ABSTRACT
Fundamental solutions and Green's functions of the operator are calculated in the half-space
1. Introduction and notation
The goal of this paper consists in deriving, in the framework of Schwartz' distribution theory [Citation1], fundamental solutions and Green's functions of the operator of ‘generalized axially symmetric potential theory’ (GASPT), i.e. of (1) (1) Fundamental solutions of were presented first by Weinstein (see [Citation2,Citation3]). His method of derivation was based on classical analysis and did not involve distribution theory, which at that time was not yet state of the art. Weinstein's method ran along the following five steps:
assume first that is a natural number;
use the known fundamental solution of the Laplacean operator in variables;
introduce polar coordinates with respect to the first variables;
integrate with respect to the sphere
replace by
Finally , Weinstein checked the ‘nature of singularity’ of the function found by the procedure in (a) to (e). The resulting fundamental solutions are expressed by definite integrals, and as customary until 1951, they are defined only up to multiplicative constants.
However ingenious Weinstein's approach may be, it does not seem satisfactory from the view-point of modern analysis. In the literature, there are two further treatments of the GASPT operator we know of. In [Citation4, Ch. VIII: Degenerate elliptic operators], a formulation of existence and uniqueness results for a class of operators including the one in GASPT is given. However, the approach is based on Sobolev spaces and Green's functions are not expressed by special functions but only as definite integrals.
A different attempt at constructing fundamental solutions of in a distributionally correct way is contained in the studies [Citation5,Citation6]. Therein, fundamental solutions are set up as infinite series motivated by the derivation of the fundamental solution of the EPD-operator in [Citation7] (see also [Citation8]). However, no effort is made of deriving uniqueness results or Green's functions; furthermore, the result in the case n=1 [Citation6, Theorem 3.4, Equation (3.27), p.507] seems to be incorrect.
For the reasons explained above, we have taken up anew the study of fundamental solutions and Green's functions of the operator in (Equation1(1) (1) ). In Definition 2.1, we define the notions of temperate fundamental solutions and Green's functions of the Dirichlet problem and the Neumann problem in the half-space for the singular operator in (Equation1(1) (1) ). The uniqueness of Green's functions is investigated in Proposition 2.2, and we represent Green's functions and temperate fundamental solutions of by hypergeometric functions in Theorem 2.3. In the case of even n, we represent these fundamental solutions by elementary transcendental functions in Corollary 2.4. Of course, some of our formulas can be found already in [Citation2–6] (see the remarks following Theorem 2.3).
We derive our results by employing the partial Fourier transform with respect to the -variables and by using suitable identities for the hypergeometric function. We also make use of the theory of distribution-valued analytic functions as expounded in [Citation9].
Let us introduce some notation. Besides the spaces open, and of distributions and temperate distributions, respectively, we also use the space of temperate distributions on the half-space H defined above. Note that the partial Fourier transform which is extended by continuity from yields also an isomorphism on The Heaviside function is denoted by Y, and we write for the delta distribution with support in τ, i.e. for the derivative of
2. Temperate fundamental solutions and Green's functions in GASPT
As mentioned already in the introduction, the operator arises in the so-called GASPT (see [Citation2] for historical remarks and connections to physics). Let us first introduce the notions of temperate fundamental solution and Green's functions for
Definition 2.1
Set and fix and
is called temperate fundamental solution of if and only if holds in H.
is called Green's function of the Dirichlet problem for if and only if E is a temperate fundamental solution of that satisfies in
is called Green's function of the Neumann problem for if and only if E is a temperate fundamental solution of that satisfies in
Remark 2.1
Note that a fundamental solution E of is in due to [Citation10, Theorem 13.4.1, p.191]. Hence we can fix t in for For example, the hypothesis in in Definition 2.1 then means that belongs, for fixed large t, to and converges therein to 0 if
The next proposition will show that the Green functions of the Dirichlet problem and the Neumann problem, respectively, for are uniquely determined in those cases where they exist.
Proposition 2.2
Fix and let fulfill in H and in If, additionally, either or hold in then T vanishes identically.
Proof.
