On the Green Function of the Annulus

Massimo Grossiand Djordjije Vujadinovi´c

1Massimo Grossi,Dipartimento di Matematica,Universit`a di Roma La Sapienza, P.le A.Moro 2–00185 Roma,Italy

2Djordjije Vujadinovi´c,Faculty of Mathematics,University of Montenegro,Dzordza Vˇasingtona bb,81000 Podgorica,Montenegro



On the Green Function of the Annulus

Massimo Grossi1,∗and Djordjije Vujadinovi´c

1Massimo Grossi,Dipartimento di Matematica,Universit`a di Roma La Sapienza, P.le A.Moro 2–00185 Roma,Italy

2Djordjije Vujadinovi´c,Faculty of Mathematics,University of Montenegro,Dzordza Vˇasingtona bb,81000 Podgorica,Montenegro

Abstract.Using the Gegenbauer polynomials and the zonal harmonics functions we give some representation formula of the Green function in the annulus.We apply this result to prove some uniqueness results for some nonlinear elliptic problems.

Green’s function,symmetries,uniqueness.

AMS Subject Classifications:35B09

Analysis in Theory and Applications

Anal.Theory Appl.,Vol.32,No.1(2016),pp.52-64

1　Introduction and statement of the main results

The classical Green function of the operator−∆with Dirichlet boundary conditions is defined by

where δyis the Dirac function centered at y and Ω is a bounded domain of IRn,n≥2. It is well known that the Green function can be written as

where H(x,y)is a smooth function in Ω×Ω which is harmonic in both variables x and y. Finally the Robin function is defined as

The knowledge of the Green(or the Robin)function is of great importance in applications(we mention the paper[2]and the rich list of references therein).Indeed the explicitcalculation of the Green function is an old problem(see for example the book by Courant and Hilbert,[5])but it can be solved only in special cases(like the ball or half-space).

For these reason,even if it is not possible to have the explicit expression,it is very important to deduce any properties of the Green function.

In this paper we are interested to the case where the domain is the annulus in IRn, namely Ω={x∈IRn:a＜|x|＜b}(in the rest of the paper by simplicity we assume that b=1).Even if the annuls possesses many symmetries,you can not explicitly write the Green function.If n=2 in[7]it was given a representation for the Green function as trigonometrical series.In this paper we give a representation formula of the Green function when n≥3 using the zonal spherical harmonics.Our first result is the following,

Theorem 1.1.Let A be the annulus A={x∈IRn:a＜|x|＜1}.Then we have that the Green function in A is given by,

Moreover the Robin function is given by,setting d0=1 and

for m≥1,

Here Zm(x,y)are the zonal spherical harmonics(see Section 2 or[1]for the definition and main properties).

Next corollary gives an alternative expression of the Green function which does not involve the Newtonian potential.

Corollary 1.1.Let A be the annulus A={x∈IRn:a＜|x|＜1}.Then we have that the Green function is given by,

These results are useful to derive some properties of the Robin function of the annulus.Actually we will show that the Robin function is a radial function which admits only one critical point which is nondegenerate in the radial direction.

Theorem 1.2.Let A be the annulus{x∈Rn|a＜|x|＜1}for n≥2 and RA(x)the corresponding Robin function.Then,if r=|x|,we have that RA(x)=RA(r)and RA(r)has a unique critical point r0,which satisfies

Note that this result was proved,when n=2,in[4]using different techniques.In Proposition 3.1 we give an alternative proof of this result.

Finally we apply these results to deduce some propertiesof nonlinear PDE’s problem. A straightforward application is a uniqueness results of concentrating solutions.Let us consider the problem

and

It is well known(see[3])that there exists solutions uεwhich achieve Sεand satisfying

where S is the best constant in Sobolev inequalities.In the next theorem we show the uniqueness of this solution(up to a suitable rotation).

Theorem 1.3.Let us suppose that u1,εand u2,εare two solutions of(1.7)satisfying(1.9).Then, up to a suitable rotation,we have that

for ε small enough.

