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Some properties for a subfunction associated with the stationary Schrödinger operator in a cone
Journal of Inequalities and Applications volume 2012, Article number: 295 (2012)
Abstract
For a subfunction u associated with the stationary Schrödinger operator, which is dominated on the boundary by a certain function on a cone, we correct Theorem 1 in (Qiao and Deng in Glasg. Math. J. 53(3):599-610, 2011). Then by the theorem we generalize some theorems of Phragmén-Lindelöf type for a subfunction in a cone. Meanwhile, we obtain some results about the existence of solutions of the Dirichlet problem associated with the stationary Schrödinger operator in a cone and about the type of their uniqueness.
MSC:31B05.
1 Introduction and main results
To begin with, let us agree to some basic conventions. As usual, let S be an open set in (), where is the n-dimensional Euclidean space. The boundary and the closure of S are denoted by ∂ S and , respectively. Let , where , and let be the Euclidean norm of P and be the Euclidean distance of two points P and Q in . The unit sphere and the upper half unit sphere are denoted by and , respectively. For simplicity, the point on and the set for a set Ω, are often identified with Θ and Ω, respectively. For 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 and call it a cone. For an interval and , set , and . By we denote , which is . Furthermore, we denote by the -dimensional volume elements induced by the Euclidean metric on . For and , let denote the open ball with center at P and radius r in , then .
We introduce the system of spherical coordinates , for in via the following formulas:
and if ,
where , () and .
Relative to the system of spherical coordinates, the Laplace operator Δ may be written
where the explicit form of the Beltrami operator is given by Azarin (see [1]).
For an arbitrary domain D in , denotes the class of non-negative radial potentials (i.e., for ) such that with some if and with if or .
If , the stationary Schrödinger operator with a potential ,
can be extended in the usual way from the space to an essentially self-adjoint operator on , where Δ is the Laplace operator and I is the identical operator (see Reed and Simon [2], Chapter 13). Furthermore, has a Green a-function . Here is positive on D and its inner normal derivative , where denotes the differentiation at Q along the inward normal into D. We denote this derivative by , which is called the Poisson a-kernel with respect to D. There is an inequality between the Green a-function and that of the Laplacian, hereafter denoted by . It is well known that for any potential ,
The inverse inequality is much more elaborate when D is a bounded domain in . For a bounded domain D in , Cranston, Fabes and Zhao (see [3], the case is implicitly contained in [4]) have proved
where is positive and independent of points in D. If , obviously .
Suppose that a function is upper semi-continuous in D. is called a subfunction of the Schrödinger operator if the generalized mean-value inequality
is satisfied at each point with , where , is the Green a-function of in and is the surface area element on .
Denote the class of subfunctions in D by . If , we call u a superfunction and denote the class of superfunctions by . If a function u is both subfunction and superfunction, clearly, it is continuous and called an a-harmonic function associated with the operator . The class of a-harmonic functions is denoted by . In terminology we follow Levin and Kheyfits (see [5] or [6]). From now on, we always assume . For the sake of brevity, we shall write instead of , instead of , (resp. ) instead of (resp. ) and instead of .
For positive functions and , we say that if for some constant . If and , we say that .
Let Ω be a domain on with a smooth boundary, and let λ be the least positive eigenvalue for on Ω (see [7], p.41),
The corresponding eigenfunction is denoted by satisfying . In order to ensure the existence of λ and , 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 Gilbarg and Trudinger [8], pp.88-89 for the definition of -domain). We denote the non-decreasing sequence of positive eigenvalues of (1.5) by . In the expression, we write the same number of times as the dimension of the corresponding eigenspace. When the normalized eigenfunction corresponding to is denoted by , the set of sequential eigenfunctions corresponding to the same value of in the sequence makes an orthonormal basis for the eigenspace of the eigenvalue . Hence, for each , there is a sequence of positive integers such that , ,
and is an orthonormal basis for the eigenspace of the eigenvalue (). It is well known that and for any (see Courant and Hilbert [9]). For the case (), () when , and the situation is more complicated when (see the Remark in [10] for the detailed information). For a domain Ω and the sequence mentioned above, by we denote the set of all positive integers less than (). In spite of the fact that , the summation over of a function of a variable k is used by promising
If Ω is an -dimensional compact Riemannian manifold with its boundary to be sufficiently regular, we see that
(e.g., see Cheng and Li [11]) and
uniformly with respect to Θ (e.g., see Minakshisundaram and Pleijel [12] or Essén and Lewis [13], p.120 and pp.126-128), where and are both constants depending on Ω and n. Hence, there exist two positive , such that
and
Solutions of an ordinary differential equation
are known (see [14] for more references) when the potential . We know equation (1.11) has a fundamental system of positive solutions such that V is non-decreasing with
and W is monotonically decreasing with
We remark that both and () are a-harmonic on and vanish continuously on , where and are the solutions of equation (1.11) with .
