Sharp geometrical properties of a-rarefied sets via fixed point index for the Schrödinger operator equations
- Zhiqiang Li^{1} and
- Beatriz Ychussie^{2}Email author
https://doi.org/10.1186/s13663-015-0342-1
© Li and Ychussie 2015
Received: 23 January 2015
Accepted: 28 May 2015
Published: 16 June 2015
Abstract
In this paper, we use the theory of fixed point index for the Schrödinger operator equations to obtain a geometrical property of a-rarefied sets at infinity on cones. Meanwhile, we give an example to show that the reverse of this property is not true.
Keywords
1 Introduction and main theorem
Let R and \({\mathbf{R}}_{+}\) be the set of all real numbers and the set of all positive real numbers, respectively. We denote by \({\mathbf{R}}^{n}\) (\(n\geq2\)) the n-dimensional Euclidean space. A point in \({\mathbf{R}}^{n}\) is denoted by \(P=(X,x_{n})\), \(X=(x_{1},x_{2},\ldots,x_{n-1})\). The Euclidean distance between two points P and Q in \({\mathbf{R}}^{n}\) is denoted by \(|P-Q|\). Also \(|P-O|\) with the origin O of \({\mathbf{R}}^{n}\) is simply denoted by \(|P|\). The boundary and the closure of a set S in \({\mathbf{R}}^{n}\) are denoted by ∂S and \(\overline{S}\), respectively. For \(P\in{\mathbf{R}}^{n}\) and \(r>0\), let \(B(P,r)\) denote the open ball with center at P and radius r in \({\mathbf{R}}^{n}\).
We introduce a system of spherical coordinates \((r,\Theta)\), \(\Theta=(\theta_{1},\theta_{2},\ldots, \theta_{n-1})\), in \({\mathbf{R}}^{n}\) which are related to Cartesian coordinates \((x_{1},x_{2},\ldots,x_{n-1},x_{n})\) by \(x_{n}=r\cos\theta_{1}\).
Let D be an arbitrary domain in \({\mathbf{R}}^{n}\) and \(\mathscr{A}_{a}\) denote the class of nonnegative radial potentials \(a(P)\), i.e. \(0\leq a(P)=a(r)\), \(P=(r,\Theta)\in D\), such that \(a\in L_{\mathrm{loc}}^{b}(D)\) with some \(b> {n}/{2}\) if \(n\geq4\) and with \(b=2\) if \(n=2\) or \(n=3\).
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 a generalized harmonic function (with respect to the Schrödinger operator \(\mathit{Sch}_{a}\)).
The unit sphere and the upper half unit sphere in \({\mathbf{R}}^{n}\) are denoted by \({\mathbf{S}}^{n-1}\) and \({\mathbf{S}}_{+}^{n-1}\), respectively. For simplicity, a point \((1,\Theta)\) on \({\mathbf{S}}^{n-1}\) and the set \(\{\Theta; (1,\Theta)\in\Omega\}\) for a set Ω, \(\Omega\subset{\mathbf{S}}^{n-1}\), are often identified with Θ and Ω, respectively. For two sets \(\Xi\subset{\mathbf{R}}_{+}\) and \(\Omega\subset{\mathbf{S}}^{n-1}\), the set \(\{(r,\Theta)\in{\mathbf{R}}^{n}; r\in\Xi,(1,\Theta)\in\Omega\}\) in \({\mathbf{R}}^{n}\) is simply denoted by \(\Xi\times\Omega\). By \(C_{n}(\Omega)\), we denote the set \({\mathbf{R}}_{+}\times\Omega\) in \({\mathbf{R}}^{n}\) with the domain Ω on \({\mathbf{S}}^{n-1}\). We call it a cone. We denote the set \(I\times\Omega\) with an interval on R by \(C_{n}(\Omega;I)\).
We shall say that a set \(H\subset C_{n}(\Omega)\) has a covering \(\{r_{j}, R_{j}\}\) if there exists a sequence of balls \(\{B_{j}\}\) with centers in \(C_{n}(\Omega)\) such that \(H\subset\bigcup_{j=0}^{\infty} B_{j}\), where \(r_{j}\) is the radius of \(B_{j}\) and \(R_{j}\) is the distance from the origin to the center of \(B_{j}\). For positive functions \(h_{1}\) and \(h_{2}\), we say that \(h_{1}\lesssim h_{2}\) if \(h_{1}\leq Mh_{2}\) for some constant \(M>0\). If \(h_{1}\lesssim h_{2}\) and \(h_{2}\lesssim h_{1}\), we say that \(h_{1}\approx h_{2}\).
