Some new results on the boundary behaviors of harmonic functions with integral boundary conditions
- Xiaozhen Xie^{1} and
- Costanza T Viouonu^{2}Email author
Received: 21 March 2016
Accepted: 11 July 2016
Published: 27 July 2016
Abstract
In this paper, using a generalized Carleman formula, we prove two new results on the boundary behaviors of harmonic functions with integral boundary conditions in a smooth cone, which generalize some recent results.
Keywords
1 Introduction
Let \(\mathbf{R}^{n} \) (\(n\geq2\)) be the n-dimensional Euclidean space. A point in \(\mathbf{R}^{n}\) is denoted by \(V=(X,y)\), where \(X=(x_{1},x_{2},\ldots,x_{n-1})\). The boundary and the closure of a set E in \(\mathbf{R}^{n}\) are denoted by ∂E and E̅, respectively.
We introduce a system of spherical coordinates \((l,\Lambda)\), \(\Lambda=(\theta_{1},\theta_{2},\ldots,\theta_{n-1})\), in \(\mathbf{R}^{n}\) that are related to Cartesian coordinates \((x_{1},x_{2},\ldots,x_{n-1},y)\) by \(y=l\cos\theta_{1}\).
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 \((1,\Lambda)\) on \(\mathbf{S}^{n-1}\) and the set \(\{\Lambda; (1,\Lambda)\in\Gamma\}\) for a set \(\Gamma\subset\mathbf{S}^{n-1}\) are often identified with Λ and Γ, respectively. For two sets \(\Xi\subset\mathbf{R}_{+}\) and \(\Gamma\subset \mathbf{S}^{n-1}\), the set \(\{(l,\Lambda)\in\mathbf{R}^{n}; l\in\Xi,(1,\Lambda)\in\Gamma\}\) in \(\mathbf{R}^{n}\) is simply denoted by \(\Xi\times\Gamma\).
We denote the set \(\mathbf{R}_{+}\times\Gamma\) in \(\mathbf{R}^{n}\) with the domain Γ on \(\mathbf{S}^{n-1}\) by \(T_{n}(\Gamma)\). We call it a cone. In particular, the half-space \(\mathbf{R}_{+}\times\mathbf{S}_{+}^{n-1}\) is denoted by \(T_{n}(\mathbf{S}_{+}^{n-1})\). The sets \(I\times\Gamma\) and \(I\times\partial{\Gamma}\) with an interval on R are denoted by \(T_{n}(\Gamma;I)\) and \(\mathcal{S}_{n}(\Gamma;I)\), respectively. We denote \(T_{n}(\Gamma)\cap S_{l}\) by \(\mathcal{S}_{n}(\Gamma ; l)\), and we denote \(\mathcal{S}_{n}(\Gamma; (0,+\infty))\) by \(\mathcal{S}_{n}(\Gamma)\).
The estimate we deal with has a long history tracing back to known Matsaev’s estimate of harmonic functions from below in the half-plane (see, e.g., Levin [2], p.209).
Theorem A
In 2014, Xu and Zhou [3] considered Theorem A in the half-space. Pan et al. [4], Theorems 1.2 and 1.4, obtained similar results, slightly different from the following Theorem B.
Theorem B
Recently, Pang and Ychussie [5], Theorem 1, further extended Theorems A and B and proved Matsaev’s estimates for harmonic functions in a smooth cone.
Theorem C
In this paper, we obtain two new results on the lower bounds of harmonic functions with integral boundary conditions in a smooth cone (Theorems 1 and 2), which further extend Theorems A, B, and C. Our proofs are essentially based on the Riesz decomposition theorem (see [6]) and a modified Carleman formula for harmonic functions in a smooth cone (see [5], Lemma 1).
In order to avoid complexity of our proofs, we assume that \(n\geq3\). However, our results in this paper are also true for \(n=2\). We use the standard notations \(h^{+}=\max\{h,0\}\) and \(h^{-}=-\min\{h,0\}\). All constants appearing further in expressions will be always denoted M because we do not need to specify them. We will always assume that \(\eta(t)\) and \(\rho(t)\) are nondecreasing real-valued functions on an interval \([1,+\infty)\) and \(\rho(t)> \aleph^{+}\) for any \(t\in[1,+\infty)\).
2 Main results
First of all, we shall state the following result, which further extends Theorem C under weak boundary integral conditions.
Theorem 1
Let \(h(V)\) (\(V=(R,\Lambda)\)) be harmonic on \(T_{n}(\Gamma)\) and continuous on \(\overline{T_{n}(\Gamma)}\).
- (I)For any \(V=(R,\Lambda)\in T_{n}(\Gamma;(1,\infty))\), we haveand$$ \int_{\mathcal{S}_{n}(\Gamma;(1,R))}h^{-}t^{\aleph^{-}}{\partial\psi }/{ \partial n}\,d\sigma_{W} \leq M\eta(R) (cR)^{\rho(cR)-\aleph^{+}} $$(2.1)$$ \chi \int_{\mathcal{S}_{n}(\Gamma ;R)}h^{-}R^{\aleph^{-}-1}\psi d S_{R} \leq M\eta(R) (cR)^{\rho(cR)-\aleph^{+}}. $$(2.2)
- (II)For any \(V=(R,\Lambda)\in T_{n}(\Gamma;(0,1])\), we haveThen$$ h(V)\geq-\eta(R). $$(2.3)where \(V\in T_{n}(\Gamma)\), N (≥1) is a sufficiently large number, and M is a constant independent of R, \(\psi(\Lambda)\), and the functions \(\eta(R)\) and \(h(V)\).$$h(V)\geq-M\eta(R) \bigl(1+(cR)^{\rho(cR)} \bigr)\psi^{1-n}( \Lambda), $$
Remark 1
From the proof of Theorem 1 it is easy to see that condition (I) in Theorem 1 is weaker than that in Theorem C in the case \(c\equiv(N+1)/{N}\) and \(\eta (R)\equiv K\), where N (≥1) is a sufficiently large number, and K is a constant.
Theorem 2
3 Proof of Theorem 1
We next distinguish three cases.
Case 1. \(V=(l,\Lambda)\in T_{n}(\Gamma;({5}/{4},\infty ))\) and \(R={5l}/{4}\).
Case 2. \(V=(l,\Lambda)\in T_{n}(\Gamma;({4}/{5},{5}/{4}])\) and \(R={5l}/{4}\).
Case 3. \(V=(l,\Lambda)\in T_{n}(\Gamma;(0,{4}/{5}])\).
4 Proof of Theorem 2
Hence, (4.5) gives (2.1), which, together with Theorem 1, gives Theorem 2.
Declarations
Acknowledgements
This work was supported by the National Natural Science Foundation of China under Grant no. 61401368. We are grateful to the editor and anonymous reviewers for their valuable comments and corrections that helped improve the original version of this 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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