European Journal of Mathematics

, Volume 1, Issue 3, pp 582–640 | Cite as

Decomposition theorem and Riesz basis for axisymmetric potentials in the right half-plane

Research Article

Abstract

We consider the Weinstein equation, also known as the equation governing generalized axisymmetric potentials (GASP), with complex coefficients \(L_mu={\Delta } u+(m/x)\partial _x u =0\), \(m\in {\mathbb {C}}\). We generalize results known for \(m\in {\mathbb {R}}\) to the case \(m\in {\mathbb {C}}\). In particular, explicit expressions of fundamental solutions for Weinstein operators and their estimates near singularities are presented, a Green’s formula for GASP in the right half-plane \({\mathbb {H}}^+\) for \(\mathrm{Re}\,m<1\) is established. We prove a new decomposition theorem for the GASP in annular domains for \(m\in {\mathbb {C}}\), which is in fact a generalization of the Bôcher’s decomposition theorem. In particular, using bipolar coordinates, it is proved for annuli that a family of solutions for the GASP equation in terms of associated Legendre functions of first and second kind is complete. This family is shown to be a Riesz basis in some non-concentric circular annuli.

Keywords

Axially symmetric solutions Fundamental solutions  Riesz basis Elliptic functions and integrals 

Mathematics Subject Classification

35B07 33E05 35J15 35E05 42C15 

1 Introduction

In this article, we study the Weinstein differential operatorwell-defined on the right half-plane \(\mathbb {H}^+\!=\{(x,y)\in {{\mathbb {R}}}^2: x>0\}=\{z\in {\mathbb {C}}: \mathrm{Re}\,z>0\}\) with the convention Open image in new window . This class of operators is also called operators governing axisymmetric potentials. They have been studied quite extensively in cases \(m\in \mathbb {N}\) and \(m\in {{\mathbb {R}}}\) in [10, 11, 12, 13, 16, 17, 18, 22, 23, 24, 29, 30, 31, 32, 33, 36, 37, 38, 41, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68]. We will focus exclusively on the case \(m\in {\mathbb {C}}\), recalling in the course of the paper some results for integer values of m. The Weinstein equation reads
$$\begin{aligned} L_mu=0. \end{aligned}$$
(1)
The main motivation for which we consider the case \(m\in {\mathbb {C}}\) is that if we complexify the coordinates by writing \(z=x+iy\), (1) takes the formwhich is a particular case of the equationconsidered with \(\alpha ,\beta \in {\mathbb {C}}\) in [49, Equation (5.7), p. 20].
Equation (1) also appears in physics in the study of the behavior of plasma in a tokamak. The role of tokamak, which has a toroidal geometry, is to control location of the plasma in its chamber by applying magnetic fields on its boundary. It is possible to assume that plasma is axially symmetric what reduces this problem to a plane section in \(\mathbb {H}^+\), where the magnetic flux in the vacuum between the plasma and the circular boundary of the chamber satisfies a second-order elliptic nonlinear partial differential equation, the so-called Grad–Shafranov equation, which reduces to the homogeneous equation (1) with \(m=-1\) (Fig. 1).
Fig. 1

Section of a Tokamak

Note that in this instance, (1) takes place in an annular domain rather than in a simply connected domain, see [8, 9, 46]. This fact motivates our decomposition theorem, Theorem 5.9.

In the sequel, the sense in which the solutions are studied will be specified. We will also look at solutions to the equation in the sense of distributions
$$\begin{aligned} L_mu=\delta _{(x,y)}, \end{aligned}$$
where \(\delta _{(x,y)}\) denotes the Dirac mass at \((x,y)\in \mathbb {H}^+\).

The mentioned above class of operators was first considered by Weinstein in [54], where he studied the case \(m\in \mathbb {N}^*\). He also established a relation between the axisymmetric potentials for \(m\in \mathbb {N}^*\) and harmonic functions on \({{\mathbb {R}}}^{m+2}\), see Proposition 2.4.

In [20, 58, 59], Weinstein and Diaz–Weinstein established the correspondence principle between axisymmetric potentials \(L_m\) and \(L_{2-m}\), see Proposition 2.3. They deduced an expression of a fundamental solution (where the singular point is taken on the y-axis) for \(m\in {{\mathbb {R}}}\) and established a link between the Weinstein equation and Tricomi equations and their fundamental solutions.

Let us return to the book [49]. Studying elliptic equations with analytic coefficients, Vekua provided means to express their fundamental solutions by using the Riemann functions, introduced earlier (see e.g. [28]) in the real hyperbolic context, he also investigated generalized elliptic equations with complex operators \({\partial }_z\) and \({\partial }_{\overline{z}}\). In heuristic words, in the same way as a harmonic function is the real part of a holomorphic function, or the sum of a holomorphic and an anti-holomorphic function, Vekua established the fact that solutions to elliptic equations, and therefore GASP, can be written as a sum of two functionals, one applied to an arbitrary holomorphic function and the other applied to an arbitrary anti-holomorphic function. These functionals can be written explicitly in terms of the Riemann function, by using the hypergeometric functions [49] or fractional derivations [16]. In [34], Henrici gave a very interesting introduction to the work of Vekua.

More recently, basing on the work of Vekua, Savina [44] gave a series representation of fundamental solutions for the operator \(\widehat{L}u={\Delta } u+a{\partial }_x u+b{\partial }_y u+cu\) and studied the convergence of this series. She also provided an application of her results to the Helmholtz equation.

In [31], Gilbert studied the non-homogeneous Weinstein equation, i.e. the case \(m\ge 0\), and gave an integral formula for this class of equations. In particular, an explicit solution was given when the second member depends only on one variable.

