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RETRACTED ARTICLE: New applications of Schrödingerean Green potential to boundary behaviors of superharmonic functions
Boundary Value Problems volume 2017, Article number: 21 (2017)
Abstract
By using the Schrödingerean continuation theorem due to Li (Bound. Value Probl. 2015:242, 2015), we obtain some new results for boundary value problems of Schrödingerean Green potentials. As new applications, the boundary behaviors of superharmonic functions at infinity are also obtained.
1 Introduction
Cartesian coordinates of a point G of \(\mathbf{R}^{n}\), \(n\geq2\), are denoted by \((X,x_{n})\), where \(\mathbf{R}^{n}\) is the n-dimensional Euclidean space and \(X=(x_{1},x_{2},\ldots,x_{n-1})\). We introduce spherical coordinates for \(G=(r,\Xi)\) \((\Xi=(\theta _{1},\theta_{2},\ldots,\theta_{n-1}))\) by \(\vert x\vert =r\),
where \(0\leq r<+\infty\), \(-\frac{1}{2}\pi\leq\theta_{n-1}< \frac{3}{2}\pi\) and \(0\leq\theta_{j}\leq\pi\) for \(1\leq j\leq n-2\) (\(n\geq3\)).
We denote the unit sphere and the upper half unit sphere by \(\mathbf{S}^{n-1}\) and \(\mathbf{S}_{+}^{n-1}\), respectively. Let \(\Sigma \subset\mathbf{S}^{n-1}\). The point \((1,\Xi)\) and the set \(\{\Xi; (1, \Xi)\in\Sigma\}\) are identified with Ξ and Σ, respectively. Let \(\Xi\times\Sigma\) denote the set \(\{(r,\Xi) \in\mathbf{R}^{n}; r\in\Xi,(1,\Xi)\in\Sigma\}\), where \(\Xi\subset \mathbf{R}_{+}\). The set \(\mathbf{R}_{+}\times\Sigma\) is denoted by \(\beth_{n}(\Sigma)\), which is called a cone. Especially, the set \(\mathbf{R}_{+}\times\mathbf{S}_{+}^{n-1}\) is called the upper-half space, which is denoted by \(\mathcal{T}_{n}\). Let \(I\subset\mathbf{R}\). Two sets \(I\times\Sigma\) and \(I\times\partial {\Sigma}\) are denoted by \(\beth_{n}(\Sigma;I)\) and \(\daleth_{n}( \Sigma;I)\), respectively. We denote \(\daleth_{n}(\Sigma; \mathbf{R} ^{+})\) by \(\daleth_{n}(\Sigma)\), which is \(\partial{\beth_{n}( \Sigma)}-\{O\}\).
Let \(B(G,l)\) denote the open ball, where \(G\in\mathbf{R}^{n}\) is the center and \(l>0\) is the radius.
Definition 1
Let \(E\subset\beth_{n}(\Sigma)\). If there exists a sequence of countable balls \(\{B_{k}\}\) (\(k=1,2,3,\ldots\)) with centers in \(\beth_{n}(\Sigma)\) satisfying
then we say that E has a covering \(\{r_{k},R_{k}\}\), where \(r_{k}\) is the radius of \(B_{k}\) and \(R_{k}\) is the distance from the origin to the center of \(B_{k}\).
In spherical coordinates the Laplace operator is
where \(\Lambda_{n}\) is the Beltrami operator. Now we consider the boundary value problem
If the least positive eigenvalue of it is denoted by \(\tau_{\Sigma}\), then we can denote by \(h_{\Sigma}(\Xi)\) the normalized positive eigenfunction corresponding to it.
We denote by \(\iota_{\Sigma}\) (>0) and \(-\kappa_{\Sigma}\) (<0) two solutions of the problem \(t^{2}+(n-2)t-\tau_{\Sigma}=0\), Then \(\iota_{\Sigma}+\kappa_{\Sigma}\) is denoted by \(\varrho_{\Sigma}\) for the sake of simplicity.
Remark 1
In the case \(\Sigma=\mathbf{S}_{+}^{n-1}\), it follows that
-
(I)
\(\iota_{\Sigma}=1\) and \(\kappa_{\Sigma}=n-1\).
-
(II)
\(h_{\Sigma}(\Xi)=\sqrt{\frac{2n}{w_{n}}}\cos\theta_{1}\), where \(w_{n}\) is the surface area of \(\mathbf{S}^{n-1}\).
