SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research

A remark on the a-minimally thin sets associated with the Schrödinger operator

Gaixian Xue

Author Affiliations

School of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou, 450046, China

Boundary Value Problems 2014, 2014:133  doi:10.1186/1687-2770-2014-133

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/133


Received:23 February 2014
Accepted:29 April 2014
Published:23 May 2014

© 2014 Xue; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Abstract

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.

Keywords:
minimally thin set; Schrödinger operator; Green a-potential

1 Introduction and results

Let R and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M1">View MathML</a> be the set of all real numbers and the set of all positive real numbers, respectively. We denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M2">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M3">View MathML</a>) the n-dimensional Euclidean space. A point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M2">View MathML</a> is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M6">View MathML</a>. The Euclidean distance between two points P and Q in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M2">View MathML</a> is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M8">View MathML</a>. Also <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M9">View MathML</a> with the origin O of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M2">View MathML</a> is simply denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M11">View MathML</a>. The boundary and the closure of a set S in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M2">View MathML</a> are denoted by ∂S and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M13">View MathML</a>, respectively. Further, intS, diamS, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M14">View MathML</a> stand for the interior of S, the diameter of S, and the distance between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M15">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M16">View MathML</a>, respectively.

We introduce a system of spherical coordinates <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M18">View MathML</a>, in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M2">View MathML</a> which are related to cartesian coordinates <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M20">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M21">View MathML</a>.

Let D be an arbitrary domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M2">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M23">View MathML</a> denote the class of non-negative radial potentials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M24">View MathML</a>, i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M26">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M27">View MathML</a> with some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M28">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M29">View MathML</a> and with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M30">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M31">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M32">View MathML</a> (see [[1], p.354] and [2]).

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M33">View MathML</a>, then the stationary Schrödinger operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M34">View MathML</a>

where Δ is the Laplace operator and I is the identical operator, can be extended in the usual way from the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M35">View MathML</a> to an essentially self-adjoint operator on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M36">View MathML</a> (see [[1], Ch. 11]). We will denote it <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M37">View MathML</a> as well. This last one has a Green a-function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M38">View MathML</a>. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M38">View MathML</a> is positive on D and its inner normal derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M40">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M41">View MathML</a> denotes the differentiation at Q along the inward normal into D.

We call a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M42">View MathML</a> that is upper semi-continuous in D a subfunction with respect to the Schrödinger operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M37">View MathML</a> if its values belong to the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M44">View MathML</a> and at each point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M45">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M46">View MathML</a> we have the generalized mean-value inequality (see [[1], Ch. 11])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M47">View MathML</a>

satisfied, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M48">View MathML</a> is the Green a-function of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M37">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M50">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M51">View MathML</a> is a surface measure on the sphere <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M52">View MathML</a>. If −u is a subfunction, then we call u a superfunction (with respect to the Schrödinger operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M37">View MathML</a>).

The unit sphere and the upper half unit sphere in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M2">View MathML</a> are denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M55">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M56">View MathML</a>, respectively. For simplicity, a point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M57">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M55">View MathML</a> and the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M59">View MathML</a> for a set Ω, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M60">View MathML</a>, are often identified with Θ and Ω, respectively. For two sets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M61">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M60">View MathML</a>, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M63">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M2">View MathML</a> is simply denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M65">View MathML</a>. By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66">View MathML</a>, we denote the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M67">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M2">View MathML</a> with the domain Ω on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M55">View MathML</a>. We call it a cone. We denote the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M70">View MathML</a> with an interval on R by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M71">View MathML</a>.

From now on, we always assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M72">View MathML</a>. For the sake of brevity, we shall write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M73">View MathML</a> instead of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M74">View MathML</a>. We shall also write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M75">View MathML</a> for two positive functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M76">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M77">View MathML</a>, if and only if there exists a positive constant c such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M78">View MathML</a>.

Let Ω be a domain on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M55">View MathML</a> with smooth boundary. Consider the Dirichlet problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M80">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M81">View MathML</a> is the spherical part of the Laplace operata <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M82">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M83">View MathML</a>

We denote the least positive eigenvalue of this boundary value problem by λ and the normalized positive eigenfunction corresponding to λ by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M84">View MathML</a>. In order to ensure the existence of λ and a smooth <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M84">View MathML</a>, we put a rather strong assumption on Ω: if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M86">View MathML</a>, then Ω is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M87">View MathML</a>-domain (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M88">View MathML</a>) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M55">View MathML</a> surrounded by a finite number of mutually disjoint closed hypersurfaces (e.g. see [[3], pp.88-89] for the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M87">View MathML</a>-domain).