Some Dirichlet problems are considered in [40, 41] in a special geometry, the so-called “geometry with separable variable”.

Even if some results presented in this paper are known for real values of m, we make a totally self-contained presentation involving elementary technics not necesseraly used in the papers mentioned above. For instance, usual arguments involving estimates of hypergeometric integrals are replaced by arguments using the Lebesgue dominated convergence theorem. Our main result is a decomposition theorem for axisymmetric potentials which is new also for real values of m. We obtain a Liouville-type result for the solutions of Weinstein equation on \(\mathbb {H}^+\), with an interesting observation that there is a loss of strict ellipticity of the Weinstein operator on the boundary of \(\mathbb {H}^+\). An application of the decomposition theorem is given by showing that an explicit family of axisymmetric potentials constructed by introduction of bipolar coordinates is a Riesz basis in some annuli.

The plan of the paper is the following. In Sect. 2, we recall preliminary information about fundamental solutions for linear partial differential operators with non-constant coefficients. Proposition 2.1 provides with a connection between fundamental solutions for \(L_m\) and fundamental solutions for \(L_m^\star \), where \(L_m^\star \) denotes the formal adjoint of \(L_m\). The Weinstein principle [59], valid for m real and complex, establishes a connection between \(L_m\) and \(L_{2-m}\). We state it without proof as Proposition 2.3. Proposition 2.4, valid only for \(m\in \mathbb {N}\), is fundamental in the sense that one can compute a fundamental solution for \(L_m\) just knowing the usual fundamental solution for the Laplacian in \({{\mathbb {R}}}^{m+2}\). The corresponding computations are done for \(m\in \mathbb {N}\) and for \(m\in \mathbb {Z}\) in Sect. 3.

The extension of formulas for fundamental solutions to the case \(m\in {\mathbb {C}}\) is the core of Sect. 4. First we describe in Proposition 4.2 in an elementary way the behavior of fundamental solutions near their singularities. Next we use the corresponding estimates to establish the main result of the section, Theorem 4.4.

Section 5 is dedicated to the decomposition theorem. First we modify the fundamental solutions built earlier in order to get fundamental solutions which vanish on the boundary of \(\mathbb {H}^+\). Next, in Proposition 5.3, we show that if u is a solution for \(L_mu=0\) which vanishes on the boundary of \(\mathbb {H}^+\), then \(u\equiv 0\) on \(\mathbb {H}^+\). Let us emphasize here that, although this statement looks obvious, this is not the case due to the loss of ellipticity of \(L_m\) on the boundary of \(\mathbb {H}^+\). Let us mention that Proposition 5.3 is a consequence of the maximum principle for pseudo-analytic functions given in a recent paper by Chalendar–Partington [14] for more general function \(\sigma \) than \(x^m\), but in [14] there is an additional assumption on \(\sigma \), which in our case corresponds to the assumption \(|m|\ge 1\). The proof of Proposition 5.3 is quite long, but not difficult, it follows careful estimates of fundamental solutions in some parts of \(\mathbb {H}^+\). Finally the decomposition theorem, Theorem 5.9, is proved. Its proof is similar to the proof of the Bôcher’s decomposition theorem presented in [4]. We end Sect. 5 with a Poisson formula for axisymmetric potentials in \(\mathbb {H}^+\), Proposition 5.10.

In Sect. 6, we consider the case where the annular domain is a kind of annulus. We introduce very classical (in physics) bipolar coordinates, cf. [39], in which the GASP equation has a form presented in Theorem 6.2. Next, applying the method of separation of variables we obtain a basis of solutions in disks and complements of disks in \(\mathbb {H}^+\), see Theorem 6.3. In Sect. 7 it is shown that this basis forms a Riesz basis.

2 Notations and preliminaries

Throughout \(\mathbb {H}^+\!=\{(x,y)\in {{\mathbb {R}}}^2: x>0\}\) stands for the right half-plane, all scalar functions are assumed to be complex valued. If \({\Omega }\) is an open set in \({{\mathbb {R}}}^n\), \(n\in \mathbb {N}^*\), let \({\mathscr {D}}({\Omega })\) designate the space of \(C^\infty \)-functions compactly supported on \({\Omega }\), where \(\mathrm{supp }\,f=\overline{\{x\in {\Omega }: f(x)\ne 0\}}\). If K is a compact set in \({\Omega }\), let \({\mathscr {D}}_K({\Omega })\) be the set of functions \(\varphi \in {\mathscr {D}}({\Omega })\) such that \(\mathrm{supp}\,\varphi \subset K\).

The partial derivatives of a differentiable function u on an open set \({\Omega }\subset {{\mathbb {R}}}^n\) will be denoted by \({\partial }u/ {\partial }x_i\), \({\partial }_{x_i}u\) or \(u_{x_i}\), \(i\in \{1,\dots ,n\}\). If \(\alpha =(\alpha _1,\ldots ,\alpha _n)\in \mathbb {N}^n\) is a multi-index, we denote
$$\begin{aligned} {\partial }^\alpha ={\partial }_{x_1}^{\alpha _1}\ldots {\partial }_{x_n}^{\alpha _n} ={{\partial }^{|\alpha |}\over {\partial }x_1^{\alpha _1}\ldots {\partial }x_n^{\alpha _n}} \end{aligned}$$
with \(|\alpha |=\alpha _1+\cdots +\alpha _n\).

It is assumed that the reader is familiar with the terminology of distributions and we refer to [35].

