Abstract
The aim of this paper is to give a new criterion for aminimally 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 apotential1 Introduction and results
Let R and
We introduce a system of spherical coordinates
Let D be an arbitrary domain in
If
where Δ is the Laplace operator and I is the identical operator, can be extended in the usual way from the space
We call a function
satisfied, where
The unit sphere and the upper half unit sphere in
From now on, we always assume
Let Ω be a domain on
where
We denote the least positive eigenvalue of this boundary value problem by λ and the normalized positive eigenfunction corresponding to λ by
For any
where
Solutions of an ordinary differential equation (see [[5], p.217])
It is well known (see, for example, [6]) that if the potential
We will also consider the class
Denote
then the solutions to equation (2) have the asymptotic (see [3])
It is well known that the Martin boundary of
for any
In [[8], p.67], Zhao introduce the notations of athin (with respect to the Schrödinger operator
where
Let H be a bounded subset of
for any
The Green aenergy
Also, we can define a measure
In [[8], Theorem 5.4.3], Long gave a criterion that characterizes aminimally thin sets at infinity in a cone.
Theorem AA subsetHof
where
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
(II) There exists a positive superfunction
and
(III) There exists a positive superfunction
Theorem CIf a subsetHof
Remark From equation (3), we immediately know that equation (6) is equivalent to
This paper aims to show that the sharpness of the characterization of an aminimally thin set in Theorem C. In order to do this, we introduce the Whitney cubes in a cone.
A cube is the form
where j,
(I)
(II)
(III)
Theorem 1IfHis a union of cubes from the Whitney cubes of
From the Remark and Theorem 1, we have the following.
Corollary 1Let
Corollary 2LetHbe a Borel measurable subset of
If
2 Lemmas
To prove our results, we need some lemmas.
Lemma 1Let
Proof If we apply a result of Long (see [[8], Theorem 6.1.3]) for compact set
for any
for any
Let
for any
for any
from equations (8), (9), (10), and (11). Since
from equations (3), (9), and (10), we have from (12)
Since
we obtain from equation (13)
On the other hand, we have from equation (1)
which, together with equation (14), gives the conclusion of Lemma 1. □
3 Proof of Theorem 1
Let
Since
for
which, together with equation (1), gives
for
for
which shows with Theorem A that H is aminimally thin at infinity with respect to
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.
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