A remark on the a-minimally thin sets associated with the Schrödinger operator
© Xue; licensee Springer. 2014
Received: 23 February 2014
Accepted: 29 April 2014
Published: 23 May 2014
The aim of this paper is to give a new criterion for a-minimally thin sets at infinity with respect to the Schrödinger operator in a cone, which supplement the results obtained by T. Zhao.
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. Further, intS, diamS, and stand for the interior of S, the diameter of S, and the distance between and , respectively.
We introduce a system of spherical coordinates , , in which are related to cartesian coordinates by .
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 the differentiation at Q along the inward normal into D.
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 (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 . We shall also write for two positive functions and , if and only if there exists a positive constant c such that .
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.
for any .
where is the regularized reduced function of relative to H (with respect to the Schrödinger operator ).
In [, Theorem 5.4.3], Long gave a criterion that characterizes a-minimally thin sets at infinity in a cone.
where and .
In recent work, Zhao (see [, Theorems 1 and 2]) proved the following results. For similar results in the half space with respect to the Schrödinger operator, we refer the reader to the papers by Ren and Su (see [9, 10]).
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 .
This paper aims to show that the sharpness of the characterization of an a-minimally thin set in Theorem C. In order to do this, we introduce the Whitney cubes in a cone.
Theorem 1 If H is a union of cubes from the Whitney cubes of , then equation (7) is also sufficient for H to be a-minimally thin at infinity with respect to .
From the Remark and Theorem 1, we have the following.
If is a non-negative superfunction on and c is a positive number such that for all , then for all .
To prove our results, we need some lemmas.
for any .
which, together with equation (14), gives the conclusion of Lemma 1. □
3 Proof of Theorem 1
Let be a family of cubes from the Whitney cubes of such that . Let be a subfamily of such that , where .
which shows with Theorem A that H is a-minimally thin at infinity with respect to .
This work was supported by the National Natural Science Foundation of China under Grants Nos. 11301140 and U1304102.
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