Minimally thin sets associated with the stationary Schrödinger operator
© Zhao; licensee Springer. 2014
Received: 29 November 2013
Accepted: 31 January 2014
Published: 13 February 2014
This paper gives some new criteria for a-minimally thin sets at infinity with respect to the Schrödinger operator in a cone, which supplement the results obtained by Long-Gao-Deng.
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 O the origin 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 denote the class of nonnegative 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 as well. This last one has a Green a-function . Here is positive on D and its inner normal derivative , where denotes 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 superfunctions (with respect to the Schrödinger operator ). 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 .
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.
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 the finite limit exists, and moreover, . If , then the (sub)superfunctions are continuous (see ). In the rest of this paper, we assume that and we shall suppress this assumption for simplicity.
for any .
where is the regularized reduced function of relative to H (with respect to the Schrödinger operator ).
Recently, Long-Gao-Deng (see [, Theorem 2.5]) gave a criterion that characterizes a-minimally thin sets at infinity in a cone.
where and .
In this paper, we shall obtain a series of new criteria for a-minimally thin sets at infinity on , which complemented Theorem A by the way completely different from theirs. Our results are essentially based on Ren and Su (see [9, 10]).
First we have the following.
A subset H of is a-minimally thin at infinity on .
- (II)There exists a positive superfunction on such that(5)and
There exists a positive superfunction on such that even if for any , there exists satisfying .
Next we shall state Theorem 2, which is the main result in this paper.
In our discussions, the following estimate for the Green a-potential is fundamental, as follows from .
for any and any satisfying .
for every (see [, Theorem 2]).
and further , where denotes the closure of in .
which gives the conclusion of Lemma 2. □
3 Proof of Theorem 1
for any and any integer j satisfying .
If we put , then this shows that is the function required in (II).
Now we shall show that (III) follows from (II). Let be the function in (II). It follows that for any . On the other hand, from equation (5) we can find a point such that . Therefore satisfies (III) with .
where is a positive constant depending only on ∞ and v. Since there exists satisfying , we note that . Now we obtain for any . Hence by a result of [, p.69], H is a-minimally thin at infinity on with respect to the Schrödinger operator, which is the statement of (I). Thus we complete the proof of Theorem 1.
4 Proof of Theorem 2
where is the set in equation (10) and is the n-dimensional Lebesgue measure of .
which is the conclusion of Theorem 2.
This work was supported by the National Natural Science Foundation of China under Grants Nos. 11301140 and U1304102. The author would like to thank two anonymous referees for numerous insightful comments and suggestions, which have greatly improved the paper.
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