Green’s Function for a Slice of the Korányi Ball in the Heisenberg Group
We give a representation formula for solution of the inhomogeneous Dirichlet problem on the upper half Korányi ball and for the slice of the Korányi ball in the Heisenberg group by obtaining explicit expressions of Green-like kernel when the given data has certain radial symmetry.
The Heisenberg groups, in discrete and continuous versions, appear in several streams of mathematics, including Fourier analysis, several complex variables, geometry, and topology. The well known concepts of Green, Neumann, and Robin functions are important for representing solutions to certain boundary value problems for elliptic equations and to realize their smoothness properties. While the existence of these particular fundamental solutions for admissible domains are presented in all textbooks (see, e.g., [1–3]), explicit expressions for these functions are rare. In case of the Laplace operator mostly just the unit ball serves as an example.
The objective of this paper is to continue the search for explicit Green’s functions for domains other than the Korányi ball in the Heisenberg group. There is little hope to get explicit kernels that work for arbitrary continuous boundary data, but it is possible to find some if one is restricted to boundary data having certain symmetry properties. This line of investigation was started by Gaveau et al. in  where they dealt with the case of the unit ball in the 3-dimensional Heisenberg group and functions invariant under a circle action. This result was extended in  to the general Heisenberg group with its natural metric, for functions invariant under the unitary group . Further, in the case of , it has been shown that the method of  works for the much larger class of circular functions, that is, functions invariant under a circle action . The Dirichlet problem on the Heisenberg group and the existence of unique solution was discussed in . Green’s function for circular data in the Heisenberg group has been studied for various domains, for example, for half space in , for quarter space in , and for annulus in .
In following sections, we obtain the circular Green’s function for the upper half Korányi ball and a slice of the Korányi ball by two parallel planes. We apply the method of infinitely many reflections along the boundaries of the domain, which was introduced by Courant and Hilbert in , generalized for the annulus in the Heisenberg group .
2. Analysis on the Heisenberg Group
We begin by recalling some notions from , which have laid the foundation for harmonic analysis on the Heisenberg group.
Definition 1. The Heisenberg group (of degree ) is the Lie group structure on whose group law is given by
We write , , and we define on the vector fields One may note the following: Equivalently, Here is the Lie bracket.
Observe that , are left invariant vector fields generated by the tangent vectors , at .
If , the natural gauge on is given by On the Heisenberg group we have an analogue of the Laplacian which was first studied by Folland and Stein . The sub-Laplacian on is explicitly given by Let denote the slightly modified subelliptic operator .
The fundamental solution for the sub-Laplacian on the Heisenberg group was first given by Folland in  with pole at identity by where is the normalization constant.
The fundamental solution of with pole at is given by From , for and , whereFor an integrable function on , we denote the average of by A function is said to be circular, that is, invariant under circle action if , for .
The average of the fundamental solution with pole at is given in  as where is the Gaussian hypergeometric function . We can easily see that the average of the fundamental solution is also -harmonic away from .
3. Green’s Function for the Upper Half Korányi Ball
In this section, the domain is defined as , where denotes the Korányi ball in ; that is, and denotes the half space in .
W. Thomson (Lord Kelvin) proved in 1847 the following fact: if is harmonic on then the function defined by (, ) is harmonic on [14, p. 232]. For this reason the transformation is called the Kelvin transform. An analogue of the classical Kelvin transform is defined for functions on the Heisenberg group and it is shown that this transform preserves the class of functions annihilated by the left invariant sub-Laplacian . For a function on , the Kelvin transform is defined by where is the inversion defined as for . The Kelvin transform sends a -harmonic function on to a -harmonic function. It was shown in  that, for a circular function on , we havefor all with .
From [5, (3.3)] we have, for where we wrote for .
For , let being the reflection of the point with respect to the boundary ; that is, .
In the next result, we take the differential operator with respect to the variable .
Theorem 2. For each , the function as defined in (19) is a smooth function on which extends continuously up to and satisfies the following:(i).(ii)Limits of the function vanish at the boundaries of the upper half Korányi ball, that is, at and .
Proof. (i) For , we have since lies outside and we know that is -harmonic away from therefore is -harmonic on .
Similarly and are -harmonic on because and lie outside .
(ii) We have where and .
At , , therefore,Hence at .
Since is circular, so, by using (17) we get Hence at .
By using the circular Green’s function, we can easily solve the inhomogeneous Dirichlet problem where and are continuous circular functions and the solution is given by the representation formula where is the Poisson kernel and denote the Riemannian surface element, being the corresponding Riemannian volume element on ; for detailed study refer .
4. Green’s Function for a Slice of the Korányi Ball
In this section denotes a slice of the Korányi ball in which is defined as follows:For and we define where , are as in (11). A differential operator, whenever applied to function , will be with respect to the variable . Now, we assert that the Green function for the domain is given by where denotes the Kelvin transform in . In the following lemma, we prove that is well defined for all , with or .
Lemma 3. For a fixed and such that or , the series on RHS of (29) are absolutely and uniformly convergent on a compact neighbourhood of .
Case 1. Suppose that , we choose a relatively compact neighbourhood of such that , . Then, and therefore, for sufficiently large , On the set , the function is bounded and so, for sufficiently large , are uniformly bounded. Also, for smaller positive values of Hence are uniformly bounded for all . For , the term is In case or , above expression is equal to zero. For and ,and therefore, is uniformly bounded on . Now, Thus, Case 2. Suppose that , we choose a compact neighbourhood of such that , , for all .So, the functions are uniformly bounded on . Now Case 3. Now, suppose that , , and considerTherefore, We can choose a suitable compact neighbourhood of such that for some . We have This implies that Clearly, Hence, in all the three cases for some positive constant . Thus, is absolutely and uniformly convergent for . Similarly, one can easily show that the other series in the RHS of (29) are uniformly and absolutely convergent for .
Therefore, is well defined for all .
In the next lemma, we show that on the boundaries of the domain .
Lemma 4. For , and for Similarly, for , and for Finally, for and for
Proof. Consider Therefore, On the same lines, one can easily prove the limits agree over the other boundaries also.
Now, we show that works as Green’s function for domain when applied to circular data.
Theorem 5. For each , the function as defined in (29) is a smooth function on which extends continuously up to and satisfies the following:(i).(ii)Limits of the function vanish at the boundaries of Korányi ball slice, that is, at , , and .
Proof. (i) For , , the functions , , , , , , , and are all harmonic at because poles of these functions lie outside the domain . Also we have is harmonic and from the definition of it follows that Therefore, by using Lemma 3 (ii) This part easily follows from Lemma 4.
Hence the theorem is established.
Now, we have circular Green’s function for a slice of the Korányi ball so we can obtain explicit expression of the Poisson kernel for this domain to solve inhomogeneous Dirichlet boundary value problem. The Poisson kernel is the normal derivative of Green’s function and, from [5, 6], is given by where
Theorem 6. Green’s function and the Poisson kernel which we have obtained solve the inhomogeneous Dirichlet boundary value problem for and the solution for BVP is given by where and are continuous circular functions.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The first author is supported by the Senior Research Fellowship of Council of Scientific and Industrial Research, India (Grant no. 09/045(1152)/2012-EMR-I), and the second author is supported by R&D grant from University of Delhi, Delhi, India. The authors would like to thank the referees for their careful reading and substantial comments on an earlier version of this paper.
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