Growth properties of Green-Sch potentials at infinity
© Zhao and Yamada; licensee Springer. 2014
Received: 19 September 2014
Accepted: 12 November 2014
Published: 27 November 2014
This paper gives growth properties of Green-Sch potentials at infinity in a cone, which generalizes results obtained by Qiao-Deng. The proof is based on the fact that the estimations of Green-Sch potentials with measures are connected with a kind of densities of the measures modified by the measures.
MSC: 35J10, 35J25.
1 Introduction and main 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 of two points P and Q in is denoted by . Also with the origin O of is simply denoted by . The boundary, the closure and the complement of a set S in are denoted by ∂ S, , and , respectively. For and , let denote the open ball with center at P and radius r in .
where , , and if , then ().
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 . In particular, the half space will be denoted by .
By , we denote the set in with the domain Ω on (). We call it a cone. Then is a special cone obtained by putting . We denote the sets and with an interval on R by and . By we denote . By we denote , which is .
Let 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. 13]). We will denote it as well. This last one has a Green-Sch function . Here is positive on and its inner normal derivative , where denotes the differentiation at Q along the inward normal into . We denote this derivative by , which is called the Poisson-Sch kernel with respect to .
We shall say that a set has a covering if there exists a sequence of balls with centers in such that , where is the radius of and is the distance from the origin to the center of .
For positive functions and , we say that if for some constant . If and , we say 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.
is denoted by .
If (), then for any positive number β. So we can find .
About the growth properties of Green potentials at infinity in a cone, Qiao-Deng (see , Theorem 1]) has proved the following result.
Now we state our first result.
By comparison the condition (1.4) is fairly briefer and easily applied. Moreover, is a set of 1-finite view in the sense of ,  (see , Definition 2.1] for the definition of 1-finite view). In the case , Theorem 1 (1.6) is just the result of Theorem A.
2 Some lemmas
for anyand anysatisfying (resp. );
which gives (2.5). □
Since can be covered by the union of a family of balls (). By the Vitali lemma (see ), there exists , which is at most countable, such that are disjoint and .
where () is the ball which covers . □
3 Proof of Theorem 1
For any point , where R () is a sufficiently large number and ϵ is a sufficiently small positive number.
from Lemma 2.
Combining (3.1)-(3.3) and (3.5)-(3.7), we finally obtain the result that if R is sufficiently large and ϵ is a sufficiently small, then as , where . Finally, there exists an additional finite ball covering , which together with Lemma 3, gives the conclusion of Theorem 1.
The authors are very thankful to the anonymous referees for their valuable comments and constructive suggestions, which helped to improve the quality of the paper. This work is supported by the Academy of Finland Grant No. 176512.
- Escassut A, Tutschke W, Yang CC: Some Topics on Value Distribution and Differentiability in Complex and P-Adic Analysis. Science Press, Beijing; 2008.Google Scholar
- Gilbarg D, Trudinger NS: Elliptic Partial Differential Equations of Second Order. Springer, Berlin; 1977.View ArticleGoogle Scholar
- Miranda C: Partial Differential Equations of Elliptic Type. Springer, Berlin; 1970.View ArticleGoogle Scholar
- Verzhbinskii GM, Maz’ya VG: Asymptotic behavior of solutions of elliptic equations of the second order close to a boundary. I. Sib. Mat. Zh. 1971, 12(2):874-899.Google Scholar
- Qiao L, Deng GT: A lower bound of harmonic functions in a cone and its application. Sci. Sin., Math. 2014, 44(6):671-684. (in Chinese) 10.1360/N012013-00108View ArticleGoogle Scholar
- Qiao L, Deng GT: Minimally thin sets at infinity with respect to the Schrödinger operator. Sci. Sin., Math. 2014, 44(12):1247-1256. (in Chinese)MathSciNetView ArticleGoogle Scholar
- Xue GX: A remark on the a-minimally thin sets associated with the Schrödinger operator. Bound. Value Probl. 2014., 2014: 10.1186/1687-2770-2014-133Google Scholar
- Zhao T: Minimally thin sets associated with the stationary Schrodinger operator. J. Inequal. Appl. 2014., 2014: 10.1186/1029-242X-2014-67Google Scholar
- Simon B: Schrödinger semigroups. Bull. Am. Math. Soc. 1982, 7(2):447-526. 10.1090/S0273-0979-1982-15041-8View ArticleGoogle Scholar
- Hartman P: Ordinary Differential Equations. Wiley, New York; 1964.Google Scholar
- Qiao L, Ren YD: Integral representations for the solutions of infinite order of the stationary Schrödinger equation in a cone. Monatshefte Math. 2014, 173(4):593-603. 10.1007/s00605-013-0506-1MathSciNetView ArticleGoogle Scholar
- Qiao L, Deng GT: The Riesz decomposition theorem for superharmonic functions in a cone and its application. Sci. Sin., Math. 2012, 42(8):763-774. (in Chinese) 10.1360/012011-1018View ArticleGoogle Scholar
- Qiao L: Integral representations for harmonic functions of infinite order in a cone. Results Math. 2012, 61(4):63-74. 10.1007/s00025-010-0076-7MathSciNetView ArticleGoogle Scholar
- Qiao L, Deng GT: Integral representations of harmonic functions in a cone. Sci. Sin., Math. 2011, 41(6):535-546. (in Chinese)Google Scholar
- Miyamoto I, Yoshida H: Two criteria of Wiener type for minimally thin sets and rarefied sets in a cone. J. Math. Soc. Jpn. 2002, 54: 487-512. 10.2969/jmsj/1191593906MathSciNetView ArticleGoogle Scholar
- Yoshida H: Harmonic majorant of a radial subharmonic function on a strip and their applications. Int. J. Pure Appl. Math. 2006, 30(2):259-286.MathSciNetGoogle Scholar
- Stein EM: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton; 1970.Google Scholar
This article is published under license to BioMed Central Ltd.Open Access 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.