Abstract
Motivated by (Xu et al. in Bound. Value Probl. 2013:262, 2013) and (Yang and Ren in Proc. Indian Acad. Sci. Math. Sci. 124(2):175-178, 2014), in this paper we aim to construct a modified Green function in the upper-half space of the n-dimensional Euclidean space, which generalizes the boundary property of general Green potential.
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1 Introduction and main results
Let \({\mathbf{R}}^{n}\) (\(n\geq2\)) denote the n-dimensional Euclidean space. The upper half-space H is the set \(H=\{x=(x_{1},x_{2},\ldots,x_{n})\in{\mathbf{R}}^{n}: x_{n}>0\}\), whose boundary and closure are ∂H and \(\overline{H}\) respectively.
For \(x\in{\mathbf{R}}^{n}\) and \(r>0\), let \(B(x,r)\) denote the open ball with center at x and radius r.
Set
Let \(G_{\alpha}\) be the Green function of order α for H, that is,
where ∗ denotes reflection in the boundary plane ∂H just as \(y^{\ast}=(y_{1},y_{2},\ldots,-y_{n})\).
In case \(\alpha=n=2\), we consider the modified kernel function, which is defined by
In case \(0<\alpha<n\), we define
where m is a non-negative integer, \(C^{\omega}_{k}(t)\) (\(\omega=\frac{n-\alpha}{2}\)) is the ultraspherical (or Gegenbauer) polynomial (see [1]). The expression arises from the generating function for Gegenbauer polynomials
where \(|r|<1\), \(|t|\leq1\) and \(\omega>0\). The coefficient \(C^{\omega}_{k}(t) \) is called the ultraspherical (or Gegenbauer) polynomial of degree k associated with ω, the function \(C^{\omega}_{k}(t) \) is a polynomial of degree k in t.
Then we define the modified Green function \(G_{\alpha,m}(x,y)\) by
where \(x, y\in\overline{H}\) and \(x\neq y\). We remark that this modified Green function is also used to give unique solutions of the Neumann and Dirichlet problem in the upper-half space [2–4].
Write
where μ is a non-negative measure on H. Here note that \(G_{2,0}(x,\mu)\) is nothing but the general Green potential.
Let k be a non-negative Borel measurable function on \({\mathbf{R}}^{n}\times{\mathbf{R}}^{n}\), and set
for a non-negative measure μ on a Borel set \(E\subset{\mathbf{R}}^{n}\). We define a capacity \(C_{k}\) by
where the supremum is taken over all non-negative measures μ such that \(S_{\mu}\) (the support of μ) is contained in E and \(k(y,\mu) \leq1\) for every \(y\in H\).
For \(\beta\leq0\), \(\delta\leq0\) and \(\beta\leq\delta\), we consider the kernel function
Now we prove the following result. For related results in a smooth cone and tube, we refer the reader to the papers by Qiao (see [5, 6]) and Liao-Su (see [7]), respectively. The readers may also find some related interesting results with respect to the Schrödinger operator in the papers by Su (see [8]), by Polidoro and Ragusa (see [9]) and the references therein.
Theorem
Let \(n+m-\alpha+\delta+2\geq0\). Ifμis a non-negative measure onHsatisfying
then there exists a Borel set \(E\subset H\)with properties:
where \(E_{i}=\{x\in E: 2^{-i}\leq x_{n}<2^{-i+1}\}\).
Remark
By using Lemma 4 below, condition (2) in Theorem with \(\alpha=2\), \(\beta=0\), \(\delta=0\) means that E is 2-thin at ∂H in the sense of [10].
2 Some lemmas
Throughout this paper, let M denote various constants independent of the variables in questions, which may be different from line to line.
Lemma 1
There exists a positive constantMsuch that \(G_{\alpha}(x,y)\leq M\frac{x_{n}y_{n}}{|x-y|^{n-\alpha+2}}\), where \(0<\alpha\leq n\), \(x=(x_{1},x_{2},\ldots,x_{n})\)and \(y=(y_{1},y_{2},\ldots, y_{n})\)inH.
This can be proved by a simple calculation.
Lemma 2
Gegenbauer polynomials have the following properties:
-
(1)
\(|C_{k}^{\omega}(t)|\leq C_{k}^{\omega}(1)=\frac{\Gamma(2\omega +k)}{\Gamma(2\omega)\Gamma(k+1)}\), \(|t|\leq1 \);
-
(2)
\(\frac{d}{dt}C_{k}^{\omega}(t)=2\omega C_{k-1}^{\omega+1}(t)\), \(k \geq1\);
-
(3)
\(\sum_{k=0}^{\infty} C_{k}^{\omega}(1)r^{k}=(1-r)^{-2\omega}\);
-
(4)
\(|C^{\frac{n-\alpha}{2}}_{k} (t)-C^{\frac{n-\alpha}{2}}_{k} ( t^{\ast})| \leq(n-\alpha)C^{\frac{n-\alpha+2}{2}}_{k-1} (1)|t-t^{\ast}|\), \(|t|\leq1\), \(|t^{\ast}|\leq1\).
