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Rarefied sets at infinity associated with the Schrödinger operator
Journal of Inequalities and Applications volume 2014, Article number: 247 (2014)
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 be the set of all real numbers and the set of all positive real numbers, respectively. We denote by () the n-dimensional Euclidean space. A point in is denoted by , . The Euclidean distance between two points P and Q in is denoted by . Also with the origin O of is simply denoted by . The boundary and the closure of a set S in are denoted by ∂S and , respectively.
We introduce a system of spherical coordinates , , in which are related to Cartesian coordinates by .
Let D be an arbitrary domain in and let denote the class of non-negative radial potentials , i.e., , , such that with some if and with if or .
If , then the Schrödinger operator
where Δ is the Laplace operator and I is the identical operator, can be extended in the usual way from the space to an essentially self-adjoint operator on (see [, Ch. 11]). We will denote it by as well. This last one has a Green a-function . Here is positive on D and its inner normal derivative , where denotes the differentiation at Q along the inward normal into D.
We call a function that is upper semi-continuous in D a subfunction with respect to the Schrödinger operator if its values belong to the interval and at each point with the generalized mean-value inequality (see )
is satisfied, where is the Green a-function of in and is a surface measure on the sphere .
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 ).
The unit sphere and the upper half unit sphere in are denoted by and , respectively. For simplicity, a point on and the set for a set Ω, , are often identified with Θ and Ω, respectively. For two sets and , the set in is simply denoted by . By we denote the set in with the domain Ω on . We call it a cone. We denote the set with an interval on R by .
We shall say that a set has a covering if there exists a sequence of balls with centers in such that , where is the radius of and is the distance from the origin to the center of .
From now on, we always assume . For the sake of brevity, we shall write instead of . 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 with smooth boundary. Consider the Dirichlet problem
where is the spherical part of the Laplace operator
We denote the least positive eigenvalue of this boundary value problem by λ and the normalized positive eigenfunction corresponding to λ by . In order to ensure the existence of λ and a smooth , we put a rather strong assumption on Ω: if , then Ω is a -domain () on surrounded by a finite number of mutually disjoint closed hypersurfaces (e.g., see [, pp.88-89] for the definition of -domain).
For any , we have (see [, pp.7-8])
where and .
Solutions of an ordinary differential equation
We will also consider the class , consisting of the potentials , such that there exists the finite limit and, moreover, . If , then the (sub)superfunctions are continuous (see ).
In the rest of paper, we assume that and we shall suppress this assumption for simplicity.
Let ν be any positive measure on such that the Green a-potential
for any . Then the positive measure on is defined by
Remark 1 We remark that the total mass is finite (see [, Lemma 5]).
For each , the maximal function is defined by
where and λ is a positive measure on . The set
is denoted by .
It is known that the Martin boundary of is the set , each of which is a minimal Martin boundary point. For and , the Martin kernel can be defined by . If the reference point P is chosen suitably, then we have
for any .
In , Long et al. introduced the notations of a-thin (with respect to the Schrödinger operator ) at a point, a-polar set (with respect to the Schrödinger operator ) and a-rarefied sets at infinity (with respect to the Schrödinger operator ), which generalized earlier notations obtained by Brelot and Miyamoto (see [8, 9]). A set H in is said to be a-thin at a point Q if there is a fine neighborhood E of Q which does not intersect . Otherwise H is said to be not a-thin at Q on . A set H in is called a polar set if there is a superfunction u on some open set E such that . A subset H of is said to be a-rarefied at infinity on if there exists a positive superfunction on such that
Let H be a bounded subset of . Then is bounded on and the greatest a-harmonic minorant of is zero. We see from the Riesz decomposition theorem (see [, Theorem 2]) that there exists a unique positive measure on such that (see [, p.6])
for any and is concentrated on , where
We denote the total mass of by .
By using this positive measure (with respect to the Schrödinger operator ), we can further define another measure on by
for any . It is easy to see that .
Recently, Long et al. (see [, Theorem 2.5]) gave a criterion for a subset H of to be a-rarefied set at infinity.
Theorem A A subset H of is a-rarefied at infinity on if and only if
where and .
In this paper, we shall obtain a series of new criteria for a-rarefied sets at infinity on , which complement Theorem A. Our results are essentially based on Qiao and Deng, Ren and Zhao, Xue (see [2, 11–14]). In order to avoid complexity of our proofs, we shall assume . But our results in this paper are also true for .
First we shall state Theorem 1, which is the main result in this paper.
Theorem 1 A subset H of is a-rarefied at infinity on if and only if there exists a positive measure on such that
for any and
Next we give the geometrical property of a-rarefied sets at infinity.
Theorem 2 If a subset H of is a-rarefied at infinity on , then H has a covering () satisfying
Finally, by an example we show that the reverse of Theorem 2 is not true.
A covering satisfies
from equation (3).
Let be a subset of , i.e., . Suppose that this covering is located as follows: there is an integer such that and for . Then the set is not a-rarefied at infinity on . This fact will be proved in Section 5.
for any and any satisfying (resp. ).
Lemma 2 (see [, Lemma 5])
Let ν be a positive measure on such that there is a sequence of points , () satisfying ( ; ). Then, for a positive number L,
Lemma 3 (see [, Theorem 3])
Let ν be any positive measure on such that for any . Then, for a sufficiently large L,
Lemma 4 (see [, Lemma 6])
Let λ be any positive measure on having finite total mass. Then has a covering () satisfying
3 Proof of Theorem 1
for a positive measure on satisfying equation (6).
Now we shall show the existence of an integer N such that for any integer j (≥N), we have
for any integer j (≥N).
For any , we have
from Lemma 1.
By applying Lemma 2, we can take an integer N such that for any j (≥N),
Thus we obtain
for any , where .
Thus, if (), then we obtain
From equations (4), (7) and (11), we have
where () and
And then we obtain
for . Then we have
in which the last integral is finite by Remark 1. And hence H is a-rarefied set at infinity from Theorem A.
Consider a function on defined by
for any , where .
If we put , then from equation (5) we have that
for any .
Next we shall show that is always finite on . Take any point and a positive integer satisfying . We write
Since is concentrated on , we have that
for . Hence we have
which, together with Theorem A, shows that is finite and hence is also finite for any .
holds on and , we see that for any ()
And hence equation (15) also holds for any . Since is equal to H except a polar set , we can take another positive superfunction on such that with a positive measure on and is identically +∞ on .
Thus we complete the proof of Theorem 1.
5 Proof of an example
Since for any , we have for any , where . Hence we have
for any , where .
Take a measure δ on , , such that
for any , where Cap denotes the Newton capacity. Since
If we observe , then we have by equation (3)
from which it follows by Theorem A that H is not a-rarefied at infinity on .
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This work was supported by the National Natural Science Foundation of China under Grants Nos. 11301140 and U1304102.
The author declares that they have no competing interests.
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Cite this article
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
- rarefied set
- Schrödinger operator
- Green a-potential