The partial Fourier transform of T satisfies the ‘ordinary’ differential equation (2) (2) For let Then where Since V fulfills in we conclude that Therefore, holds in for
Let us use now the boundary conditions for T. The assumption in implies in and therefore vanishes and holds in On the other hand, either of the limits or implies Hence vanishes and i.e. (Note that U is a distribution of finite order due to
Let us assume that is such that and that does not vanish identically. Then (Equation2(2) (2) ) implies that and hence for or if In both cases, the conditions or then imply that vanishes and that leads to a contradiction. Therefore U=0 and thus also T=0 and the proof is complete.
Theorem 2.3
As before, set Let and and set (3) (3) The functions (4) (4) are in and locally integrable in H. For even n, the mapping extends to an entire function for odd n, this mapping is meromorphic on with simple poles in the set Furthermore, we set for if n is even and for if n is odd, respectively.
For those for which respectively, are defined, they are temperate fundamental solutions of Furthermore, is the uniquely determined Green function of the Neumann problem for if and is the uniquely determined Green function of the Dirichlet problem for if
(In (Equation4(4) (4) ) denotes an associated Legendre function and denotes Gauß' hypergeometric function. If α is a negative entire number not belonging to then in (Equation4(4) (4) ) has to be interpreted as a limit.)
Proof.
(a) Let us first assume and represent by a partial Fourier transform with respect to x. From we obtain From and by the Neumann boundary condition, we infer, for fixed, that with the jump conditions The ‘Wronskian’ determinant of this linear system of equations fulfills (see [Citation11, A, 17.1, p.72]), and employing the series expansions of and yields Thus and The inequalities imply that and for fixed positive due to the hypothesis These inequalities also imply that the limits and hold in by Lebesgue's theorem on dominated convergence. Hence is indeed the Green function of the Neumann problem for and and hold even uniformly in x.
(b) In order to calculate for we apply the classical Poisson–Bochner formula (see [[Citation1, (VII, 7; 22), p.259],[Citation12, Satz 56, p.186],[Citation8, (1.1)]]). For Equation 6.578.11 in [Citation13] then implies (5) (5) with z as in (Equation3(3) (3) ). Equation (Equation5(5) (5) ) also holds for either by the real analyticity of in or by using Equation 6.578.11 in [Citation13] again with t and τ interchanged. Eventually, we employ formula [Citation14, 7.3.1.72] for in order to derive the representation in (Equation4(4) (4) ) of by the hypergeometric function. (Note that and hence also are continuous functions of t with values in and hence is already determined by its restriction to
(c) Let us next investigate the analytic continuation of with respect to α. If then formula (Equation4(4) (4) ) yields Similarly, for we have and hence converges to 0 if and else grows like a multiple
In order to analyze the behavior of near we employ Equation 9.131.1 in [Citation13]. This furnishes (6) (6) Formula (Equation6(6) (6) ) clearly implies, for each that is well defined and depends on Furthermore, if then converges to 1 from below and Equation 9.122.1 in [Citation13] yields that if Hence formula (Equation6(6) (6) ) shows that is bounded by a constant multiple of near for If we use [Citation14, 7.3.1.30] and obtain that grows like near In particular, we see that is locally integrable, depending holomorphically in on and by analytic continuation, we conclude that is a temperate fundamental solution of for such α.
(d) Let us consider now the behavior of if α converges to If n is even, then is holomorphic, and hence is an entire function of α. In contrast, if n is odd, then has simple poles at In fact, [Citation13, Equation 9.134.1] yields the representation Due to this implies (7) (7) upon using the complement formula of the gamma function. Note that the residue is a polynomial in x since the hypergeometric series in (Equation7(7) (7) ) terminates, and that
(e) Let us finally discuss Clearly, From the equation we infer Hence is a temperate fundamental solution of for each Furthermore, (Equation4(4) (4) ) shows that and hold uniformly with respect to if Thus is the Green function of the Dirichlet problem for if This completes the proof.
Remark 2.2
(1) By analyzing the partial Fourier transform similarly as in the proof of Proposition 2.2, one readily sees that Green's functions E of the Neumann problem and the Dirichlet problem, respectively, for can exist only if and respectively.
(2) The Green function of the Dirichlet problem for could, albeit more laboriously, also be derived by the partial Fourier transform. Setting and yields, first for the equation Since is meromorphic with simple poles in we can conclude from this that if and only if α is entire and is odd or
(3) Let us point out that the finite parts and n odd, respectively, are not, in general, temperate fundamental solutions of and of respectively. In fact, if, e.g. then Hence is a temperate fundamental solution of if and only if R is constant with respect to t.
