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Rarefied sets at infinity associated with the Schrödinger operator

Abstract

This paper gives some criteria for a-rarefied sets at infinity associated with the Schrödinger operator in a cone. Our proofs are based on estimating Green a-potential with a positive measure by connecting with a kind of density of the modified measure. Meanwhile, the geometrical property of this a-rarefied sets at infinity is also considered. By giving an example, we show that the reverse of this property is not true.

1 Introduction and results

Let R and R + be the set of all real numbers and the set of all positive real numbers, respectively. We denote by R n (n2) the n-dimensional Euclidean space. A point in R n is denoted by P=(X, x n ), X=( x 1 , x 2 ,, x n 1 ). The Euclidean distance between two points P and Q in R n is denoted by |PQ|. Also |PO| with the origin O of R n is simply denoted by |P|. The boundary and the closure of a set S in R n are denoted by ∂S and S ¯ , respectively.

We introduce a system of spherical coordinates (r,Θ), Θ=( θ 1 , θ 2 ,, θ n 1 ), in R n which are related to Cartesian coordinates ( x 1 , x 2 ,, x n 1 , x n ) by x n =rcos θ 1 .

Let D be an arbitrary domain in R n and let A a denote the class of non-negative radial potentials a(P), i.e., 0a(P)=a(r), P=(r,Θ)D, such that a L loc b (D) with some b>n/2 if n4 and with b=2 if n=2 or n=3.

If a A a , then the Schrödinger operator

Sch a =Δ+a(P)I=0,

where Δ is the Laplace operator and I is the identical operator, can be extended in the usual way from the space C 0 (D) to an essentially self-adjoint operator on L 2 (D) (see [[1], Ch. 11]). We will denote it by Sch a as well. This last one has a Green a-function G D a (P,Q). Here G D a (P,Q) is positive on D and its inner normal derivative G D a (P,Q)/ n Q 0, where / n Q denotes the differentiation at Q along the inward normal into D.

We call a function u that is upper semi-continuous in D a subfunction with respect to the Schrödinger operator Sch a if its values belong to the interval [,) and at each point PD with 0<r<r(P) the generalized mean-value inequality (see [2])

u(P) S ( P , r ) u(Q) G B ( P , r ) a ( P , Q ) n Q dσ(Q)

is satisfied, where G B ( P , r ) a (P,Q) is the Green a-function of Sch a in B(P,r) and dσ(Q) is a surface measure on the sphere S(P,r)=B(P,r).

If −u is a subfunction, then we call u a superfunction. If a function u is both subfunction and superfunction, it is, clearly, continuous and is called an a-harmonic function (with respect to the Schrödinger operator Sch a ).

The unit sphere and the upper half unit sphere in R n are denoted by S n 1 and S + n 1 , respectively. For simplicity, a point (1,Θ) on S n 1 and the set {Θ;(1,Θ)Ω} for a set Ω, Ω S n 1 , are often identified with Θ and Ω, respectively. For two sets Ξ R + and Ω S n 1 , the set {(r,Θ) R n ;rΞ,(1,Θ)Ω} in R n is simply denoted by Ξ×Ω. By C n (Ω) we denote the set R + ×Ω in R n with the domain Ω on S n 1 . We call it a cone. We denote the set I×Ω with an interval on R by C n (Ω;I).

We shall say that a set H C n (Ω) has a covering { r j , R j } if there exists a sequence of balls { B j } with centers in C n (Ω) such that H j = 0 B j , where r j is the radius of B j and R j is the distance from the origin to the center of B j .

From now on, we always assume D= C n (Ω). For the sake of brevity, we shall write G Ω a (P,Q) instead of G C n ( Ω ) a (P,Q). Throughout this paper, let c denote various positive constants, because we do not need to specify them. Moreover, ϵ appearing in the expression in the following sections will be a sufficiently small positive number.

Let Ω be a domain on S n 1 with smooth boundary. Consider the Dirichlet problem

( Λ n + λ ) φ = 0 on  Ω , φ = 0 on  Ω ,

where Λ n is the spherical part of the Laplace operator Δ n

Δ n = n 1 r r + 2 r 2 + Λ n r 2 .

We denote the least positive eigenvalue of this boundary value problem by λ and the normalized positive eigenfunction corresponding to λ by φ(Θ). In order to ensure the existence of λ and a smooth φ(Θ), we put a rather strong assumption on Ω: if n3, then Ω is a C 2 , α -domain (0<α<1) on S n 1 surrounded by a finite number of mutually disjoint closed hypersurfaces (e.g., see [[3], pp.88-89] for the definition of C 2 , α -domain).