It is easy to see that the set \(\partial{\beth_{n}(\Sigma)}\cup\{ \infty\}\) is the Martin boundary of \(\beth_{n}(\Sigma)\). For any \(G\in\beth_{n}(\Sigma)\) and any \(H\in\partial{\beth_{n}(\Sigma)} \cup\{\infty\}\), if the Martin kernel is denoted by \(\mathcal{MK}(G,H)\), where a reference point is chosen in advance, then we see that (see [2], p.292)
where \(G=(r,\Xi)\in\beth_{n}(\Sigma)\) and c is a positive number.
We shall say that two positive real valued functions f and g are comparable and write \(f\approx g\) if there exist two positive constants \(c_{1}\leq c_{2}\) such that \(c_{1}g\leq f\leq c_{2}g\).
Remark 2
Let \(\Xi\in\Sigma\). Then \(h_{\Sigma}(\Xi)\) and \(\operatorname{dist}(\Xi,\partial{\Sigma})\) are comparable (see [3]).
Remark 3
Let \(\varrho(G)=\operatorname{dist}(G,\partial {\beth_{n}(\Sigma)})\). Then \(h_{\Sigma}(\Xi)\) and \(\varrho(G) \) are comparable for any \((1,\Xi)\in\Sigma\) (see [4]).
Remark 4
Let \(0\leq\alpha\leq n\). Then \(h_{\Sigma}(\Xi)\leq c_{3}(\Sigma,n)\{h_{\Sigma}(\Xi)\}^{1- \alpha}\), where \(c_{3}(\Sigma,n)\) is a constant depending on Σ and n (e.g. see [5], pp.126-128).
Definition 2
For any \(G\in\beth_{n}(\Sigma)\) and any \(H\in\beth_{n}(\Sigma)\). If the Green function in \(\beth_{n}(\Sigma)\) is defined by \(\mathcal{GF}_{\Sigma}(G,H)\), then:
-
(I)
The Poisson kernel can be defined by
$$\mathcal{POI}_{\Sigma}(G,H)=\frac{\partial}{\partial n_{H}} \mathcal{GF}_{\Sigma}(G,H), $$where \(\frac{\partial}{\partial n_{H}}\) denotes the differentiation at H along the inward normal into \(\beth_{n}(\Sigma)\).
-
(II)
The Green potential on \(\beth_{n}(\Sigma)\) can be defined by
$$\mathcal{GF}_{\Sigma} \nu(G)= \int_{\beth_{n}(\Sigma)}\mathcal{GF} _{\Sigma}(G,H)\,d\nu(H), $$where \(G\in\beth_{n}(\Sigma)\) and ν is a positive measure in \(\beth_{n}(\Sigma)\).
Definition 3
For any \(G\in\beth_{n}(\Sigma)\) and any \(H\in\daleth_{n}(\Sigma)\). Let μ be a positive measure on \(\daleth_{n}(\Sigma)\) and g be a continuous function on \(\daleth_{n}(\Sigma)\). Then (see [6]):
-
(I)
The Poisson integral with μ can be defined by
$$\mathcal{POI}_{\Sigma} \mu(G)= \int_{\daleth_{n}(\Sigma)} \mathcal{POI}_{\Sigma}(G,H)\,d\mu(H). $$ -
(II)
The Poisson integral with g can be defined by
$$\mathcal{POI}_{\Sigma} [g](G)= \int_{\daleth_{n}(\Sigma)} \mathcal{POI}_{\Sigma}(G,H)g(H)\,d\sigma_{H}, $$
where \(d\sigma_{H}\) is the surface area element on \(\daleth_{n}( \Sigma)\).