When Ω is a generic domain of IRn,Theorem 1.3 was proved by Glangetas(see[6]) under the assumption that the critical point of the Robin function is nondegenerate.Of course,due to the rotationally invariance of the annulus,any critical point is degenerate and so Glangetas’result is therefore not applicable(note that the author conjectured the uniqueness result in the annulus at page 573 in[6]).Indeed the meaning of Theorem 1.3 is that just the nondegeneracy in the radial direction is necessary to have the uniqueness of the solution up to a suitable rotation.

The paper is organized as follows:in Section 2 we recall some preliminaries about the zonal harmonics and the Gegenbauer polynomials.In Section 3 we prove the Theorem 1.1,Corollary 1.1 and some properties of the Robin function(proof of Theorem 1.2). Finally in Section 4 we prove Theorem 1.3.

2　Preliminaries

In this Section we would like to point out the basic properties of zonal harmonics which are going to be used trough the paper.A good reference for the interested reader is the book[1].

By Hm(Rn)weare goingtodenotethefinitedimensionalHilbertspace ofall harmonic homogeneous polynomials of degree m.

Let us denote by S the unit sphere of IRn.A spherical harmonic of degree m is the restriction to S of an element of Hm(IRn).The collection of all spherical harmonics of degree m will be denoted by Hm(S).

Now we consider an important subset of Hm(S),the so-called zonal harmonics.They can be defined in different ways.We choose the equivalent definition given in Theorem 5.38 in[1].

For x∈IRnwith n≥2 and ξ∈S we define the zonal harmonic Zm(x,ξ)of degree m as

as m＞0.Several properties of the zonal harmonic can be found in Chapter 5 of[1].Let us emphasize that there is an explicit formula for the zonal harmonic as n=2,

The zonal harmonics have a particularly simple expression in terms of the Gegenbauer(or ultraspherical)polynomialsThe latter can be defined in terms of generating functions.If we write(see[11,pp.148])

where 0≤|r|＜1,|t|≤1 and λ＞0,then the coefficientis called Gegenbauer polynomial of degree m associated with λ.

The next theorem(see[11,pages 146–150])is related to representation of the zonal harmonics.

Theorem 2.1.If n＞2 is an integer,λ=and k=0,1,2,···,then we have that for all x′,y′∈S it holds

Proof.In Corollary 2.13 in[11]it is proved that

Let us compute the constant cn,m.If we put x′=y′∈S in(2.3)we get

In[1],Proposition 5.27 and Proposition 5.8 showed that

On the other hand in[9],it was shown that

We end this section pointing out the result(see[1,pp.217,Theorem 10.13])related to the solution of Dirichlet problem in annulus.Recall that A={x∈Rn|a＜|x|＜1}and set

where

and

Both series PA(x,ξ),PA(x,aξ)converge absolutely and uniformly on K×S,K⊂A(K is some compact subset).We have the following

Theorem 2.2.Suppose n＞2 and that f is continuous function on∂A.Define u on¯A by

3　The representation formula for the Green function

Our first aim is to write the Green function for the annulus in terms of Zm(x,y).The starting point for our results is going to be the next easy lemma which will play an important role in proving Theorem 1.1.

Lemma 3.1.We have that,for any|ξ|=1,|y|≤1 and y6=ξ,

and

Proof.Since|ξ|=1,by using the formula(2.2)we obtain

In a similar way we prove(3.2):

Thus,we complete the proof.

Now we are in position to prove our representation formula for the Green function. Proof of Theorem 1.1.By(1.2)we have to write H(x,y),where H(x,y)is an harmonic function satisfying

on∂A.By Theorem 2.2 we have that H(x,y)=PA[fy](x)with

where y∈A is fixed.Using Lemma 3.1,we obtain

So we have that

Similarly,for the second integral,we get

So,we obtain

The Robin function is

By direct calculation we get the formulas(1.4)and(1.5).□

Proof of Corollary 1.1.As in the proof of Lemma 3.1 we have,for|x|＞|y|,

From(3.10)and(1.4)the claim follows.□

We have the following,

Corollary 3.1.We have that

Proof.It follows directly by Theorem 1.1.