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M91">View MathML</a>, we have (see [[4], pp.7-8])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M92">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M93">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M94">View MathML</a>.

Solutions of an ordinary differential equation (see [[5], p.217])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M95">View MathML</a>

(2)

It is well known (see, for example, [6]) that if the potential <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M33">View MathML</a>, then equation (2) has a fundamental system of positive solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M97">View MathML</a> such that V and W are increasing and decreasing, respectively.

We will also consider the class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M98">View MathML</a>, consisting of the potentials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M33">View MathML</a> such that there exists the finite limit <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M100">View MathML</a>, and, moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M101">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M102">View MathML</a>, then the (sub)superfunctions are continuous (see [7]). In the rest of paper, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M103">View MathML</a> and we shall suppress this assumption for simplicity.

Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M104">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M105">View MathML</a>

(3)

It is well known that the Martin boundary of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66">View MathML</a> is the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M107">View MathML</a>, each of which is a minimal Martin boundary point. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M108">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M109">View MathML</a>, the Martin kernel can be defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M110">View MathML</a>. If the reference point P is chosen suitably, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M111">View MathML</a>

(4)

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M93">View MathML</a>.

In [[8], p.67], Zhao introduce the notations of a-thin (with respect to the Schrödinger operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M37">View MathML</a>) at a point, a-polar set (with respect to the Schrödinger operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M37">View MathML</a>) and a-minimal thin sets at infinity (with respect to the Schrödinger operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M37">View MathML</a>). A set H in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M2">View MathML</a> is said to be a-thin at a point Q if there is a fine neighborhood E of Q which does not intersect <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M117">View MathML</a>. Otherwise H is said to be not a-thin at Q on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66">View MathML</a>. A set H in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M2">View MathML</a> is called a polar set if there is a superfunction u on some open set E such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M120">View MathML</a>. A subset H of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66">View MathML</a> is said to be a-minimal thin at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M109">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66">View MathML</a>, if there exists a point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M108">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M125">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M126">View MathML</a> is the regularized reduced function of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M127">View MathML</a> relative to H (with respect to the Schrödinger operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M37">View MathML</a>).

Let H be a bounded subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M130">View MathML</a> is bounded on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66">View MathML</a> and hence the greatest a-harmonic minorant of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M132">View MathML</a> is zero. When by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M133">View MathML</a> we denote the Green a-potential with a positive measure μ on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66">View MathML</a>, we see from the Riesz decomposition theorem that there exists a unique positive measure <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M135">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M137">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M108">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M135">View MathML</a> is concentrated on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M140">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M141">View MathML</a>

The Green a-energy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M142">View MathML</a> (with respect to the Schrödinger operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M37">View MathML</a>) of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M135">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M145">View MathML</a>

Also, we can define a measure <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M146">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M148">View MathML</a>

In [[8], Theorem 5.4.3], Long gave a criterion that characterizes a-minimally thin sets at infinity in a cone.

Theorem AA subsetHof<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66">View MathML</a>is a-minimally thin at infinity on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66">View MathML</a>if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M151">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M152">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M153">View MathML</a> .

In recent work, Zhao (see [[2], 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]).

Theorem BThe following statements are equivalent.

(I) A subsetHof<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66">View MathML</a>is a-minimally thin at infinity on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66">View MathML</a>.

(II) There exists a positive superfunction<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M156">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M158">View MathML</a>

(5)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M159">View MathML</a>

(III) There exists a positive superfunction<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M156">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66">View MathML</a>such that even if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M162">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M163">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M164">View MathML</a>satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M165">View MathML</a>.

Theorem CIf a subsetHof<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66">View MathML</a>is a-minimally thin at infinity on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M168">View MathML</a>

(6)

Remark From equation (3), we immediately know that equation (6) is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M169">View MathML</a>

(7)

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.

A cube is the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M170">View MathML</a>

where j, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M171">View MathML</a> are integers. The Whitney cubes of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66">View MathML</a> are a family of cubes having the following properties:

(I) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M173">View MathML</a>.

(II) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M174">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M175">View MathML</a>).

(III) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M176">View MathML</a>.

Theorem 1IfHis a union of cubes from the Whitney cubes of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66">View MathML</a>, then equation (7) is also sufficient forHto be a-minimally thin at infinity with respect to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66">View MathML</a>.