Definition 4
Let μ be defined in Definition 3. Then the positive measure \(\mu'\) is defined by
Definition 5
Let ν be any positive measure in \(\beth_{n}(\Sigma)\) satisfying
for any \(G\in\beth_{n}(\Sigma)\). Then the positive measure \(\nu'\) is defined by
Definition 6
Let ν be any positive measure in \(\mathcal{T}_{n}\) such that (1.1) holds for any \(G\in\beth _{n}(\Sigma)\). Then the positive measure \(\nu_{1}\) is defined by
Definition 7
Let μ and ν be defined in Definitions 3 and 4, respectively. Then the positive measure ξ is defined by
where
Remark 5
Let \(\Sigma=\mathbf{S}_{+}^{n-1}\). Then
where \(G=(X,x_{n})\), \(H^{\ast}=(Y,-y_{n})\), that is, \(H^{\ast}\) is the mirror image of \(H=(Y,y_{n})\) with respect to \(\partial{\mathcal{T} _{n}}\). Hence, for the two points \(G=(X,x_{n})\in\mathcal{T}_{n}\) and \(H=(Y,y_{n})\in\partial{\mathcal{T}_{n}}\), we have
Remark 6
Let \(\Sigma=\mathbf{S}_{+}^{n-1}\). Then we define
where
Definition 8
Let λ be any positive measure on \(\mathbf{R}^{n}\) having finite total mass. Then the maximal function \(M(G;\lambda,\beta)\) is defined by
for any \(G=(r,\Xi)\in\mathbf{R}^{n}-\{O\}\), where \(\beta\geq0\). The exceptional set can be defined by
where ϵ is a sufficiently small positive number.
Remark 7
Let \(\beta>0\) and \(\lambda(\{P\})>0\) for any \(P\neq O\). Then:
-
(I)
\(\mathfrak{M}(G;\lambda,\beta)=+\infty\).
-
(II)
\(\{G\in\mathbf{R}^{n}-\{O\}; \lambda(\{P\})>0\}\subset \mathbb{EX}(\epsilon; \lambda, \beta)\).
The boundary behavior of classical Green potential in \(\mathcal{T} _{n}\) was proved by Huang in [7], Corollary and Remark 5.
Theorem A
Let g be a measurable function on \(\partial{\mathcal{T}_{n}}\) satisfying
Then
for any \(x\in\mathcal{T}_{n}-\mathbb{EX}(\epsilon;\nu_{1},n-1)\), where \(\mathbb{EX}(\epsilon;\nu_{1},n-1)\) is a subset of \(\mathcal{T}_{n}\) and has a covering \(\{r_{k},R_{k}\}\) satisfying
2 Results
Our first aim in this paper is also to consider boundary value problems for Green potential in a cone, which generalize Theorem A to the conical case. For similar results for Green-Sch potentials, we refer the reader to the paper by Li (see [1]).
The estimation of the Green potential at infinity is the following.
Theorem 1
If ν is a positive measure on \(\beth_{n}(\Sigma)\) such that (1.1) holds for any \(G\in\beth_{n}(\Sigma)\). Then
for any \(G=(r,\Xi)\in\beth_{n}(\Sigma)-\mathbb{EX}(\epsilon; \nu',n-\alpha)\) as \(r \rightarrow\infty\), where \(\mathbb{EX}( \epsilon;\nu',n-\alpha)\) is a subset of \(\beth_{n}(\Sigma)\) and has a covering \(\{r_{k},R_{k}\}\) satisfying
Corollary 1
Under the conditions of Theorem 1, \(\mathcal{GF} _{\Sigma} \nu(G)=o(r^{\iota_{\Sigma}})\) for any \(G=(r,\Xi)\in \beth_{n}(\Sigma)-\mathbb{EX}(\epsilon;\nu',n-1)\) as \(r \rightarrow \infty\), where \(\mathbb{EX}(\epsilon;\nu',n-1)\) is a subset of \(\beth_{n}(\Sigma)\) and has a covering \(\{r_{k},R_{k}\}\) satisfying (1.4).
Corollary 2
Under the conditions of Theorem 1, \(\mathcal{GF} _{\Sigma} \nu(G)=o(r^{\iota_{\Sigma}}h_{\Sigma}(\Xi))\) for any \(G=(r,\Xi)\in\beth_{n}(\Sigma)-\mathbb{EX}(\epsilon;\nu',n)\) as \(r \rightarrow\infty\), where \(\mathbb{EX}(\epsilon;\nu',n)\) is a subset of \(\beth_{n}(\Sigma)\) and has a covering \(\{r_{k},R_{k}\}\) satisfying
Theorem B
see [8], Chapter 6, Theorem 6.2.1
Let \(0< w(G)\) be a superharmonic function in \(\mathcal{T}_{n}\). Then there exist a positive measure μ on \(\partial\mathcal{T}_{n}\) and a positive measure ν on \(\mathcal{T}_{n}\) such that \(w(G)\) can be uniquely decomposed as
where \(G\in\mathcal{T}_{n}\) and c is a nonnegative constant.