Let be the class of the potentials such that
When , the (super)subfunctions are continuous (e.g., see [15]). In the rest of paper, we assume that and we suppress this assumption for simplicity.
Denote
When , the solutions , to equation (1.11) normalized by have the asymptotic (see [8])
and
where is their Wronskian at .
Remark 1.1 If and , , and , where is the surface area of .
Let be a function on . We introduce , and .
We say that () satisfies the Phragmén-Lindelöf boundary condition on , namely
Let be a function on . For any given positive real number r, the integral
is denoted by , when it exists. For two non-negative integers p and q, the finite or infinite limit
is denoted by (resp. ), when it exists.
If is a real finite-valued function defined on an interval , for any given , () and , we have
if and only if
where has the following expression:
We say that is -convex on if () for any , ().
Remark 1.2 A function is -convex on if and only if is a convex function of on or, equivalently, if and only if is a convex function of on ; refer to Dinghas [16] for the relevant properties of a convex function with respect to an ODE.
The Poisson a-integral of g relative to is defined by
where
denotes the differentiation at Q along the inward normal into and is the surface area element on .
For two non-negative integers l, m and two points and , we put
and
We introduce two functions of and as follows:
and
The kernel with respect to is defined by
In fact
and
Based on the elaborate research, Yoshida ([17] and [18]) has considered the subharmonic function defined on a cone or a cylinder which is dominated on the boundary by a certain function and generalized the classical Phragmén-Lindelöf theorem by making a harmonic majorant. Later Yoshida [19] proved the property of a harmonic function defined on a half-space which is represented by the generalized Poisson integral with a slowly growing continuous function on the boundary. In [20] or [10] Yoshida and Miyamoto generalized some theorems (from [19]) to the conical case and extended the results (from [17] and [18]) given particular solutions and a type of general solutions of the Dirichlet problem on a cone by introducing conical generalized Poisson kernels and Poisson integrals. On the other hand, Qiao and Deng [21] extended Yoshida’s results (from [18]) to the situation for the stationary Schrödinger operator; for the relevant research on the stationary Schrödinger operator, we may refer to Bramanti [22], Kheyfits [23–30] and Levin et al.[6, 31]. However, we find a falsehood in [21] and have to make a correction. In [5] or [6] we also know the Green function associated with the stationary Schrödinger operator. Dependent on the related statement above, we are to be concerned with the solutions of the Dirichlet problem for the stationary Schrödinger operator on and with their growth properties. Furthermore, we note the existence of solutions of the Dirichlet problem for the stationary Schrödinger operator in a cone and the type of their uniqueness. First of all, we start with the following result.
Theorem A Let be a continuous function on satisfying
and
Then the function () satisfies
and
Remark 1.3 As to Theorem 1 in the paper [21], the factor can be replaced with such that it is true, that is, Theorem A corrects Theorem 1 (from [21]) which is a generalization for a result from Siegel and Talvila (see [32]). Moreover, as to Theorem A we may follow the proof procedure of Theorem 1 in [21].
Next, we state our main results as follows.
Theorem 1.4 Let l, m be two non-negative integers andbe a continuous function onsatisfying
and
Then
is a solution of the Dirichlet problem for the stationary Schrödinger operator on with g satisfying
Theorem 1.5 Let l be a non-negative integer andbe a continuous function onsatisfying
Then
is a solution of the Dirichlet problem for the stationary Schrödinger operator on with g satisfying
Remark 1.6 When , Theorem 1.4 is equal to Theorem 1.5. Since Theorem 1.4 may follow the proof for Theorem 1.5, for convenience, we only prove the latter.