We end this section by proving Theorem 1.2.As we mention in the introduction this result generalizes that of Chen and Lin[4]to higher dimensions.

Proof of Theorem 1.2.By(3.11)we have the the Robin function is radial.

The inequality(3.12)implies that the function f(r)is strictly decreasing.Since we have that

and

weconclude thatthereexistsexactlyoner0,a＜r0＜1,suchthat f(r0)=0and then R′A(r0)= 0.

On the other hand,

Following the line of the proof of the previous theorem we have the following alternative proof to the result in[4].

Proposition 3.1.The Robin function of the 2-dimensional annulus has a unique nondegenerate critical point.

Proof.Let us recall the formula for the Robin function in the plane(see[7])

As in the previous theorem we have,

So,

On the other hand we get

On the other hand,while

So,we can conclude that there exist unique r0,a＜r0＜1,for which=0 and＞0.

4　A uniqueness result for a nonlinear elliptic equation

Let us consider the problem

and solutions satisfying

where S is the best constant in Sobolev inequalities and

It is well known that a family of solutions satisfying(1.7)and(1.9)concentrates at one point,i.e.,

weakly in the sense of measure as ε→0.It was proved by Han(see[8])that P is a critical point of the Robin function.

Using Theorem 1.2 we have the following”asymptotic”uniqueness result.

Theorem 4.1.Let A={x∈Rn|a＜|x|＜1}and uεa family of solutions to(1.7)satisfying(1.9). Then

where r0is the unique root of the equation

Next aim is to improve the previous result.Indeed we will show that not only our problemhas a unique pointof concentration,but evenit has a unique solutionfor ǫ small. Of course,since the problem is rotationally invariant we can have uniqueness only up to a suitable rotation.

Now let us consider a solution to(1.7)satisfying(1.9).Up to a suitable rotation we can assume that its maximum point is given by(y1,0,···,0)with y1∈(a,1).Then we want to give a representation formula for this solution.This involves classical results which we recall below.Basically we follow the line of the proof of Theorem A in[6].

Let us introduce some notations.Set,for y=(y1,0,···,0)with y1∈(a,1),

which is the only positive solution to

and for λ＞0 let us define

where〈u1,u2〉=RA∇u1∇u2is the scalar product in(A).

Finally set for some δ＞0,ω0=(a+δ,1−δ)⊂(a,1)such that|∇R|＞1 in(a,a+δ)∪(1−δ,1).

The following proposition is classical for concentration problems like(1.7)as ε→0 (see[10]or[6]for example).

Proposition 4.1.There exist ε0＞0,λ0＞0 and η0＞0 such that,for ε∈(0,ε0)and for (x,λ)∈ω0×[λ0,+∞),there exists a uniquesuch thatand for any

Now we are in position to prove Theorem 1.3.

Now we use the crucial fact that u=PUy,λ+vy,λis a solution to(1.7)if and only if the pair(y,λ)is a critical point of the reduced functional

So our claim is equivalent to show that the functional Kε(y,λ):ω0×(0,+∞)→IR has exactly one critical point.

Let us introduce the function˜Kε(y,λ):ω0×(0,+∞)→IR defined as

In[6,page 576],it was proved that there exist constants A,B,C∈IR such that

By Theorem 1.2 we have that∇RA(r0)=0 andThis means that r0is a nondegenerate critical point for the function RA(y1,0,···,0).Hence Step 1 in[6]applies and thenwe have the uniquenessof the critical point of Kε(y1,λ).Thenandand the claim follows.

Acknowledgments

The first author is supportedby PRIN-2009-WRJ3W7 grant and the second author is supported by Basileus scholarship programme.

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10.4208/ata.2016.v32.n1.5

30 October 2015;Accepted(in revised version)5 November 2015

∗Corresponding author.Email addresses:massimo.grossi@uniroma1.it(M.Grossi),djordjijevuj@t-com. me(D.Vujadinovi´c)