Let L be a linear differential operator on \({\Omega }\),
$$\begin{aligned} L=\!\sum _{|\alpha |\le N}\!a_{\alpha }{\partial }^\alpha ,\qquad N\in \mathbb {N}, \end{aligned}$$
where the summation runs over the multi-indices \(\alpha \) of length \(|\alpha |\le N\), \(a_\alpha \) are \(C^\infty ({\Omega })\)-functions. If T is a distribution, then \(LT=\sum _{|\alpha |\le N}a_\alpha {\partial }^\alpha T\). Denote by \(L^\star \) the adjoint operator of L in the sense of distributions, namely,
$$\begin{aligned} L^\star T=\!\sum _{|\alpha |\le N}\!(-1)^{|\alpha |}{\partial }^\alpha (a_\alpha T). \end{aligned}$$
One can easily check, if \(f,g\in {\mathscr {D}}({\Omega })\), we have
$$\begin{aligned} \langle Lf,g\rangle =\langle f,L^\star g\rangle . \end{aligned}$$
Let \(a\in {\Omega }\) and L be a differential operator on \({\Omega }\). A fundamental solution for L on \({\Omega }\) at \(a\in {\Omega }\) is a distribution \(T_a\) such that
$$\begin{aligned} LT_a=\delta _a, \end{aligned}$$
where the equality is understood in the sense of distributions on \({\Omega }\). This equality can be rewritten as
$$\begin{aligned} \varphi (a)=\langle LT_a,\varphi \rangle =\langle T_a,L^\star \varphi \rangle , \qquad \varphi \in {\mathscr {D}}({\Omega }). \end{aligned}$$
In particular, if \(a\in {\Omega }\) and \(T_a\) is a fundamental solution to \(L^\star \) at a on \({\Omega }\) and if \(g\in {\mathscr {D}}({\Omega })\) is such that \(g=L(\varphi )\) with \(\varphi \in {\mathscr {D}}({\Omega })\), then
$$\begin{aligned} \varphi (a)=\langle T_a,g\rangle , \qquad a\in {\Omega }. \end{aligned}$$
Indeed, we have
$$\begin{aligned} \varphi (a)=\langle \delta _a,\varphi \rangle =\langle L^\star T_a,\varphi \rangle =\langle T_a,L\varphi \rangle =\langle T_a,g\rangle , \qquad a\in {\Omega }. \end{aligned}$$
These fundamental solutions are therefore a good tool for solving equations \(L\varphi =g\) on \({\mathscr {D}}({\Omega })\) if \(g\in {\mathscr {D}}({\Omega })\).
If \(m\in \mathbb {N}^*\), the Laplacian in \({{\mathbb {R}}}^m\) will be denoted by \({\Delta }_m\), or \({\Delta }\) when \(m=2\). For \(m\in {\mathbb {C}}\), \(L_m\) denotes the Weinstein operator:If \(f(x,y)=(f_1(x,y),f_2(x,y))\) is a \({C}^1\)-vector \({\mathbb {C}}^2\)-valued function on an open set in \({{\mathbb {R}}}^2\),Similarly, if \(f:{{\mathbb {R}}}^2\rightarrow {\mathbb {C}}\) is a \({C^1}\)-scalar \({\mathbb {C}}^2\)-valued function on an open set in \({{\mathbb {R}}}^2\),With these notations, the operator \(L_m\) can be written as follows:By the Schwarz rule, if u is a function defined on a connected open set in \(\mathbb {H}^+\) such that Open image in new window , where \(\sigma :\mathbb {H}^+\rightarrow {\mathbb {C}}^*\) is a \({C}^1\)-function, then there is a function v which satisfies the well-known generalized Cauchy–Riemann system of equationsand v satisfies the conjugate equation Open image in new window , see for example [7]. This observation justifies the fact that we call \(L_{-m}\), \(m\in {\mathbb {C}}\), the conjugate operator of \(L_m\).
The adjoint operator of \(L_m\) iswhere \(u\in {C}^2(\mathbb {H}^+)\), \((x,y)\in \mathbb {H}^+\). This definition is given on \(\mathbb {H}^+\) but it is easily transposed to the case of an open set \({\Omega }\) of \(\mathbb {H}^+\).

In the case where the functions involved do not depend only on x and y, we will write \(L_{m,x, y}\) instead of \(L_{m}\), which means that the partial derivatives are related to the variables x and y, and all other variables are considered to be fixed.

If \(u\in {\mathscr {D}}(\mathbb {H}^+)\), we define \(S_m u,Du\in {\mathscr {D}}(\mathbb {H}^+)\) asThese operators satisfy the following property.

Proposition 2.1

The operator \(S_m\) conjugates \(L_m^\star \) and \(L_m\), D conjugates \(L_{-m}^\star \) and \(L_m\), which means that
$$\begin{aligned} S_mL_m^\star =L_mS_m,\qquad L_{-m}^\star D=DL_m. \end{aligned}$$

Remark 2.2

  1. 1.

    Let \(m\in {\mathbb {C}}\), \(S_m\) and \(L_m S_m\) are self-adjoint operators, i.e. \(S_m=S_m^\star \) and \(L_m S_m=(L_m S_m)^\star \).

     
  2. 2.
    Let \(\sigma :{\Omega }\rightarrow {\mathbb {C}}\) be a \({C}^1\)-function which does not vanish, consider the operator defined on \({C}^2({\Omega })\) as follows: where \({\Omega }\) is an open set in \({{\mathbb {R}}}^2\). Then Indeed, if \(u,v\in {\mathscr {D}}({\Omega })\), then, by using the derivation in the sense of distributions, we have
    $$\begin{aligned} \langle P_{\sigma }u,v\rangle&= \int _{{\Omega }}{1\over \sigma (x,y)}\,\mathrm{div}\bigl (\sigma (x,y)\nabla u(x,y)\bigr )v(x,y)\,dxdy\\&= -\int _{{\Omega }}\sigma \nabla u{\cdot }\nabla \biggl ({v\over \sigma }\biggr )\,dxdy= \int _{{\Omega }}u\mathrm{div}\biggl (\sigma \nabla \biggl ({v\over \sigma }\biggr )\biggr )dxdy\\&= \langle u,P_{\sigma }^\star v\rangle . \end{aligned}$$
    Define \(S_\sigma \) as
    $$\begin{aligned} (S_\sigma u)(x,y)={1\over \sigma (x,y)}\,u(x,y), \qquad u\in C^2({\Omega }). \end{aligned}$$
    Then, one can easily check that \(S_\sigma P_\sigma ^\star =P_\sigma S_\sigma \), hence \(S_\sigma \) conjugates \(P_\sigma \) and \(P_\sigma ^\star \). The operators \(P_\sigma \) and \(S_\sigma \) are a generalization of \(L_m\) and \(S_m\) with the conjugation relation preserved.
     