It is natural to ask if 0 in (1.14) can be replaced with a general function on . The following Theorem 1.7 gives an affirmative answer to this question. For related results, we refer the readers to the paper by Levin and Kheyfits (see [6], Section 3 or [5], Chapter 11).
Theorem 1.7 Let p, q be two positive integers satisfying. Letbe a continuous function onsatisfying (1.25) and (1.26) andbe a subfunction onsuch that
Then all of the limits, , and (, , , ) exist. Moreover, when
for any, where () and () are all constants.
As an application of Theorems A and 1.7, we obtain the following result.
Theorem 1.8 Let p, q be two positive integers satisfying. Letbe defined as in Theorem 1.7 andbe any solution of the Dirichlet problem for the stationary Schrödinger operatoronwith g. Then all of the limits, , and (, , , ) exist. Moreover, when
for any, where () and () are all constants.
Remark 1.9 For , Theorems 1.7 and 1.8 come from Qiao and Deng [21]. Furthermore, when and , Theorems 1.7 and 1.8 are due to Yoshida (see [18], Theorems 2 and 3(II)). In fact, for we know , (or , ) are equal to the corresponding (or , ), respectively. Without the potential function, we may refer to Yoshida (see [33]).
Theorem 1.10 Let l, m be two non-negative integers and p, q be two positive integers satisfying. Letbe defined as in Theorem 1.7 satisfying (1.25) with l and (1.26) with m, respectively. Ifis any solution of the Dirichlet problem for the stationary Schrödinger operatoronwith g satisfying
for any, where () and () are all constants.
Theorem 1.11 Let l be a non-negative integer and p be a positive integer satisfying. Ifis a generalized harmonic function onand continuous onsuch that the restrictionof h tosatisfies
for some non-negative integer l, and for a positive integer
then, for some positive integer,
for any, where () are all constants.
Remark 1.12 If we take , Theorems 1.10 and 1.11 are similar to Theorems 7 and 9 in [20], respectively. In [33] Yoshida and Miyamoto considered the case when , and about Theorem 1.10 and gave the proof. In addition, with Theorem 1.10 we easily get the conclusion of Theorem 1.11, then we do not have to prove it.
2 Some lemmas
In our arguments, we need some important results and techniques, which result from [5, 6, 27, 34], and [1] (Lemma 4 and Remark).
Lemma 2.1
for anyand anysatisfying (resp. ). In addition,
for anyand any.
Lemma 2.2 Let. For a non-negative integer, we have
for anyandsatisfying (), whereis a constant dependent on n, and s.
Lemma 2.3 Letbe a locally integrable and upper semicontinuous function on. Letbe a function ofandsuch that for any fixedthe functionofis a locally integrable function on. Put
Suppose that the following (I) and (II) are satisfied.
-
(I)
For any and any , there exist a neighborhood of in and a number R () such that
(2.6)
for any
-
(II)
For any and any R (),
(2.7)
Then
for any.
Remark 2.4 When , Lemma 2.3 is due to Yoshida (see [10], Lemma 5). By (1.2), obviously we reduce to Yoshida’s results. Therefore, we may omit the proof.
Lemma 2.5 Ifis an a-harmonic function onvanishing continuously on,
for any, () and every r ().
Lemma 2.6 Ifis-convex on (), then
exists. Further, if, is non-decreasing on.
It is known that is regular, the Dirichlet problem for Δ and is solvable in it (see [6]). Based on this fact, Lemmas 2.7, 2.8 and 2.9 could be derived from (1.2), (1.3), (1.12), Remark 1.2, Lemmas 2.3 and 2.5 with their means of proof essentially due to Yoshida (see [17], Theorems 3.1 and 5.1, and [18], Lemma 3).
Lemma 2.7 Ifis a subfunction onsatisfying the Phragmén-Lindelöf boundary condition on, then
forandis-convex on. If there are three numbers, andsatisfyingsuch that
we have that
-
(1)
();
-
(2)
is an a-harmonic function on and vanishes continuously on .
Lemma 2.8 Letbe defined as in Theorem 1.7. Then (resp. ) is an a-harmonic function onsuch that both of the limitsand (resp. and) exist, and
Lemma 2.9 Letbe a subfunction onsatisfying the Phragmén-Lindelöf boundary condition on. If (1.32) is satisfied forand,
for any.