From the Remark and Theorem 1, we have the following.

Corollary 1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M156">View MathML</a>be a positive superfunction on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66">View MathML</a>such that equation (5) holds. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M181">View MathML</a>

Corollary 2LetHbe a Borel measurable subset of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66">View MathML</a>satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M183">View MathML</a>

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M156">View MathML</a>is a non-negative superfunction on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66">View MathML</a>andcis a positive number such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M162">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M163">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M162">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M108">View MathML</a>.

2 Lemmas

To prove our results, we need some lemmas.

Lemma 1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M190">View MathML</a>be a cube from the Whitney cubes of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66">View MathML</a>. Then there exists a constant cindependent ofksuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M192">View MathML</a>

Proof If we apply a result of Long (see [[8], Theorem 6.1.3]) for compact set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M193">View MathML</a>, we obtain a measure μ on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M195">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M196">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M197">View MathML</a>

(8)

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M198">View MathML</a>. Also there exists a positive measure <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M199">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M201">View MathML</a>

(9)

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M108">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M203">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M204">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M205">View MathML</a> be the center of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M190">View MathML</a>, the diameter of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M207">View MathML</a>, the distance between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M190">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M209">View MathML</a>, respectively. Then we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M210">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M211">View MathML</a>. Then from equation (1) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M212">View MathML</a>

(10)

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M198">View MathML</a>. We can also prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M214">View MathML</a>

(11)

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M198">View MathML</a> and any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M216">View MathML</a>. Hence we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M217">View MathML</a>

(12)

from equations (8), (9), (10), and (11). Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M218">View MathML</a>

from equations (3), (9), and (10), we have from (12)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M219">View MathML</a>

(13)

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M220">View MathML</a>

we obtain from equation (13)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M221">View MathML</a>

(14)

On the other hand, we have from equation (1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M222">View MathML</a>

which, together with equation (14), gives the conclusion of Lemma 1. □

3 Proof of Theorem 1

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M223">View MathML</a> be a family of cubes from the Whitney cubes of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M225">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M226">View MathML</a> be a subfamily of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M223">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M228">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M229">View MathML</a> .

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M230">View MathML</a> is a countably subadditive set function (see [[8], p.49]), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M231">View MathML</a>

(15)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M232">View MathML</a> . Hence for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M232">View MathML</a> we see from Lemma 1

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M234">View MathML</a>

(16)

which, together with equation (1), gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M235">View MathML</a>

(17)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M232">View MathML</a> . Thus equations (15), (16), and (17) give

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M237">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M232">View MathML</a> . Finally we obtain from equation (1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M239">View MathML</a>

which shows with Theorem A that H is a-minimally thin at infinity with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/133/mathml/M66">View MathML</a>.

Competing interests

The author declares that they have no competing interests.

Acknowledgements

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

References

  1. Levin, B, Kheyfits, A: Asymptotic behavior of subfunctions of time-independent Schrödinger operator. Some Topics on Value Distribution and Differentiability in Complex and P-Adic Analysis, pp. 323–397. Science Press, Beijing (2008)

  2. Zhao, T: Minimally thin sets at infinity with respect to the Schrödinger operator. J. Inequal. Appl.. 2014, (2014) Article ID 67

  3. Gilbarg, D, Trudinger, NS: Elliptic Partial Differential Equations of Second Order, Springer, Berlin (1977)

  4. Courant, R, Hilbert, D: Methods of Mathematical Physics, Interscience, New York (2008)

  5. Miranda, C: Partial Differential Equations of Elliptic Type, Springer, London (1970)

  6. Verzhbinskii, GM, Maz’ya, VG: Asymptotic behavior of solutions of elliptic equations of the second order close to a boundary. I. Sib. Mat. Zh.. 12, 874–899 (1971)

  7. Simon, B: Schrödinger semigroups. Bull. Am. Math. Soc.. 7, 447–526 (1982). Publisher Full Text OpenURL

  8. Long, PH: The Characterizations of Exceptional Sets and Growth Properties in Classical or Nonlinear Potential Theory. Dissertation of Beijing Normal University, Beijing (2012)

  9. Ren, YD: Solving integral representations problems for the stationary Schrödinger equation. Abstr. Appl. Anal.. 2013, (2013) Article ID 715252

  10. Su, BY: Dirichlet problem for the Schrödinger operator in a half space. Abstr. Appl. Anal.. 2012, (2012) Article ID 578197