If m is a positive integer, introduce an operator \(T_m:u\mapsto v\) defined as follows: \(T_m\) maps a function u defined on an open set \({\Omega }\) of \(\mathbb {H}^+\) to the function
$$\begin{aligned} v(x_1,\ldots ,x_{m+2})=u\Bigl (\sqrt{x_1^2 +\cdots +x_{m+1}^2},x_{m+2}\Bigr ). \end{aligned}$$
The following two propositions can be found in the Weinstein paper [59] in the case \(m\in {{\mathbb {R}}}\). They can be checked by a direct computation for m real and complex, so we omit the proofs.

Proposition 2.3

(Weinstein principle [59]) Let \({\Omega }\) be a relatively compact open set in \(\mathbb {H}^+\), if \(u:{\Omega }\rightarrow {\mathbb {C}}\) is \({C}^2\), then for all \(m\in {\mathbb {C}}\),
$$\begin{aligned} L_mu=x^{1-m}L_{2-m}\bigl [x^{m-1}u\bigr ]. \end{aligned}$$

Proposition 2.4

([54]) Let \({\Omega }\) be a relatively compact open set in \(\mathbb {H}^+\). If \(u\in {C}^2({\Omega })\) and \(m\in \mathbb {N}\), then \({\Delta }_{m+2}(T_mu)=T_m(L_mu)\).

These properties will allow us to calculate fundamental solutions for \(L_m\) and \(L_m^\star \) for \(m\in \mathbb {N}\), and, thereafter, for \(m\in \mathbb {Z}\). Finally, estimates of formulas for \(L_m,L_m^\star \), \(m\in \mathbb {Z}\), will show that these formulas actually provide fundamental solutions for \(L_m\) and \(L_m^\star \) in the case \(m\in {\mathbb {C}}\).

3 Integral expressions of fundamental solutions for integer values of m

Let us recall the definition of the Dirac mass in a point \((x,y)\in {{\mathbb {R}}}^2\):
$$\begin{aligned} \langle \delta _{(x,y)},\varphi \rangle =\varphi (x,y),\qquad \varphi \in {\mathscr {D}}({{\mathbb {R}}}^2). \end{aligned}$$

Proposition 3.1

(partially in [20, 53, 54]) Let \(m\in \mathbb {N}^\star \). For \((x,y)\in \mathbb {H}^+\) and \((\xi ,\eta )\in \mathbb {H}^+\),is a fundamental solution on \(\mathbb {H}^+\) for the operator \(L^\star _{m,\xi ,\eta }\) at the fixed point \((x,y)\in \mathbb {H}^+\), which meansin the sense of distributions. Moreover, if \((\xi ,\eta )\in \mathbb {H}^+\) is fixed, thenin the sense of distributions, which means that \(E_m\) is a fundamental solution on \(\mathbb {H}^+\) for the operator \(L_{m,x,y}\) at the fixed point \((\xi ,\eta )\in \mathbb {H}^+\).

Proof

Let \(m\in \mathbb {N}^*\). Recall thatis a fundamental solution for the Laplacian on \({{\mathbb {R}}}^{m+2}\), i.e. in the sense of distributions \({\Delta }_{m+2} E=\delta _0\), where \(\omega _{m+2}\) is the area of the unit sphere in \({{\mathbb {R}}}^{m+2}\). Thus, for all \(v\in {\mathscr {D}}({{\mathbb {R}}}^{m+2})\),where \(\tau =(\tau _1,\dots ,\tau _{m+2})\).
Applying this relation to the function \(v=T_mu\), where \(u\in {\mathscr {D}}(\mathbb {H}^+)\), and, due to Proposition 2.4, for all \((x,y)\in \mathbb {H}^+\) we haveWe will simplify this integral expression. For this, we will consider the following hyper-spherical coordinates:
where \(\xi =\sqrt{\xi _1^2+\cdots +\xi _{m+1}^2}\ge 0\), \(\theta _m\in (-\pi ,\pi )\) and \(\theta _1,\ldots ,\theta _{m-1}\in (0,\pi )\). The absolute value of the determinant of the Jacobian matrix defined by this system of coordinates isThen, for all \((x,y)\in \mathbb {H}^+\),with
Let
then \(E_m\) can be written as
Also, due to (2) we haveMoreover, since for all \((x,y),(\xi ,\eta )\in \mathbb {H}^+\),
$$\begin{aligned} E_m(x,y,\xi ,\eta )=\biggl ({ x\over \xi }\biggr )^{\!-m}\! E_m(\xi ,\eta ,x,y), \end{aligned}$$
and by Proposition 2.1, \(S_m\) conjugates \(L_m^\star \) and \(L_m\), we have
in the sense of distributions. Henceand the proof is complete. \(\square \)

This proposition and the Weinstein principle imply the following result.

