Rarefied sets at infinity associated with the Schrödinger operator
© Xue; licensee Springer. 2014
Received: 26 March 2014
Accepted: 30 May 2014
Published: 18 July 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 .
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.
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.
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).
where and .
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.
Remark 1 We remark that the total mass is finite (see [, Lemma 5]).
is denoted by .
for any .
We denote the total mass of 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.
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.
Next we give the geometrical property of a-rarefied sets at infinity.
Finally, by an example we show that the reverse of Theorem 2 is not true.
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])
Lemma 3 (see [, Theorem 3])
Lemma 4 (see [, Lemma 6])
3 Proof of Theorem 1
for a positive measure on satisfying equation (6).
for any integer j (≥N).
from Lemma 1.
for any , where .
in which the last integral is finite by Remark 1. And hence H is a-rarefied set at infinity from Theorem A.
for any , where .
for any .
which, together with Theorem A, shows that is finite and hence is also finite 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.
4 Proof of Theorem 2
5 Proof of an example
for any , where .
from which it follows by Theorem A that H is not a-rarefied at infinity on .
This work was supported by the National Natural Science Foundation of China under Grants Nos. 11301140 and U1304102.
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