By the Kelvin transformation (see [35], p.59), Lemmas 2.6 and 2.7, we immediately have the following result, which is due to Yoshida in the case (see [17], Theorem 3.3).
Lemma 2.10 Letbe defined as in Lemma 2.9. Then
-
(1)
Both of the limits and (, ) exist.
-
(2)
If , then is non-decreasing on .
-
(3)
If , then is non-increasing on .
Lemma 2.11 Letbe an a-harmonic function invanishing continuously on, and p, q be two positive integers. h satisfies
then
for any, where () and () are all constants.
Remark 2.12 When and , Yoshida states the result in [10]. Later Qiao [34] proves Lemma 2.11 when . Similarly, we may complete the proof of Lemma 2.11. Herein we leave out the detailed information for the proof.
3 Proofs of the theorems
Proof of Theorem 1.5 For any fixed , we take a number R satisfying (). Then from Lemma 2.2 and (1.25)
Then is absolutely convergent and finite for any . We remark that is a harmonic function of for any . Thus, is a generalized harmonic function of .
Next, we consider the boundary behavior of . To prove that
for any , we may apply Lemma 2.3 to and by putting
which is locally integrable on for any fixed . Let δ be a positive number and take and any . Then from (1.25) and (3.1) we can choose a number R () () such that for any , where and δ is a positive number
which is (I) in Lemma 2.3.
Because
as , we know that for any and ,
According to Lemma 2.3, we get the required results.
Next, we note the inequality
where
and
for , . For any positive number ε, from (1.25) we can take a sufficiently large number such that
where M is the constant in Lemma 2.2 and
Then from Lemma 2.2 we have
which gives .
Following this, we see the inequality
where
and
for and . First, we know from (1.10) and (1.16) that if
where
We claim that
First we note increasing and Lemma C.1 in [5], Chapter 13 or [6], then by (1.12) we get
Hence, we can conclude that if , then
Next, we see and note that
where
and
Since
we see from (3.3) and (3.7) that
If we can show that
we can finally conclude from (3.8) and (3.9) that
which gives the required result. To prove (3.10), we recall that is an a-harmonic function on satisfying
for any . Hence, we know
and so
Thus we obtain that if , then
Therefore,
and so we conclude that
Similarly, we apply the method to , then we have
Since
we get the desired conclusion. □
Proof of Theorem 1.7 We see from Theorem A that
Set the two subfunctions on as follows:
We have from (1.31) and (3.12)
Hence, it follows from Lemma 2.10 that all of the limits , , and (, , , ) for any exist. Since
according to Lemma 2.8, we know all of the limits , , and exist and that
for any . Since
we have from Lemma 2.8 and (1.32) that
Applying Lemma 2.9 to U, we can obtain from (3.13)
for , which is required. □
Proof of Theorem 1.8 We put and in Theorem 1.7, respectively. Then Theorem 1.7 gives the existence of all limits , , ,
for any . Since
it follows that both limits and exist. If
then we see from (3.14), (3.15) and (1.33) that
Hence, by applying Theorem 1.7 to and again, we obtain from (1.34) that
and
respectively, which gives the required result. □
Proof of Theorem 1.10 From Theorem 1.4 we have the solution of the Dirichlet problem for the stationary Schrödinger operator on with g satisfying (1.28). We consider the function . Then we see that it is an a-harmonic function in and vanishes continuously on . Since
for any ,
and
from (1.28), (1.36) gives that
and
From Lemma 2.11 we see that
for any , where () and () are all constants. Thus, we obtain the conclusion of Theorem 1.10. □
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Acknowledgements
We wish to express our appreciation to the referee for their careful reading and some useful suggestions which led to an improvement of our original manuscript. Supported by SRFDP (No. 20100003110004) and NSF of China (No. 11071020 and No. 11271045).
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PL carried out the study, participated in the design and drafted the manuscript, GD conceived the study and participated in the design. All authors read and approve the final manuscript.
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Long, P., Deng, G. Some properties for a subfunction associated with the stationary Schrödinger operator in a cone. J Inequal Appl 2012, 295 (2012). https://doi.org/10.1186/1029-242X-2012-295
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DOI: https://doi.org/10.1186/1029-242X-2012-295