Proposition 3.2

(partially in [20, 53, 54]) Let Open image in new window . For \((x,y),(\xi ,\eta )\in \mathbb {H}^+\),
is a fundamental solution on \(\mathbb {H}^+\) for the operator \(L^\star _{m,\xi ,\eta }\) at the fixed point \((x,y)\in \mathbb {H}^+\) and it is also a fundamental solution on \(\mathbb {H}^+\) for the operator \(L_{m,x,y}\) at the fixed point \((\xi ,\eta )\in \mathbb {H}^+\).

Proof

For all \(m\in \mathbb {N}^*\), \(u\in {\mathscr {D}}(\mathbb {H}^+)\) and \((x,y)\in \mathbb {H}^+\) we haveand by the Weinstein principle, Proposition 2.3, we haveDenoting \(v(x,y)=x^{m-1}u(x,y)\), we obtainthen, for all Open image in new window , \(v\in {\mathscr {D}}(\mathbb {H}^+)\) and \((x,y)\in \mathbb {H}^+\), putting \(m=2-m'\), we haveThe proof of the second statement is similar. \(\square \)

4 Fundamental solutions for the Weinstein equation with complex coefficients

In this section, we will generalize the result obtained in the previous section for \(m\in \mathbb {Z}\) to the case \(m\in {\mathbb {C}}\).

Let \(m\in {\mathbb {C}}\). If \(\mathrm{Re}\,m\ge 1\) putand if \(\mathrm{Re}\,m<1\) puthere, if \(\alpha >0\) is a real number and \(\mu \) is a complex number, Open image in new window . Both values are well defined as the integrals on the right-hand side converge in the Lebesgue sense.

Proposition 4.1

For \(m\in {\mathbb {C}}\) and \((\xi ,\eta )\in \mathbb {H}^+\) fixed, we haveand for \((x,y)\in \mathbb {H}^+\) fixed, we have

Proof

For convenience, denoteTo prove the first equality of the proposition, it suffices to show thatLet us compute the partial derivatives of the function \(f_m\):
We then have
Note that
hence
Noting thatwe haveIntegrating by parts, we have
and the result is deduced in the case \(\mathrm{Re} \,m\ge 1\). The same argument is valid for \(\mathrm{Re}\,m<1\). The second equality of the proposition can be deduced immediately from the fact that \(S_m\) conjugates \(L_m^\star \) and \(L_m\), see Proposition 2.1. \(\square \)
In the sequel, we will denoteThe following proposition describes the behavior of \(E_m\) defined by (3) and (4) near its singularity. In particular, we show that the behavior of \(E_m\) is close to the behavior of fundamental solutions for the Laplacian. This fact is well known for elliptic operators. But we emphasize here that in our proof the estimates of elliptic integrals are elementary (obtained using the dominated convergence theorem) and we do not use estimates arising from classical estimates of hypergeometric functions.

Proposition 4.2

Let \(m\in {\mathbb {C}}\). For \((x,y)\in \mathbb {H}^+\) fixed,

Proof

We start with \(\mathrm{Re}\,m\ge 1\). In this case, we have
Note that when \(d\rightarrow 0\), k tends to \(+\infty \).

Proof

Putting \(u=\sin \theta /2\), we have
Note that
where, due to monotone convergence, the right-hand side tends toas \(k\rightarrow +\infty \). The change of variable \(u={\text{ sh }}\,t/\sqrt{k}\) gives usSince \(\mathrm{th}^{m-1}t\) tends to 1 as \(t\rightarrow +\infty \), we deduce that as \(k\rightarrow +\infty \),The proof is complete. \(\blacksquare \)
Due to Claim 4.3, we haveas \(d\rightarrow 0+\). The case \(\mathrm{Re }\,m<1\) is analogous. \(\square \)

Now, we can prove the main result of this section which shows that \(E_m\) are fundamental solutions not only for \(m\in \mathbb {N}\) but for all \(m\in {\mathbb {C}}\).

Theorem 4.4

Let \(m\in {\mathbb {C}}\). For \((x,y),(\xi ,\eta )\in \mathbb {H}^+\), \(E_m\) defined by (3) and (4) is a fundamental solution on \(\mathbb {H}^+\) for the operator \(L^\star _{m,\xi ,\eta }\) at the fixed point \((x,y)\in \mathbb {H}^+\), which means that on \(\mathbb {H}^+\) in the sense of distributions. Moreover, if \((\xi ,\eta )\in \mathbb {H}^+\) is fixed, then on \(\mathbb {H}^+\) in the sense of distributions, which means that \(E_m\) is a fundamental solution on \(\mathbb {H}^+\) for the operator \(L_{m,x,y}\) at the fixed point \((\xi ,\eta )\in \mathbb {H}^+\).

Proof

Let \(m\in {\mathbb {C}}\) and \(u\in \mathscr { D}(\mathbb {H}^+)\). Let \((x,y)\in \mathbb {H}^+\) and \({\varepsilon }>0\) be such that \(D((x,y),{\varepsilon })\subset \mathbb {H}^+\), where \(D((x,y),{\varepsilon })\) is a disk with center (xy) and radius \({\varepsilon }\). Put
We use the fact that \(L_m^\star (E_m)=0\) on Open image in new window . An elementary computation gives
Let us recall the Green formula: If \({\Omega }\) is an open set in \({{\mathbb {R}}}^2\) with a piecewise \({C}^1\)-differentiable boundary, thenwhere \(\mathbf {n}\) is the outer unit normal vector to \({\partial }{\Omega }\) and ds the arc length element on \({\partial }{\Omega }\) (positively oriented), \(X=(X_1,X_2):\overline{\Omega }\rightarrow {\mathbb {C}}^2\) is a \({C}^1\)-vector field.
Applying this formula to the open set Open image in new window , where U is a regular open set in \(\mathbb {H}^+\) containing the support of u, we have
Proposition 4.2 impliesas \({\varepsilon }\rightarrow 0+\) because \(\lim _{{\varepsilon }\rightarrow 0}{\varepsilon }\ln {\varepsilon }=0\). Then, if we want to prove that \(\lim _{{\varepsilon }\rightarrow 0} \mathrm{I}_{\varepsilon }\) exists, we have to prove the existence ofand this limit will be equal to the limit of \(\mathrm{I}_{\varepsilon }\).
Now, assume that \(\mathrm{Re}\,m\ge 1\). Denote by \(\mathrm{J}_{\varepsilon }\) the integral in the previous expression. A computation gives
where \(k=4x\xi /{\varepsilon }^2\).