Theorem C
Let \(0< w(G)\) be a superharmonic function in \(\beth_{n}(\Sigma)\). Then there exist a positive measure μ on \(\daleth_{n}(\Sigma)\) and a positive measure ν on \(\beth_{n}( \Sigma)\) such that \(w(G)\) can be uniquely decomposed as
where \(G\in\beth_{n}(\Sigma)\), \(c_{5}(w)\) and \(c_{6}(w)\) are two constants dependent on w satisfying
As an application of Theorem 1 and Lemma 3 in Section 2, we prove the following result.
Theorem 2
Let \(0\leq\alpha< n\), ϵ be defined as in Theorem 1 and \(w(G)\) (\(\not\equiv+\infty\)) (\(G=(r,\Xi)\in\beth _{n}(\Sigma)\)) be defined by (2.4). Then
for any \(G\in\beth_{n}(\Sigma)- \mathbb{EX}(\epsilon;\xi,n-\alpha)\) as \(r \rightarrow\infty\), where \(\mathbb{EX}(\epsilon;\xi,n- \alpha)\) is a subset of \(\beth_{n}(\Sigma)\) and has a covering \(\{r_{k},R_{k}\}\) satisfying (2.1).
Corollary 3
Under the conditions of Theorem 2,
for any \(G\in\beth_{n}(\Sigma)- \mathbb{EX}(\epsilon;\xi,n-1)\) as \(r \rightarrow\infty\), where \(\mathbb{EX}(\epsilon;\xi,n-1)\) is a subset of \(\beth_{n}(\Sigma)\) and has a covering \(\{r_{k},R_{k}\}\) satisfying (2.2).
Corollary 4
Under the conditions of Theorem 2,
for any \(G\in\beth_{n}(\Sigma)- \mathbb{EX}(\epsilon;\xi,n)\) as \(r \rightarrow\infty\), where \(\mathbb{EX}(\epsilon;\xi,n)\) is a subset of \(\beth_{n}(\Sigma)\) and has a covering \(\{r_{k},R_{k}\}\) satisfying (1.4).
In \(\mathcal{T}_{n}\), we have
Corollary 5
Let \(w(x)\) (\(\not\equiv+\infty\)) (\(x=(X,x_{n}) \in\mathcal{T}_{n}\)) be defined by (2.3). Then
for any \(x\in\mathcal{T}_{n}- \mathbb{EX}(\epsilon;\varrho,n-1)\) as \(\vert x\vert \rightarrow\infty\), where \(\mathbb{EX}(\epsilon;\varrho,n-1)\) has a covering satisfying (2.2).
Corollary 6
Under the conditions of Corollary 5,
for any \(x\in\mathcal{T}_{n}- \mathbb{EX}(\epsilon;\varrho,n)\) as \(\vert x\vert \rightarrow\infty\), where \(\mathbb{EX}(\epsilon;\varrho,n)\) has a covering satisfying (1.4).
3 Lemmas
In order to prove our main results we need following lemmas.
Lemma 1
Let any \(G=(r,\Xi)\in\beth_{n}(\Sigma)\) and any \(H=(t,\Omega)\in\daleth_{n}(\Sigma)\), we have the following estimates:
for \(0<\frac{t}{r}\leq\frac{4}{5}\),
for \(0<\frac{r}{t}\leq\frac{4}{5}\) and
for \(\frac{4r}{5}< t\leq\frac{5r}{4}\), where
Lemma 2
see [10], Lemma 2
If λ is positive measure on \(\mathbf{R}^{n}\) having finite total mass, then exceptional set \(\mathbb{EX}(\epsilon; \lambda, \beta)\) has a covering \(\{r_{k},R_{k}\}\) (\(k=1,2,\ldots\)) satisfying
The following result is due to Jiang et al. (see [10], Theorem 1), who are concerned with the boundary behaviors of Poisson integrals and their applications. For similar results in a half space, we refer the reader to the paper by Jiang and Huang (see [7]).
Lemma 3
Let \(\mathcal{POI}_{\Sigma}\mu(G) \not\equiv+\infty\) for any \(G=(r,\Xi)\in\beth_{n}(\Sigma)\), where μ is a positive measure on \(\daleth_{n}(\Sigma)\). Then
for any \(G\in\beth_{n}(\Sigma)-\mathbb{EX}(\epsilon; \mu',n- \alpha)\) as \(r \rightarrow\infty\), where \(\mathbb{EX}(\epsilon; \mu',n-\alpha)\) is a subset of \(\beth_{n}(\Sigma)\) and has a covering \(\{r_{k},R_{k}\}\) of satisfying (2.1).