For example, if then this is the case (see (Equation7(7) (7) )) and therefore and respectively, are temperate fundamental solutions of and of respectively. For example, if we obtain from [Citation14, 7.3.1.30] that which of course fulfills the two-dimensional Laplace equation in Similarly, fulfills in H.
(4) Let us now refer to the literature. The relation connecting two fundamental solutions E of and F of respectively, can be found in [Citation2, (2.11), p.106], where it is traced back to G. Darboux. Furthermore, the Green function of the Neumann problem is given (up to a multiplicative constant) in the form of Euler's definite integral of the hypergeometric function in [Citation2, (3.4), p.108], and some hints regarding uniqueness are also given at the bottom of page 108. The Green function of the Dirichlet problem appears in [Citation2, (4.1), p.109] and is referred to M. Olevskii. Green's functions for also appear in [Citation4, (8.4), p.217, and Theorem 8.2, p.219].
As discussed in the introduction, the paper [Citation6] already contains some of the above results, albeit in a less systematic way. First note that the notation in [Citation6] slightly differs from ours: there, are written for our In [Citation6, Theorem 3.1, Equation (3.2), p.503], in the case of even n, a fundamental solution ‘’ of is given by a hypergeometric function, which is verified by termwise differentiation of the series expansion. In our notation, ‘’ corresponds to the fundamental solution and Equation (3.2) in [Citation6] follows from formula (Equation4(4) (4) ) by using [Citation13, 9.132.2]:
In [Citation6, Theorems 3.2, 3.3], the case of odd is treated and the representation of ‘’ in [Citation6, Theorem 3.3, Equation (3.21), p.506] corresponds to the one of in (Equation4(4) (4) ) above. (Note that the values are excluded in [Citation6, Theorem 3.3, p.506] although has poles only for and is a fundamental solution of for all other complex values of
Finally, in [Citation6, Theorem 3.4], the case n=1 is considered. We observe that the fundamental solution ‘’ in [Citation6, Theorem 3.4, Equation (3.27), p.507] does not seem to be correct. For example, for Equation (Equation4(4) (4) ) in Theorem 2.3 yields (8) (8) whereas ‘’ in [Citation6, Theorem 3.4, Equation (3.27), p.507] would furnish the function However, f cannot be a fundamental solution of since it is finite at due to (The letters K, E denote, as usually, complete elliptic integrals.)
Note that the fundamental solution for n=1 in (Equation8(8) (8) ) coincides, up to a multiplicative constant, with the expressions given in [Citation15, Equation (5.35), p.149],[Citation16, Equation (2.2.16), p.9],[Citation17, p.1655]. For see [Citation18, Equation (17), p.146].
Let us finally express by elementary transcendental functions if the dimension n is even. That this is impossible in the case of odd dimensions is plainly shown by the example in (Equation8(8) (8) ).
Corollary 2.4
Let be the function given by Equation (Equation5(5) (5) ), which represents according to Theorem 2.3 for Then the recursion formula (9) (9) holds. Furthermore, for even we have (10) (10) In particular, with the notation and we obtain (if and (11) (11)
More generally, for even and we have (12) (12)
Proof.
(a) The integral representation in [Citation13, Equation 8.712] for the associated Legendre function implies that From this and the representation of in (Equation5(5) (5) ), we infer that holds for and this is the recursion relation (Equation9(9) (9) ).
According to [Citation13, Equation 8.777.2], we have and hence Together with (Equation9(9) (9) ), this implies formula (Equation10(10) (10) ) for The equations in (Equation11(11) (11) ) follow from (Equation10(10) (10) ) taking into account that and
(b) Obviously, the general formula for n=2m in (Equation12(12) (12) ) could be proven by induction over m by employing the recursion formula (Equation9(9) (9) ). We prefer to give a direct proof based on one of Kummer's transformation formulas for the hypergeometric function.
Let us apply [Citation13, Equation 9.134.3] to formula (Equation6(6) (6) ). If we set and assume we obtain and and hence (13) (13) If n=2m is even, then the hypergeometric series in (Equation13(13) (13) ) terminates, and it readily yields the finite sum in Equation (Equation12(12) (12) ). This completes the proof.
Disclosure statement
No potential conflict of interest was reported by the authors.
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