mη(R)ρ(cR)Rρ(cR)ψ() (.)

from Lemma  and (.). We consider the inequality

U(V)≤U(V) +U(V), (.)

where

U(V) =M

Sn(;(l/,R))

–h(W)ψ()

tn–

∂φ()

∂n

dσW

and

U(V) =Mrψ()

Sn(;(l/,R))

–h(W)lψ()

|V–W|n

∂φ()

∂n

dσW.

We ﬁrst have

U(V)≤Mη(R)ρ(cR)Rρ(cR)ψ() (.)

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We shall estimateU(V). Take a suﬃciently small positive numberdsuch that

Sn

; (l/,R)⊂B(P,l/)

for anyV= (l,)∈(d), where

(d) =

V= (l,)∈Tn(); inf

(,z)∈∂

(,) – (,z)<d,  <r<∞

,

and divideTn() into two sets(d) andTn() –(d).

IfV= (l,)∈Tn() –(d), then there exists a positivedsuch that|V–W| ≥dlfor

anyQ∈Sn(), and hence

U(V)≤Mη(R)ρ(cR)Rρ(cR)ψ(), (.)

which is similar to the estimate ofU(V).

We shall consider the caseV= (l,)∈(d). Now put

Hi(V) =

Q∈Sn

; (l/,R); i–δ(V)≤ |V–W|< iδ(V),

where

δ(V) = inf

Q∈∂Tn()|V–W|.

Since

Sn()∩

Q∈Rn:|V–W|<δ(V)=∅, we have

U(V) =M

i(V)

i=

Hi(V)

–h(W)rψ()

|V–W|n

∂ψ()

∂n

dσW,

wherei(V) is a positive integer satisfying i(V)–_δ₍_V₎_≤r

< 

i(V)_δ₍_V_).

Since

rψ()≤Mδ(V),

whereV= (l,)∈Tn(), similar to the estimate ofU(V) we obtain

Hi(V)

–h(W)rψ()

|V–W|n

∂ψ()

∂n

dσW≤Mη(R)ρ(cR)Rρ(cR)ψ–n()

fori= , , , . . . ,i(V). So

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From (.), (.), (.), and (.) we see that

U(V)≤Mη(R)ρ(cR)Rρ(cR)ψ–n(). (.)

On the other hand, we have from Lemma  and (.)

U(V)≤Mη(R)Rρ(cR)ψ(). (.)

We thus obtain from (.), (.), (.), and (.)

–h(V)≤Mη(R) +ρ(cR)Rρ(cR)ψ–n(). (.)

Case.V= (l,)∈Tn(; (/, /]) andR= l/.

It follows from (.) that

–h(V) =U(V) +U(V) +U(V),

whereU(V) andU(V) are deﬁned in Case  and

U(V) =

Sn(;(,R))

PI(V,W) –h(W)dσW.

Similar to the estimate ofU(V) in Case  we have

U(V)≤Mη(R)ρ(cR)Rρ(cR)ψ–n(),

which together with (.) and (.) gives (.).

Case.V= (l,)∈Tn(; (, /]).

It is evident from (.) that we have

–h≤η(R),

which also gives (.). We ﬁnally have

h(V)≥–η(R)M +ρ(cR)Rρ(cR)ψ–n()

from (.), which is the conclusion of Theorem .

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

MZ completed the main study. BH carried out the results of this article. JW veriﬁed the calculation. All the authors read and approved the ﬁnal manuscript.

Author details

1_{Mathematical and Statistical Sciences, Quzhou University, Quzhou, 324000, China.}2_{Mathematical and Statistical}

Sciences, Quzhou Vocational and Technical College, Quzhou, 324000, China.3_{Departamento de Matemática,}

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Acknowledgements

The authors are grateful to the two anonymous referees for their valuable comments, which led to a much improved version of the manuscript.

Received: 30 April 2016 Accepted: 21 June 2016

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