Measure Estimates of Nodal Sets of Polyharmonic Functions?

Long TIAN

Abstract This paper deals with the function u which satisfies△ku=0,where k≥ 2 is an integer.Such a function u is called a polyharmonic function.The author gives an upper bound of the measure of the nodal set of u,and shows some growth property of u.

Keywords Polyharmonic function,Nodal set,Frequency,Measure estimate,Growth property

1 Introduction

The nodal sets are zero level sets.We want to study the measure estimates of nodal sets of polyharmonic functions in this paper.In 1979,Almgren[1]introduced the frequency concept of harmonic functions.Then in 1986 and 1987,Garofalo and Lin[4–5]established the monotonicity formula of the frequency and the doubling conditions for solutions of the uniformly second order elliptic equations,and showed the unique continuation of such solutions by using the doubling conditions.In 2000,Han[6]studied the structure of the nodal sets of solutions of a class of uniformly high order elliptic equations.In 2003,Han,Hardt and Lin in[7]investigated structures and measure estimates of singular sets of solutions of high order uniformly elliptic equations.In 2014,the author and Yang in[13]gave the measure estimates of nodal sets for bi-harmonic functions.

The classical frequency of a harmonic function is defined as follows.

Definition 1.1If u is a harmonic function in B1,then for any r≤1,one can define the frequency function of u centered at the origin with radius r as follows:

where dσ means the(n?1)-Hausdor ffmeasure on the sphere?Br.Similarly,one can define the frequency centered at other point.

Based on this idea,we define the frequency of a polyharmonic function as follows.We first show some notations in this paper as follows:

Definition 1.2Suppose that u satisfies that△ku=0,where k is a positive integer more than or equal to 2.Such a function u is called a k-polyharmonic function in the rest of this paper.Then we define

where

The function N(r)is called the frequency of u centered at the origin with radius r.Similarly,we can define the frequency centered at other point.

Remark 1.1Noting that for any j=1,2,···,k,ujis a(k?j+1)-polyharmonic function,and ukis a harmonic function.Thus one can also define the frequency for ujas above.We denote such frequency as Nj(r).It is easy to see that N1(r)=N(r),and Nk(r)is just the classical frequency of a harmonic function as in Definition 1.1.

Remark 1.2This frequency is in fact the following form

Here uiνis ?u ·ν and ν is the outer unit normal on ?Br.

Now we state the main results of this paper.

Theorem 1.1Let u be a polyharmonic function inThen

where C is a positive constant depending only on n and k.

Theorem 1.2Let u be a k-polyharmonic function in the whole space Rn.

(1)If the frequency of u centered at the origin is bounded in Rn,then u is a polynomial.Moreover,if N(r)0,then it holds that

where deg(u)means the order of degree of u and C is a positive constant depending only on n and k.In this case,for any i=2,·,k,the functions uiare also polynomials.

(2)If u is a polynomial,then the frequency of u is bounded by the order of degree of u in the whole space Rn.

The rest of this paper is organized as follows.In the second section we introduce some interesting properties of the frequency and prove the monotonicity formula of the frequency.In the third section,the doubling conditions of the polyharmonic functions are proved.The forth section gives the measure estimates of nodal sets of polyharmonic functions,i.e.,the proof of Theorem 1.1.The last section shows the growth property of polyharmonic functions.

2 Monotonicity Formula of Frequency

In this section,we will give some interesting properties for the frequency of polyharmonic functions,and then prove the monotonicity formula for this frequency function.

Lemma 2.1If u satisfies△ku=0,where k∈N and the vanishing order of u at the origin is l≥2(k?1),then

ProofNote that u is k-polyharmonic.So each uiis analytic near the origin,thus we may assume that for eachwhere Pi(x)is a homogeneous polynomial.Assume that the order of degree of Pi(x)is li,and thenasBecause the vanishing order of u at the origin is l,it is known that l1=l,and for each i=2,3,···,k,li≥ l? 2(i? 1).Let l0=inf{l1,l2,···,lk}.Because each Pi(x)is a homogeneous polynomial of degree li,Pi(x)can be written aswhere(r,θ)is the polar coordinate system.Then

where dσ =rn?1dω,dω is the(n?1)-Hausdor ffmeasure on the unit sphere Sn?1.Letin the above form,one can getThat is the desired result.

In order to prove some properties of the proposed frequency,we need the following two lemmas which can be seen in[9,13].

Lemma 2.2If u is a harmonic function in Br,then

Lemma 2.3For anyit holds that

Now we show some properties of such frequency.

Lemma 2.4If n≥2,r≤1,and u is a k-polyharmonic function as above,then the frequency of u satisfies that

where C is a positive constant depending only on n.

ProofFor any fixed i and r,define the functionandas follows:

which is presented in[8,Chapter 2].On the other hand,the functionsare all inso from the Poincar′e’s inequality,we have

Because

and

we have

Now we will give the estimates of|I|,|II|,|III|and|IV|separately.First consider the term|I|.

By using the form(2.5),we have

For|IV|,by using(2.4),we have

Now we focus on|II|.Also from the forms(2.4)–(2.5),we have for any ?>0,it holds that

Similarly,for|III|,we have

So

So

Thus

which is the desired result.

Remark 2.1It is obvious that the result of the above lemmas also hold for the frequency centered at other points.

Remark 2.2The frequency of a harmonic function is obviously nonnegative.For a polyharmonic function,the frequency may not be nonnegative,but from Lemma 2.4,one knows that it also has a lower bound.

Next we will show the monotonicity formula for this frequency.

