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A new application of boundary integral behaviors of harmonic functions to the least harmonic majorant
- Minghua Han^{1},
- Jianguo Sun^{2} and
- Gaoying Xue^{3}Email author
- Received: 17 November 2016
- Accepted: 27 April 2017
- Published: 11 May 2017
Abstract
Our main aim in this paper is to obtain a new type of boundary integral behaviors of harmonic functions in a smooth cone. As an application, the least harmonic majorant of a nonnegative subharmonic function is also given.
Keywords
- boundary integral behavior
- subharmonic function
- harmonic majorant
1 Introduction
Let \(B(P,R)\) denote the open ball with center at P and radius R in \(\mathbf{R}^{n}\), where \(\mathbf{R}^{n}\) is the n-dimensional Euclidean space, \(P\in\mathbf{R}^{n}\) and \(R>0\). Let \(B(P)\) denote the neighborhood of P and \(S_{R}=B(O,R)\) for simplicity. The unit sphere and the upper half unit sphere in \(\mathbf{R}^{n}\) are denoted by \(\mathbf{S}_{1}\) and \(\mathbf{S}_{1}^{+}\), respectively. For simplicity, a point \((1,\Theta)\) on \(\mathbf{S}_{1}\) and the set \(\{\Theta; (1,\Theta)\in\Gamma\}\) for a set Γ, \(\Gamma\subset\mathbf{S}_{1}\), are often identified with Θ and Γ, respectively. Let \(\Lambda\times\Gamma\) denote the set \(\{(r,\Theta)\in \mathbf{R}^{n}; r\in\Lambda,(1,\Theta)\in\Gamma\}\), where \(\Lambda\subset\mathbf{R}_{+}\) and \(\Gamma\subset\mathbf{S}_{1}\). We denote the set \(\mathbf{R}_{+}\times\mathbf{S}_{1}^{+}=\{(X,x_{n})\in\mathbf{R}^{n}; x_{n}>0\}\) by \(\mathbf{T}_{n}\), which is called the half space.
We shall also write \(h_{1}\approx h_{2}\) for two positive functions \(h_{1}\) and \(h_{2}\) if and only if there exists a positive constant a such that \(a^{-1}h_{1}\leq h_{2}\leq ah_{1}\). We denote \(\max\{u(r,\Theta),0\}\) and \(\max\{-u(r,\Theta),0\}\) by \(u^{+}(r,\Theta)\) and \(u^{-}(r,\Theta)\), respectively.
The set \(\mathbf{R}_{+}\times\Gamma\) in \(\mathbf{R}^{n}\) is called a cone. We denote it by \(\mathfrak{C}_{n}(\Gamma)\), where \(\Gamma\subset\mathbf{S}_{1}\). The sets \(I\times\Gamma\) and \(I\times \partial{\Gamma}\) with an interval on R are denoted by \(\mathfrak {C}_{n}(\Gamma;I)\) and \(\mathfrak{S}_{n}(\Gamma;I)\), respectively. We denote \(\mathfrak {C}_{n}(\Gamma)\cap S_{R}\) and \(\mathfrak{S}_{n}(\Gamma; (0,+\infty))\) by \(\mathfrak {S}_{n}(\Gamma; R)\) and \(\mathfrak{S}_{n}(\Gamma)\), respectively.
Furthermore, we denote by dσ (resp. \(dS_{R}\)) the \((n-1)\)-dimensional volume elements induced by the Euclidean metric on \(\partial{\mathfrak{C}_{n}(\Gamma)}\) (resp. \(S_{R}\)) and by dw the elements of the Euclidean volume in \(\mathbf{R}^{n}\).
Remark 1
A function \(g(t)\) on \((0,\infty)\) is \(\mathbb{A}_{d_{1},d_{2}}\)-convex if and only if \(g(t)t^{d_{2}}\) is a convex function of \(t^{d}\) (\(d=d_{1}+d_{2}\)) on \((0,\infty)\) or, equivalently, if and only if \(g(t)t^{-d_{1}}\) is a convex function of \(t^{-d}\) on \((0,\infty)\).
Remark 2
We will also consider the class of all continuous functions \(u(t,\Phi)\) (\((t,\Phi)\in\overline{\mathfrak{C}_{n}(\Gamma)}\)) harmonic in \(\mathfrak{C}_{n}(\Gamma)\) with \(u^{+}(t,\Phi)\in \mathcal{A}_{\Gamma}\) (\((t,\Phi)\in\mathfrak{C}_{n}(\Gamma)\)), and \(u^{+}(t,\Phi)\in\mathcal{B}_{\Gamma}\) (\((t,\Phi)\in\mathfrak {S}_{n}(\Gamma)\)) is denoted by \(\mathcal{C}_{\Gamma}\).
Remark 3
Recently Zhao and Yamada (see [6]) obtained the following result.
Theorem A
Recently Wang and Qiao (see [7]) generalized Theorem A to the conical case.
Theorem B
2 Results
Our main aim in this paper is to give the least harmonic majorant of a nonnegative subharmonic function on \(\mathfrak{C}_{n}(\Gamma)\). For related results, we refer the reader to the papers [8, 9].
Theorem 1
3 Main lemmas
Lemma 1
Let u be a function subharmonic on \(\mathfrak{C}_{n}(\Gamma)\) satisfying (1.4) for any \(Q\in\partial {\mathfrak{C}_{n}(\Gamma)}\). Then the limit \(\mathscr{U}_{u}\) (\(-\infty<\mathscr{U}_{u}\leq+\infty\)) exists.
Proof
Lemma 2
Proof
As \(R_{1}\rightarrow0\) and \(R_{2}\rightarrow+\infty\) in (3.2), we complete the proof by (3.1). □
Lemma 3
Proof
Lemma 4
Proof
Lemma 5
Let u be subharmonic on a domain containing \(\overline{\mathfrak{C}_{n}(\Gamma)}\) such that \(u'=u\vert\partial{\mathfrak{C}_{n}(\Gamma)}\) satisfies (1.5) and \(u\geq0\) on \(\mathfrak{C}_{n}(\Gamma)\). Then \(\mathbb{PI}_{\mathfrak{C}_{n}(\Gamma )}[u'](P)\leq h(P)\) on \(\mathfrak{C}_{n}(\Gamma)\), where \(h(P)\) is any harmonic majorant of u on \(\mathfrak{C}_{n}(\Gamma)\).
Proof
4 Proof of Theorem 1
Hence we have from (4.2) and (4.3) that (2.1) holds.
We see from Lemma 2 that \(h^{\ast}(P)\leq0\) on \(\mathfrak {C}_{n}(\Gamma)\), which shows that \(h_{u}(P)\) is the least harmonic majorant of \(u(P)\) on \(\mathfrak{C}_{n}(\Gamma)\). Theorem 1 is proved.
5 Conclusions
In this article, we have obtained a new type of boundary integral behaviors of harmonic functions in a smooth cone. As an application, we also gave the least harmonic majorant of a nonnegative subharmonic function.
6 Ethics approval and consent to participate
Not applicable.
7 Consent for publication
Not applicable.
8 List of abbreviations
Not applicable.
9 Availability of data and materials
Not applicable.
Declarations
Acknowledgements
The authors would like to thank the editor, the associate editor and the anonymous referees for their careful reading and constructive comments which have helped us to significantly improve the presentation of the paper.
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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