Proof

We put \(u=\sin \theta /2\), then
Note that
as \(k\rightarrow +\infty \). The change of variable \(u={\text{ sh }}\, t/\sqrt{k}\) gives usSince \(\mathrm{th}^{m+1}t\) tends to 1 as \(t\rightarrow +\infty \), it follows that as \(k\rightarrow +\infty \)

Proof

Putting as previously \(u=\sin \theta /2\), we have
Note that
Let us estimate the right-hand side of this equality:as \(k\rightarrow +\infty \). As seen in the proof of Claim 4.5, we haveas \(k\rightarrow +\infty \). Due to (\(\star \)) and (\(\star \star \)), we haveas \(k\rightarrow +\infty \). The change of variable \(u={\text{ sh }}\,t/\sqrt{k}\) givesIt follows that as \(k\rightarrow +\infty \),Thusandas \(k\rightarrow +\infty \). This completes the proof. \(\blacksquare \)
Let us return to the proof of Theorem 4.4. Claim 4.3 implies
which tends to 0 as \({\varepsilon }\rightarrow 0+\).
Claim 4.5 implies
which tends to 0 as \({\varepsilon }\rightarrow 0+\).
Finally, Claim 4.6 implies
which tends to u(xy) as \({\varepsilon }\rightarrow 0+\).
So we have proved that for all \(m\in {\mathbb {C}}\) such that \(\mathrm{Re}\,m>0\),
therefore \(E_m\) indeed is a fundamental solution for the operator \(L_m^\star \) for all \(m\in {\mathbb {C}}\) with \(\mathrm{Re}\,m>0\). The case \(m\in {\mathbb {C}}\) with \(\mathrm{Re}\,m\le 1\) is similar.

Due to Proposition 2.1, we also have dual assertions for fundamental solutions for the operator \(L_m\). \(\square \)

The following proposition is roughly a consequence of the previous theorem. Of course, it is a classical statement, but we would like to present its short proof.

Proposition 4.7

Let \(m\in {\mathbb {C}}\) and let \({\Omega }\) be a relatively compact open set in \(\mathbb {H}^+\) whose boundary is piecewise \(C^1\)-differentiable. Then, for \((x,y)\in {\Omega }\) and \(u\in {C}^2(\overline{\Omega })\), we have
where \(u=u(\xi ,\eta )\), \(E_m=E_m(x,y,\xi ,\eta )\), \(\mathbf {n}\) is the outer unit normal vector to \({\partial }{\Omega }\) and ds is the arc length element on \({\partial }{\Omega }\) (positively oriented).

Proof

Indeed, if \(u\in {C}^2(\overline{\Omega })\), for \((x,y)\in {\Omega }\) and \({\varepsilon }>0\) such that \(\overline{D((x,y),{\varepsilon })}\subset {\Omega }\), we haveBy the Green formula, the latter integral is equal to
and, as we saw in the previous proof, it tends toas \({\varepsilon }\rightarrow 0\). Due to integrability of \(E_m\) near (xy) we haveand the proof is complete. \(\square \)

5 Liouville-type result and decomposition theorem for axisymmetric potentials

In the previous section, we have seen that fundamental solutions \(E_m\) in the complex case have different expressions depending on whether \(\mathrm{Re}\,m<1\) or \(\mathrm{Re}\,m\ge 1\). Hence the behavior of \(E_m\) will be different in each case.

We will modify fundamental solutions so that they vanish at the boundary of \(\mathbb {H}^+\), which means that they tend to zero on the y-axis and at infinity. Expression (4) satisfies this property: \(E_m(x,y,{\cdot },{\cdot })\) tends to 0 as \(x\rightarrow 0+\) and \(\Vert (x,y)\Vert \rightarrow +\infty \); whereas (3) does not. ConsiderIt is also a fundamental solution on \(\mathbb {H}^+\) and it satisfies the required property. Let us put
  • for \(\mathrm{Re}\,m<1\),
    $$\begin{aligned} F_m(x,y,\xi ,\eta )=E_m(x,y,\xi ,\eta ), \end{aligned}$$
  • for \(\mathrm{Re}\,m\ge 1\),
We will need the following definition of convergence on the boundary of \(\mathbb {H}^+\).

Definition 5.1

Let \(u:\mathbb {H}^+\!\rightarrow {{\mathbb {R}}}\) be a function defined on \(\mathbb {H}^+\). We write
$$\begin{aligned} \lim _{{\partial }\mathbb {H}^+}u=0 \end{aligned}$$
if and only if for all \({\varepsilon }>0\) there exists \(N\in \mathbb {N}\) such that for all \(n>N\) and all \((x,y)\in H^+\), \(x\le {1/n}\) or \(\Vert (x,y)\Vert \ge n\) implies \(|u(x,y)|\le {\varepsilon }\).