For any (1,Θ)Ω, we have (see [[4], pp.7-8])

c 1 rφ(Θ)δ(P)crφ(Θ),
(1)

where P=(r,Θ) C n (Ω) and δ(P)=dist(P, C n (Ω)).

Solutions of an ordinary differential equation

Q (r) n 1 r Q (r)+ ( λ r 2 + a ( r ) ) Q(r)=0,0<r<.
(2)

It is known (see, for example, [5]) that if the potential a A a , then equation (2) has a fundamental system of positive solutions {V,W} such that V and W are increasing and decreasing, respectively.

We will also consider the class B a , consisting of the potentials a A a , such that there exists the finite limit lim r r 2 a(r)=k[0,) and, moreover, r 1 | r 2 a(r)k|L(1,). If a B a , then the (sub)superfunctions are continuous (see [6]).

In the rest of paper, we assume that a B a and we shall suppress this assumption for simplicity.

Denote

ι k ± = 2 n ± ( n 2 ) 2 + 4 ( k + λ ) 2 ,

then the solutions to equation (2) have the asymptotic (see [3])

c 1 r ι k + V(r)c r ι k + , c 1 r ι k W(r)c r ι k ,as r.
(3)

Let ν be any positive measure on C n (Ω) such that the Green a-potential

G Ω a ν(P)= C n ( Ω ) G Ω a (P,Q)dν(Q)+

for any P C n (Ω). Then the positive measure m(ν) on R n is defined by

dm(ν)(Q)= { W ( t ) φ ( Φ ) d ν ( Q ) , Q = ( t , Φ ) C n ( Ω ; ( 1 , + ) ) , 0 , Q R n C n ( Ω ; ( 1 , + ) ) .

Remark 1 We remark that the total mass m(ν) is finite (see [[2], Lemma 5]).

For each P=(r,Θ) R n {O}, the maximal function M(P;λ,β) is defined by

M(P;λ,β)= sup 0 < ρ < r 2 λ ( B ( P , ρ ) ) ρ β ,

where β0 and λ is a positive measure on R n . The set

{ P = ( r , Θ ) R n { O } ; M ( P ; λ , β ) r β > ϵ }

is denoted by E(ϵ;λ,β).

It is known that the Martin boundary of C n (Ω) is the set C n (Ω){}, each of which is a minimal Martin boundary point. For P C n (Ω) and Q C n (Ω){}, the Martin kernel can be defined by M Ω a (P,Q). If the reference point P is chosen suitably, then we have

M Ω a (P,)=V(r)φ(Θ)and M Ω a (P,O)=cW(r)φ(Θ)
(4)

for any P=(r,Θ) C n (Ω).

In [7], Long et al. introduced the notations of a-thin (with respect to the Schrödinger operator Sch a ) at a point, a-polar set (with respect to the Schrödinger operator Sch a ) and a-rarefied sets at infinity (with respect to the Schrödinger operator Sch a ), which generalized earlier notations obtained by Brelot and Miyamoto (see [8, 9]). A set H in R n is said to be a-thin at a point Q if there is a fine neighborhood E of Q which does not intersect H{Q}. Otherwise H is said to be not a-thin at Q on C n (Ω). A set H in R n is called a polar set if there is a superfunction u on some open set E such that H{PE;u(P)=}. A subset H of C n (Ω) is said to be a-rarefied at infinity on C n (Ω) if there exists a positive superfunction v(P) on C n (Ω) such that

inf P C n ( Ω ) v ( P ) M Ω a ( P , ) 0

and

H { P = ( r , Θ ) C n ( Ω ) ; v ( P ) V ( r ) } .

Let H be a bounded subset of C n (Ω). Then R ˆ M Ω a ( , ) H is bounded on C n (Ω) and the greatest a-harmonic minorant of R ˆ M Ω a ( , ) H is zero. We see from the Riesz decomposition theorem (see [[10], Theorem 2]) that there exists a unique positive measure λ H a on C n (Ω) such that (see [[7], p.6])

R ˆ M Ω a ( , ) H (P)= G Ω a λ H a (P)
(5)

for any P C n (Ω) and λ H a is concentrated on I H , where

I H = { P C n ( Ω ) ; H  is not  a -thin at  P } .

We denote the total mass λ H a ( C n (Ω)) of λ H a by λ Ω a (H).