4 Proof of Theorem 1
Let \(G=(r,\Xi)\) be any point in \(\beth_{n}(\Sigma; (L,+ \infty))-\mathbb{EX}(\epsilon; \nu', n-\alpha)\), where L is a sufficiently large number satisfying \(r \geq\frac{5L}{4}\).
Put
where
We have the following estimates:
from (3.1), (3.2), and [11], Lemma 1.
By (3.3), we have
where
Then by [11], Lemma 1 we immediately get
In order to give the growth properties of \(\mathcal{GF}_{\Sigma}^{22}(G)\). Take a sufficiently small positive number \(k_{2}\) independent of G such that
The set \(\beth_{n}(\Sigma;(\frac{4}{5}r,\frac{5}{4}r))\) can be split into two sets \(\Gamma(G)\) and \(\Gamma'(G)\), where \(\Gamma'(G)=\beth _{n}(\Sigma;(\frac{4}{5}r,\frac{5}{4}r))-\Gamma(G)\). Write
where
For any \(H\in\Gamma'(G)\) we have \(\vert G-H\vert \geq k_{2}'r\), where \(k_{2}'\) is a positive number. So [11], Lemma 1 gives
To estimate \(\mathcal{GF}_{\Sigma}^{221}(G)\). Set
where \(l=0,\pm1,\pm2,\ldots\) and \(\varrho(G)=\inf_{H\in\partial{ \beth_{n}(\Sigma)}}\vert G-H\vert \).
From Remark 7 it is easy to see that \(\nu'(\{P\})=0\) for any \(G=(r,\Xi)\notin\mathbb{EX}(\epsilon; \nu', n-\alpha)\). The function \(\mathcal{GF}_{\Sigma}^{221}(G)\) can be divided into \(\mathcal{GF}_{\Sigma}^{221}(G)=\mathcal{GF}_{\Sigma}^{2211}(G)+ \mathcal{GF}_{\Sigma}^{2212}(G)\), where
For any \(H=(t,\Omega)\in I_{l}(p)\), we have \(2^{-1}\varrho(G)\leq \varrho(H)\leq Mth_{\Sigma}(\Omega)\), because \(\varrho(H)+\vert G-H\vert \geq\varrho(G)\). Then by Remark 3
for \(l=-1,-2,\ldots\) .
Moreover, we have
for any \(G=(r,\Xi)\notin\mathbb{EX}(\epsilon; \nu', n-\alpha)\).
Equation (4.4) shows that there exists an integer \(l(G)>0\) such that \(2^{l(G)}\varrho(G)\leq r<2^{l(G)+1}\varrho(G)\) and \(I_{l}(G)= \varnothing\) for \(l=l(G)+1,l(G)+2,\ldots\) . And Remark 3 shows that
for \(l=0,1,2,\ldots,l(G)\).
We have for any \(G=(r,\Xi)\notin\mathbb{EX}(\epsilon; \nu', n- \alpha)\)
for \(l=0,1,2,\ldots,l(G)-1\) and
So
From (4.1), (4.2), (4.3), (4.5), (4.6), and (4.7) we obtain \(\mathcal{GF}_{\Sigma}\nu(G)=o(r^{ \iota_{\Sigma}}\{h_{\Sigma}(\Xi)\}^{1-\alpha})\) for any \(G=(r,\Xi)\in\beth_{n}(\Sigma; (L,+\infty))-\mathbb{EX}(\epsilon; \nu', n-\alpha)\) as \(r\rightarrow\infty\), where L is sufficiently large number. Finally, Lemma 2 gives the conclusion of Theorem 1.
Change history
19 July 2021
A Correction to this paper has been published: https://doi.org/10.1186/s13661-021-01541-6
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The authors would like to thank the referee for invaluable comments and insightful suggestions.
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HW participated in the design and theoretical analysis of the study, drafted the manuscript. JM conceived the study, and participated in its design and coordination. KL participated in the design and the revision of the study. All authors read and approved the final manuscript.
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Lai, K., Mu, J. & Wang, H. RETRACTED ARTICLE: New applications of Schrödingerean Green potential to boundary behaviors of superharmonic functions. Bound Value Probl 2017, 21 (2017). https://doi.org/10.1186/s13661-017-0746-4
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DOI: https://doi.org/10.1186/s13661-017-0746-4
Keywords
- boundary value problem
- Green potential
- asymptotic behavior