Proof
(1) and (2) can be derived from [1], p.232. Equality (3) follows from expression (1.1) by taking \(t=1\); property (4) is an easy consequence of the mean value theorem, (1) and also (2). □
Lemma 3
For \(x, y\in{\mathbf{R}}^{n}\) (\(\alpha=n=2\)), we have the following properties:
-
(1)
\(|\Im \sum_{k=0}^{m}\frac{x^{k}}{y^{k+1}}|\leq\sum_{k=0}^{m-1} \frac{2^{k} x_{n} |x|^{k}}{|y|^{k+2}} \);
-
(2)
\(|\Im\sum_{k=0}^{\infty}\frac{x^{k+m+1}}{y^{k}}|\leq 2^{m+1}x_{n} |x|^{m}\);
-
(3)
\(|G_{n,m}(x,y)-G_{n}(x,y)|\leq M \sum_{k=1}^{m} \frac{k x_{n} y_{n} |x|^{k-1}}{|y|^{k+1}}\);
-
(4)
\(|G_{n,m}(x,y)|\leq M \sum_{k=m+1}^{\infty} \frac{k x_{n} y_{n} |x|^{k-1}}{|y|^{k+1}}\).
The following lemma can be proved by using Fuglede (see [11], Théorèm 7.8).
Lemma 4
For any Borel setEinH, we have \(C_{k_{\alpha}}(E)=\hat{C}_{k_{\alpha}}(E)\), where \(\hat{C}_{k_{\alpha}}(E)=\inf\lambda(H)\), \(k_{\alpha}=k_{\alpha,0,0}\), the infimum being taken over all non-negative measuresλonHsuch that \(k_{\alpha}(\lambda,x)\geq1\)for every \(x \in E\).
Following [10], we say that a set \(E\subset H\) is α-thin at the boundary ∂H if
where \(E_{i}=\{x\in E: 2^{-i}\leq x_{n} <2^{-i+1}\}\).
3 Proof of Theorem
We write
where
We distinguish the following two cases.
Case 1. \(0<\alpha<n\).
By assumption (1.2) we can find a sequence \(\{a_{i}\}\) of positive numbers such that \(\lim_{i\rightarrow\infty} a_{i}=\infty\) and \(\sum_{i=1}^{\infty}a_{i}b_{i}<\infty\), where
Consider the sets
for \(i=1,2,\ldots\) . Set
Then \(G\subset\{y \in H:2^{-i-1}< y_{n}< 2^{-i+2}\}\). Let ν be a non-negative measure on H such that \(S_{\nu}\subset E_{i}\), where \(S_{\nu}\) is the support of ν. Then we have \(k_{\alpha,\beta,\delta}(y,\nu)\leq1\) for \(y\in H\) and
So that
which yields
Setting \(E=\bigcup_{i=1}^{\infty}E_{i}\), we see that (2) in Theorem is satisfied and
For \(U_{2}(x)\), by Lemma 1 we have
Note that \(C^{\omega}_{0} ( t )\equiv1\). By (3) and (4) in Lemma 2, we take \(t=\frac{x\cdot y}{|x||y|}\), \(t^{\ast}=\frac{x\cdot y^{\ast}}{|x||y^{\ast}|}\) in Lemma 2(4) and obtain
Similarly, we have by (3) and (4) in Lemma 2
Finally, by Lemma 1, we have
Combining (3.1), (3.2), (3.3), (3.4) and (3.5), by Lebesgue’s dominated convergence theorem, we prove Case 1.
Case 2. \(\alpha=n=2\).
In this case, \(U_{1}(x)\), \(U_{2}(x)\) and \(U_{5}(x)\) can be proved similarly as in Case 1. Here we omit the details and state the following facts:
where \(E=\bigcup_{i=1}^{\infty}E_{i}\) and \(\sum_{i=1}^{\infty}2^{i(\beta+\delta)}C_{k_{\alpha,\beta,\delta }}(E_{i})<\infty\),
By Lemma 3(3), we obtain
By Lemma 3(4), we have
Combining (3.6), (3.7), (3.8) and (3.9), we prove Case 2.
Hence the proof of the theorem is completed.
Change history
08 October 2020
A Correction to this paper has been published: https://doi.org/10.1186/s13661-020-01458-6
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Acknowledgements
The authors are highly grateful for the referees’ careful reading and comments on this paper. This work was completed while the authors were visiting the Department of Mathematical Sciences at the University of Wollongong, and they are grateful for the kind hospitality of the Department.
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Zhang, Y., Piskarev, V. RETRACTED ARTICLE: A note on the boundary behavior for a modified Green function in the upper-half space. Bound Value Probl 2015, 114 (2015). https://doi.org/10.1186/s13661-015-0363-z
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DOI: https://doi.org/10.1186/s13661-015-0363-z