By using this positive measure λ H a (with respect to the Schrödinger operator Sch a ), we can further define another measure η H a on C n (Ω) by

d η H a (P)= M Ω a (P,)d λ H a (P)

for any P C n (Ω). It is easy to see that η H a ( C n (Ω))<+.

Recently, Long et al. (see [[7], Theorem 2.5]) gave a criterion for a subset H of C n (Ω) to be a-rarefied set at infinity.

Theorem A A subset H of C n (Ω) is a-rarefied at infinity on C n (Ω) if and only if

j = 0 λ Ω a ( H j )W ( 2 j ) <,

where H j =H C n (Ω;[ 2 j , 2 j + 1 )) and j=0,1,2, .

In this paper, we shall obtain a series of new criteria for a-rarefied sets at infinity on C n (Ω), which complement Theorem A. Our results are essentially based on Qiao and Deng, Ren and Zhao, Xue (see [2, 1114]). In order to avoid complexity of our proofs, we shall assume n3. But our results in this paper are also true for n=2.

First we shall state Theorem 1, which is the main result in this paper.

Theorem 1 A subset H of C n (Ω) is a-rarefied at infinity on C n (Ω) if and only if there exists a positive measure ξ H a on C n (Ω) such that

G Ω a ξ H a (P)+
(6)

for any P C n (Ω) and

H { P = ( r , Θ ) C n ( Ω ) ; G Ω a ξ H a ( P ) V ( r ) } .
(7)

Next we give the geometrical property of a-rarefied sets at infinity.

Theorem 2 If a subset H of C n (Ω) is a-rarefied at infinity on C n (Ω), then H has a covering { r j , R j } (j=0,1,2,) satisfying

j = 0 ( r j R j ) V ( R j r j ) W ( R j r j ) <.
(8)

Finally, by an example we show that the reverse of Theorem 2 is not true.

Example Put

r j =3 2 j 1 j 1 2 n and R j =3 2 j 1 (j=1,2,3,).

A covering { r j , R j } satisfies

j = 1 ( r j R j ) V ( R j r j ) W ( R j r j ) c j = 1 ( r j R j ) n 1 =c j = 1 j n 1 2 n <+

from equation (3).

Let C n ( Ω ) be a subset of C n (Ω), i.e., Ω ¯ Ω. Suppose that this covering is located as follows: there is an integer j 0 such that B j C n ( Ω ) and R j >2 r j for j j 0 . Then the set H= j = j 0 B j is not a-rarefied at infinity on C n (Ω). This fact will be proved in Section 5.

2 Lemmas

Lemma 1 (see [[1], Ch. 11] and [[15], Lemma 4])

G Ω a ( P , Q ) c V ( t ) W ( r ) φ ( Θ ) φ ( Φ ) ( resp.  G Ω a ( P , Q ) c V ( r ) W ( t ) φ ( Θ ) φ ( Φ ) )

for any P=(r,Θ) C n (Ω) and any Q=(t,Φ) C n (Ω) satisfying r2t (resp. t2r).

Lemma 2 (see [[2], Lemma 5])

Let ν be a positive measure on C n (Ω) such that there is a sequence of points P i =( r i , Θ i ) C n (Ω), r i + (i+) satisfying G Ω a ν( P i )<+ (i=1,2, ; Q C n (Ω)). Then, for a positive number L,

C n ( Ω ; ( L , + ) ) W(t)φ(Φ)dν(Q)<+

and

lim R + W ( R ) V ( R ) C n ( Ω ; ( 0 , R ) ) V(t)φ(Φ)dν(Q)=0.

Lemma 3 (see [[2], Theorem 3])

Let ν be any positive measure on C n (Ω) such that G Ω a ν(P)+ for any P C n (Ω). Then, for a sufficiently large L,

{ P = ( r , Θ ) C n ( Ω ; ( L , + ) ) ; G Ω a ν ( P ) V ( r ) φ ( Θ ) } E ( ϵ ; m ( ν ) , n 1 ) .

Lemma 4 (see [[2], Lemma 6])

Let λ be any positive measure on R n having finite total mass. Then E(ϵ;λ,n1) has a covering { r j , R j } (j=1,2,) satisfying

j = 1 ( r j R j ) V ( R j r j ) W ( R j r j ) <.
(9)

3 Proof of Theorem 1

Suppose that

HΠ ( ξ H a ) = { P = ( r , Θ ) C n ( Ω ) ; G Ω a ξ H a ( P ) V ( r ) }
(10)

for a positive measure ξ H a on C n (Ω) satisfying equation (6).