From now on, we always assume \(D=C_{n}(\Omega)\). For the sake of brevity, we shall write \(G_{\Omega}^{a}(P,Q)\) instead of \(G_{C_{n}(\Omega)}^{a}(P,Q)\). 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 all sections will be a sufficiently small positive number.
We will also consider the class \(\mathscr{B}_{a}\), consisting of the potentials \(a\in\mathscr{A}_{a}\) such that there exists the finite limit \(\lim_{r\rightarrow\infty}r^{2} a(r)=k\in[0,\infty)\), and moreover, \(r^{-1}|r^{2} a(r)-k|\in L(1,\infty)\). If \(a\in \mathscr{B}_{a}\), then the (sub)superfunctions are continuous (see [8]).
In the rest of paper, we assume that \(a\in\mathscr{B}_{a}\) and we shall suppress this assumption for simplicity.
Remark
We remark that the total masses of \(\mu'\) and \(\nu'\) are finite (see [2], Lemma 5 and [6], Lemma 4).
The following Theorems A and B give a way to estimate the Green-Sch potential and the Poisson-Sch integrals with measures on \(C_{n}(\Omega)\) and \(S_{n}(\Omega)\), respectively.
Theorem A
Theorem B
In [7, 10], Xue and Zhao-Yamada introduce the notations of a-thin (with respect to the Schrödinger operator \(\mathit{Sch}_{a}\)) at a point and a-rarefied sets at infinity (with respect to the Schrödinger operator \(\mathit{Sch}_{a}\)), which generalized the earlier notations obtained by Miyamoto, Hoshida, Brelot (see [11–14]).
Definition 1
(see [7])
A set H in \({\mathbf{R}}^{n}\) is said to be a-thin at a point Q if there is a fine neighborhood E of Q which does not intersect \(H\backslash\{Q\}\). Otherwise H is said to be not a-thin at Q on cones.
Definition 2
(see [10])
Recently, GX Xue (see [7], Theorem 2.5) gave a criterion for a subset H of \(C_{n}(\Omega)\) to be a-rarefied set at infinity.
Theorem C
Our aim in this paper is to characterize the geometrical property of a-rarefied sets at infinity.
Theorem 1
Next, we immediately have the following result from Theorem 1.
Corollary 1
Finally, we prove the following result.
Theorem 2
If a subset H of \(C_{n}(\Omega)\) has a covering \(\{r_{j},R_{j}\}\) (\(j=0,1,2,\ldots\)) satisfying (1.5), then it is possible that H is not a-rarefied at infinity on cones.
2 Main lemmas
Lemma 1
Proof
3 Proof of Theorem 1
Since H is a-rarefied at infinity on cones, by Definition 2 there exists a positive superfunction \(v(P)\) on cones such that (1.3) and (1.4) hold.
Then \(H_{2}\) and \(H_{3}\) also have coverings \(\{r_{j}^{(2)},R_{j}^{(2)}\}\) (\(j=1,2,\ldots\)) and \(\{r_{j}^{(3)},R_{j}^{(3)}\}\) (\(j=1,2,\ldots\)) satisfying (3.4) and (3.5), respectively.
Thus by rearranging coverings \(\{r_{1},R_{1}\}\), \(\{r_{j}^{(2)},R_{j}^{(2)}\}\) (\(j=1,2,\ldots\)) and \(\{r_{j}^{(3)},R_{j}^{(3)}\}\) (\(j=1,2,\ldots\)), we know that the set H has a covering \(\{r_{j},R_{j}\}\) (\(j=0,1,2,\ldots\)) from (3.2) and satisfies (1.5) from (3.3), (3.4), and (3.5).
Thus we complete the proof of Theorem 1.
4 Proof of Theorem 2
Let \(C_{n}(\Omega')\) be a subset of \(C_{n}(\Omega)\), i.e. \(\overline{\Omega}'\subset\Omega\). Suppose that this covering is so located: there is an integer \(j_{0}\) such that \(B_{j}\subset C_{n}(\Omega')\) and \(R_{j}>2r_{j}\) for \(j\geq j_{0}\).
Declarations
Acknowledgements
This work was completed while the second author was visiting the Department of Mathematical Sciences at the Columbia University, and he is grateful for the kind hospitality of the Department. This work was partially supported by NSF Grant DMS-0913205.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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