Proposition 5.2

Let \(u:\mathbb {H}^+\!\rightarrow {\mathbb {C}}\). We have \(\lim _{{\partial }\mathbb {H}^+}u=0\) if and only if
$$\begin{aligned} \lim _{\Vert (x,y)\Vert \rightarrow +\infty }\!\!u(x,y)=0\qquad \hbox {and}\qquad \lim _{(0,y)}u=0,\quad y\in {{\mathbb {R}}}. \end{aligned}$$

Proof

The direct implication is easy. Conversely, assume \(\lim _{\Vert (x,y)\Vert \rightarrow +\infty }u(x,y)=0\) and \(\lim _{(0,y)}u=0\), \(y\in {{\mathbb {R}}}\). Let \({\varepsilon }>0\), then there is \(A>0\) such that for all \((\xi ,\eta )\in \mathbb {H}^+\), \(\sqrt{\xi ^2+\eta ^2}\ge A\) implies \(|u(\xi ,\eta )|\le {\varepsilon }\). Similarly, for all \(y\in {{\mathbb {R}}}\), there is \(\alpha _y\in (0,1)\) such that for all \((\xi ,\eta )\in \mathbb {H}^+\), \(\sqrt{\xi ^2+(\eta -y)^2}<\alpha _y\) implies \(|u(\xi ,\eta )|\le {\varepsilon }\).

The interval \([-A,A]\) is compact. By the Lebesgue covering lemma, there is \(\alpha >0\) such that for all \(y'\in [-A,A]\), the ball \(B(y',\alpha )\) is included in one of the balls \(B(y,\alpha _{y})\) with \(y\in [-A,A]\). In particular, if \((\xi ,\eta )\in \mathbb {H}^+\) is such that \(0<\xi <\alpha \), then \(|u(\xi ,\eta )|\le {\varepsilon }\). This completes the proof. \(\square \)

The following proposition is a Liouville-type result for axisymmetric potentials in the right half-plane. As we mentioned in the introduction, this result is not trivial due to the loss of strict ellipticity of the Weinstein operator on the y-axis. Let us mention that in [5, Theorem 7.1] one can find an interesting result on the description of a class of non-strictly elliptic equations with unbounded coefficients.

Proposition 5.3

Let \(u\in C^2(\mathbb {H}^+)\) be such that \(L_mu=0\) and \(\lim _{{\partial }\mathbb {H}^+}u=0\). Then \(u\equiv 0\) on \(\mathbb {H}^+\).

Proof

For \((\xi ,\eta )\in \mathbb {H}^+\) and \(N\in \mathbb {N}^*\), definewhere \(\theta _1\) and \(\theta _2\) are smooth functions on \({{\mathbb {R}}}\), valued on [0, 1] and such that \(\theta _1(t)=1\) for \(t\ge 1\), \(\theta _1(t)= 0\) for \(t\le {1/2}\), \(\theta _2(t)=1\) for \(t\in [-{1/2},{1/2}]\), and \(\theta _2(t)=0\) for \(t\in {{\mathbb {R}}}{\setminus }(-1,1)\). Assume also that all derivatives of \(\theta _1\) and \(\theta _2\) vanish at \(\{-1,-{1/2},\) \({1/2},1\}\) (Fig. 2).
Fig. 2

The functions \(\theta _1\) and \(\theta _2\)

If \(u\in {C}^2(\mathbb {H}^+)\) satisfies \(L_mu=0\), then \(u\phi _N\in {C}^2(\mathbb {H}^+)\) and it is compactly supported on \(\mathbb {H}^+\). Throughout the following, we fix \((x,y)\in \mathbb {H}^+\). For N sufficiently large, due to Proposition 4.7 (true if \(E_m\) is replaced by \(F_m\)), we have(because the function \(L_m(u\phi _N)\) is identically zero in a neighborhood of the singularity of \(F_m\)), thuswhere \(D_1,\ldots , D_8\) are the following domains (which depend on N) (Fig. 3):
Fig. 3

Domains \(D_{i}\)

Since \(\lim _{{\partial }\mathbb {H}^+}\!u=0\),We will estimate integrals over sets \(D_1,\ldots ,D_8\) separately, see auxiliary lemmas below. Recall that, if \((u_N)_N\) and \((v_N)_N\) are complex sequences, \(u_N=\mathrm{O}(v_N)\) means that there exists a constant M such that, for every N sufficiently large, \(|u_N|\le M|v_N|\); \(u_N=\mathrm{o}(v_N)\) means that for every \({\varepsilon }>0\), for every N sufficiently large, \(|u_N|\le {\varepsilon }|v_N|\).

Lemma 5.4

On \(D_1\), we have
$$\begin{aligned} \sup {\biggl |{{\partial }\phi _N\over {\partial }\xi }\biggr |}=\mathrm{O}(N)\qquad \hbox {and} \qquad \sup {\biggl |{{\partial }\phi _N\over {\partial }\eta }\biggr |}=0. \end{aligned}$$
On \(D_2\cup D_4\), we have
$$\begin{aligned} \sup {\biggl |{{\partial }\phi _N\over {\partial }\xi }\biggr |}=0\qquad \hbox {and} \qquad \sup {\biggl |{{\partial }\phi _N\over {\partial }\eta }\biggr |}=\mathrm{O}\biggl ({1\over N}\biggr ). \end{aligned}$$
On \(D_3\), we have
$$\begin{aligned} \sup {\biggl |{{\partial }\phi _N\over {\partial }\xi }\biggr |}=\mathrm{O}\biggl ({1\over N}\biggr )\qquad \hbox {and}\qquad \sup {\biggl |{{\partial }\phi _N \over {\partial }\eta }\biggr |}=0. \end{aligned}$$
On \(D_5\cup D_8\), we have
$$\begin{aligned} \sup {\biggl |{{\partial }\phi _N\over {\partial }\xi }\biggr |}=\mathrm{O}(N) \qquad \hbox {and}\qquad \sup {\biggl |{{\partial }\phi _N\over {\partial }\eta }\biggr |} =\mathrm{O}\biggl ({1\over N}\biggr ). \end{aligned}$$
On \(D_6\cup D_7\), we have
$$\begin{aligned} \sup {\biggl |{{\partial }\phi _N\over {\partial }\xi }\biggr |}=\mathrm{O}\biggl ({1\over N}\biggr )\qquad \hbox {and}\qquad \sup {\biggl |{{\partial }\phi _N\over {\partial }\eta }\biggr |}=\mathrm{O}\biggl ({1\over N}\biggr ). \end{aligned}$$
On \(D_1\cup D_5\cup D_8\), we have
$$\begin{aligned} \sup |L_{-m}(\phi _N)|=\mathrm{O}(N^2). \end{aligned}$$
On \(D_2\cup D_3\cup D_4\cup D_6\cup D_7\), we have
$$\begin{aligned} \sup |L_{-m}(\phi _N)|=\mathrm{O}\biggl ({1\over N^2}\biggr ). \end{aligned}$$