We write

G Ω a ν(P)= G Ω a (1,j)(P)+ G Ω a (2,j)(P)+ G Ω a (3,j)(P),

where

G Ω a ( 1 , j ) ( P ) = C n ( Ω ; ( 0 , 2 j 1 ) ) G Ω a ( P , Q ) d ν ( Q ) , G Ω a ( 2 , j ) ( P ) = C n ( Ω ; [ 2 j 1 , 2 j + 2 ) ) G Ω a ( P , Q ) d ν ( Q )

and

G Ω a (3,j)(P)= C n ( Ω ; [ 2 j + 2 , ) ) G Ω a (P,Q)dν(Q).

Now we shall show the existence of an integer N such that for any integer j (≥N), we have

Π ( ξ H a ) (j) { P = ( r , Θ ) C n ( Ω ; [ 2 j , 2 j + 1 ) ) ; 2 G Ω a ( 2 , j ) ( P ) V ( r ) }
(11)

for any integer j (≥N).

For any P=(r,Θ) C n (Ω;[ 2 j , 2 j + 1 )), we have

G Ω a (1,j)(P)cW(r)φ(Θ) C n ( Ω ; ( 0 , 2 j 1 ) ) V(t)φ(Φ)dν(Q)

and

G Ω a (3,j)(P)cV(r)φ(Θ) C n ( Ω ; [ 2 j + 2 , ) ) dm(ν)(Q)

from Lemma 1.

By applying Lemma 2, we can take an integer N such that for any j (≥N),

W ( 2 j ) V 1 ( 2 j ) C n ( Ω ; ( 0 , 2 j 1 ) ) V(t)φ(Φ)dν(Q) 1 4 c

and

C n ( Ω ; [ 2 j + 2 , ) ) dm(ν)(Q) 1 4 c .

Thus we obtain

4 G Ω a (1,j)(P)V(r)φ(Θ)
(12)

and

4 G Ω a (3,j)(P)V(r)φ(Θ)
(13)

for any P=(r,Θ) C n (Ω;[ 2 j , 2 j + 1 )), where jN.

Thus, if P=(r,Θ)Π(ν)(j) (jN), then we obtain

2 G Ω a (1,j)(P)V(r)φ(Θ)

from equations (12) and (13), which gives equation (11).

From equations (4), (7) and (11), we have

G Ω a (2,j)(P)= C n ( Ω ) G Ω a (P,Q)d τ j a (Q) M Ω a (P,),

where P I j (jN) and

d τ j a (Q)= { 2 1 j d ξ H a ( Q ) , Q C n ( Ω ; [ 2 j 1 , 2 j + 2 ) ) , 0 , Q C n ( Ω ; ( 0 , 2 j 1 ) ) C n ( Ω ; [ 2 j + 2 , ) ) .

And then we obtain

η H j a ( C n ( Ω ) ) C n ( Ω ) V(t)φ(Φ)d τ j a (Q)= C n ( Ω ; [ 2 j 1 , 2 j + 2 ) ) V(t)φ(Φ)d ξ H a (Q)

for jN. Then we have

j = N λ Ω a ( H j )W ( 2 j ) = j = N η H j a ( C n ( Ω ) ) W ( 2 j ) c C n ( Ω ; [ 2 N 1 , ) ) dm ( ξ H a ) ,

in which the last integral is finite by Remark 1. And hence H is a-rarefied set at infinity from Theorem A.

Suppose that

j = 0 λ Ω a ( H j )W ( 2 j ) <.

Consider a function f H a (P) on C n (Ω) defined by

f H a (P)= j = 1 R ˆ M Ω a ( , ) H j (P)

for any P C n (Ω), where H 1 =H C n (Ω;(0,1)).

If we put μ H a (1)(P)= j = 1 λ H j a (P), then from equation (5) we have that

f H a (P)= C n ( Ω ) G Ω a (P,Q)d μ H a (1)(Q)

for any P C n (Ω).

Next we shall show that f H a (P) is always finite on C n (Ω). Take any point P=(r,Θ) C n (Ω) and a positive integer j(P) satisfying r 2 j ( P ) + 1 . We write

f H a (P)= f H a (1)(P)+ f H a (2)(P),

where

f H a ( 1 ) ( P ) = j = 1 j ( P ) + 1 C n ( Ω ) G Ω a ( P , Q ) d λ H j a ( Q ) and f H a ( 2 ) ( P ) = j = j ( P ) + 2 C n ( Ω ) G Ω a ( P , Q ) d λ H j a ( Q ) .