Theorem 2.1Let u be a k-polyharmonic function.Then there exists two positive constants C0and C depending only on n and k such that if N(r)≥C0,then it holds that

ProofIt is easy to check that

where dσxand dσyare the(n ? 1)-Hausdor ffmeasures on the corresponding spheres.This implies that

So

Now consider D′(r)and E′(r).First note that

where

and

Thus

So

Then we will estimate|R1|,|R2|and|R3|separately.

From the similar arguments,we have

and

From the assumption that N(r)≥C0and the proof of Lemma 2.4,we have

where C is the constant in Lemma 2.4.Choose C0large enough such that

Then

So

Thus

From(2.8),we have

So we finally get

This ends the proof.

Remark 2.3The above theorem also holds for the frequency centered at other point,i.e.,if u is a polyharmonic function and N(p,r)is the frequency of u centered at the point p with radius r,then it holds that

if N(p,r)≥C0,where C0and C are two positive constants depending only on n and k.

Lemma 2.5For anywe have

where C1and C2are positive constants depending only on n and k.

ProofWe only prove the case thatOther cases are similar.Note thatand.From Theorem 2.1,we have

Now we claim that

In fact,from(1.3),(2.8),Lemma 2.4 and some direct calculation,we know that

Thus

and

So the claim(2.12)holds.Integrating(2.13)from,we obtain

From Theorem 2.1,we know that

So

Then from Theorem 2.1 again,we get

3 Doubling Conditions

In this section,we will show the doubling condition of a polyharmonic function u.In fact,from the proof of Lemma 2.5,it is easy to see that the following doubling condition holds.

Lemma 3.1Let u be a k-polyharmonic function,and assume that 2r<1.Then it holds that

and

where C and C′are two positive constants depending only on n and k.

ProofWe only need to prove the form(3.1).Because one can simply integrate(3.1)from 0 to r to get(3.2).

Integrating(2.13)from r to 2r,we know that

and thus

From Theorem 2.1,we know that N(t)≤max{CN(2r),C0}≤CN(2r)+C for any t∈(r,2r).Here C is some positive constant depending only on n and k,and C0is the same constant as in Theorem 2.1.So

which is the desired result.

It is known that the doubling condition for harmonic functions and bi-harmonic functions as follows.

Lemma 3.2Let u be a harmonic function and 2r<1.Then

where N(r)is the frequency of u and C is a positive constant depending only on n.

Lemma 3.3Let u be a bi-harmonic function and 2r<1.Then

where N1(r)is the frequency of u,N2(r)is the frequency of△u,and C is a positive constant depending only on n.

Lemmas 3.2–3.3 can be seen in[9]and[13],respectively.

Now we will show the doubling condition for a polyharmonic function.

Theorem 3.1Let u be a k-polyharmonic function,i.e.,u satisfies thatand assume that 2r<1,n≥2.Then it holds that

where C is a positive constant depending only on n and k.

ProofWe prove this lemma by the inductions.

Assume that we have already known that for any j satisfies k≥j≥l,form(3.5)and the following inequality

holds for uj.From the above two lemmas,we know that for j=k and j=k?1,these two inequalities hold.Now we will prove that the inequalities(3.5)and(3.6)hold for u replaced by ul?1and thus the theorem is proved.

Noting that

it holds that for any text function

and

Put this Ψ into(3.7),we have

Thus

Thus we have

From

and the doubling condition for ul,we have

This shows that(3.6)holds for j=l?1.Then from Lemma 3.1 and the induction assumptions,we have

and thus the desired result holds by inductions.

4 Measure Estimates of Nodal Sets

In this section,we will show the upper bound of the measure of the nodal set for a polyharmonic function u,i.e.,we will give the proof of Theorem 1.1.

To estimate the measure of the nodal set,we need an estimate for the number of zero points of analytic functions which was first proved in[2].

Lemma 4.1Suppose thatis analytic with

for some positive constant N.Then for any r∈(0,1),there holds

where C is a positive constant depending only on r.

We also need the following priori estimate.

Lemma 4.2Let u be a polyharmonic function.Then if 2r<1,we have

where C is a positive constant depending only on n and k.

ProofLet

It also holds that

So

Because ukis a harmonic function,it is known that for any y∈Br,

Thus for any x∈Br,

On the other hand,from the fact that uk?1?wk?1,2ris harmonic in B2r,we know that for any x∈Br,

Then for x∈Br,

That is the desired result for uk?1.Repeat this argument k times,the desired result can be proved.

Now we show the measure estimate of the nodal set{x:u(x)=0}.

Proof of Theorem 1.1Without loss of generality,we may assume

Then from Theorem 3.1 and Lemma 2.5,it holds that

On the other hand,from Lemma 4.2 and(3.6),one knows that for any

and

Using Lemma 4.1,we have

That means

Then from the integral geometric formula,which can be seen in[3,10],we have

and this is the desired result.

5 Growth Property of Polyharmonic Functions

In this section,we will derive a growth behavior of the polyharmonic functions in the whole space Rn.The result is written in Theorem 1.2.

Proof of Theorem 1.2First assume that N(r)is bounded,i.e.,N(r)≤N0on Rn.Then we need to show that u is a polynomial.

Without loss of generality,assume

From the mean value formula and the fact that ukis a harmonic function,we have that

holds for any r>1.For each i=1,2,···,k ? 1,write uiasas in the proof of Lemma 2.4,i.e.,

and

Then from the priori estimate ofand the mean value property of,we have

Thus for uk?1,it holds that

Continue these arguments for k times,we get

Thus from Lemma 3.1 and the assumption(5.1),we have that

holds for any r>1.Thus u must be a polynomial and the order of degree of u is less than or equal to CN0+C,where C is a positive constant depending only on n and k.

If a k-polyharmonic function u is a polynomial,then from the fact that

it is easy to check that N(r)is bounded by the order of degree of u.Of course,for any i=2,·,k,the functions uiare all polynomials.