Since λ H j a is concentrated on I H j H ¯ j C n (Ω), we have that

C n ( Ω ) G Ω a ( P , Q ) d λ H j a ( Q ) c V ( r ) φ ( Θ ) C n ( Ω ) W ( t ) φ ( Φ ) d λ H j a ( t , Φ ) c V ( r ) φ ( Θ ) W ( 2 j ) V 1 ( 2 j ) C n ( Ω ) V ( t ) φ ( Φ ) d λ H j a ( t , Φ )

for jj(P)+2. Hence we have

f H a (2)(P)cV(r)φ(Θ) j = j ( P ) + 2 η H j a ( C n ( Ω ) ) W ( 2 j ) V 1 ( 2 j ) ,
(14)

which, together with Theorem A, shows that f H a (2)(P) is finite and hence f H a (P) is also finite for any P C n (Ω).

Since

R ˆ M Ω a ( , ) H j (P)= M Ω a (P,)

holds on I H j and I H j H ¯ j C n (Ω), we see that for any P=(r,Θ) I H j (j=1,0,1,2,3,)

f H a (P)c R ˆ M Ω a ( , ) H j (P)V(r)φ(Θ).
(15)

And hence equation (15) also holds for any P=(r,Θ) H = j = 1 I H j . Since H is equal to H except a polar set H 0 , we can take another positive superfunction f H a (3)(P) on C n (Ω) such that f H a (3)(P)= G Ω a μ H a (2)(P) with a positive measure μ H a (2)(P) on C n (Ω) and f H a (3)(P) is identically +∞ on H 0 .

Finally, we can define a positive superfunction g on C n (Ω) by g(P)= f H a (P)+ f H a (3)(P)= G Ω a ξ H a (P) for any P C n (Ω) with ξ H a = μ H a (1)+ μ H a (2). Also we see from equation (15) that equations (6) and (7) hold.

Thus we complete the proof of Theorem 1.

4 Proof of Theorem 2

From Theorem 1 and Lemma 3, we have a positive number L such that

H C n ( Ω ; ( L , + ) ) E ( ϵ ; m ( ξ H a ) , n 1 ) .

Hence by Remark 1 and Lemma 4, E(ϵ;m( ξ H a ),n1) has a covering { r j , R j } (j=1,2,3,) satisfying equation (9) and hence H has also a covering { r j , R j } (j=0,1,2,3,) with an additional finite B 0 covering C n (Ω;(0,L]), satisfying equation (8), which is the conclusion of Theorem 2.

5 Proof of an example

Since φ(Θ)c for any Θ Ω , we have M Ω a (P,)cV( R j ) for any P B ¯ j , where j j 0 . Hence we have

R ˆ M Ω a ( , ) B j (P)cV( R j )
(16)

for any P B ¯ j , where j j 0 .

Take a measure δ on C n (Ω), suppδ B ¯ j , δ( B ¯ j )=1 such that

C n ( Ω ) | P Q | 2 n dδ(P)= { Cap ( B ¯ j ) } 1
(17)

for any Q B ¯ j , where Cap denotes the Newton capacity. Since

G Ω a (P,Q) | P Q | 2 n

for any P C n (Ω) and Q C n (Ω) (see [16], the case n=2 is implicitly contained in [17]),

{ Cap ( B ¯ j ) } 1 λ B j a ( C n ( Ω ) ) = ( | P Q | 2 n d δ ( P ) ) d λ B j a ( Q ) ( G Ω a ( P , Q ) d λ B j a ( Q ) ) d δ ( P ) = R ˆ M Ω a ( , ) B j d δ ( P ) c V ( R j ) δ ( B ¯ j ) = c V ( R j )

from equations (16) and (17). Hence we have

λ B j a ( C n ( Ω ) ) cCap( B ¯ j )V( R j )c r j n 2 V( R j ).
(18)

If we observe λ H j a ( C n (Ω))= λ B j a ( C n (Ω)), then we have by equation (3)

j = j 0 W ( 2 j ) λ H j a ( C n ( Ω ) ) c j = j 0 ( r j R j ) n 2 =c j = j 0 1 j =+,

from which it follows by Theorem A that H is not a-rarefied at infinity on C n (Ω).

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Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grants Nos. 11301140 and U1304102.

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Xue, G. Rarefied sets at infinity associated with the Schrödinger operator. J Inequal Appl 2014, 247 (2014). https://doi.org/10.1186/1029-242X-2014-247

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Keywords

  • rarefied set
  • Schrödinger operator
  • Green a-potential