Proof

For \((\xi ,\eta )\in D_1\), \(\phi _N(\xi ,\eta )=\theta _1(N\xi )\) and thuswhich give us
$$\begin{aligned} \sup _{D_1}{\biggl |{{\partial }\phi _N\over {\partial }\xi }\biggr |}=\mathrm{O}(N),\qquad \sup _{D_1}{\biggl |{{\partial }\phi _N\over {\partial }\eta }\biggr |}=0,\qquad \sup _{D_1}|L_{-m}(\phi _N)|=\mathrm{O}(N^2) \end{aligned}$$
since the derivatives of \(\theta _1\) are bounded and for \((\xi ,\eta )\in D_1\) one gets \(\xi \ge {1/(2N)}\).
For \((\xi ,\eta )\in D_2\), \(\phi _N(\xi ,\eta )=\theta _2({\eta / N})\) and thus
$$\begin{aligned}&\displaystyle {{\partial }\phi _N\over {\partial }\xi }(\xi ,\eta )=0,\qquad {{\partial }\phi _N\over {\partial }\eta }(\xi ,\eta )={1\over N}\,\theta _2'\biggl ({\eta \over N}\biggr ),&\\&\displaystyle L_{-m}\phi _N(\xi ,\eta )={1\over N^2}\,\theta _2''\biggl ({\eta \over N}\biggr ),&\end{aligned}$$
which give us
$$\begin{aligned} \sup _{D_2}{\biggl |{{\partial }\phi _N\over {\partial }\xi }\biggr |}=0,\qquad \sup _{D_2}{\biggl |{{\partial }\phi _N\over {\partial }\eta }\biggr |}=\mathrm{O}\biggl ({1\over N}\biggr ),\qquad \sup _{D_2}|L_{-m}(\phi _N)|=\mathrm{O}\biggl ({1\over N^2}\biggr ). \end{aligned}$$
The same works for \(D_4\).
For \((\xi ,\eta )\in D_3\), \(\phi _N(\xi ,\eta )=\theta _2({\xi / N})\) and thuswhich give us
$$\begin{aligned} \sup _{D_3}{\biggl |{{\partial }\phi _N\over {\partial }\xi }\biggr |}=\mathrm{O}\biggl ({1\over N}\biggr ),\qquad \sup _{D_3}{\biggl |{{\partial }\phi _N\over {\partial }\eta }\biggr |}=0, \qquad \sup _{D_3}|L_{-m}(\phi _N)|=\mathrm{O}\biggl ({1\over N^2}\biggr ). \end{aligned}$$
For \((\xi ,\eta )\in D_5, \phi _N(\xi ,\eta )=\theta _1(N\xi )\theta _2({\eta / N})\) and thuswhich give us
$$\begin{aligned} \sup _{D_5}{\biggl |{{\partial }\phi _N\over {\partial }\xi }\biggr |}=\mathrm{O}(N),\qquad \sup _{D_5}{\biggl |{{\partial }\phi _N\over {\partial }\eta }\biggr |}=\mathrm{O}\biggl ({1\over N}\biggr ),\qquad \sup _{D_5}|L_{-m}(\phi _N)|=\mathrm{O}(N^2). \end{aligned}$$
The same works for \(D_8\).
For \((\xi ,\eta )\in D_6\), \(\phi _N(\xi ,\eta )=\theta _2({\xi / N})\theta _2({\eta / N})\) and thuswhich give us
$$\begin{aligned} \sup _{D_6}{\biggl |{{\partial }\phi _N\over {\partial }\xi }\biggr |}=\mathrm{O}\biggl ({1\over N}\biggr ),\quad \; \sup _{D_6}{\biggl |{{\partial }\phi _N\over {\partial }\eta }\biggr |}=\mathrm{O}\biggl ({1\over N}\biggr ),\quad \;\sup _{D_6}|L_{-m}(\phi _N)|=\mathrm{O}\biggl ({1\over N^2}\biggr ). \end{aligned}$$
The same works for \(D_7\). \(\blacksquare \)
We now estimate the following quantities for \(i\in \{1,\,\ldots ,\,8\}\):
$$\begin{aligned} \int _{D_i}\!\! |F_m| \,d\xi d\eta , \qquad \int _{D_i} \!\!|{\partial }_\xi F_m| \,d\xi d\eta \qquad \int _{D_i}\!\! |{\partial }_\eta F_m| \,d\xi d\eta . \end{aligned}$$

Lemma 5.5

For \(\mathrm{Re}\, m<1\), we have:
  • for \(i=1\),
    $$\begin{aligned} \int _{D_i}\!\!|F_m|\,d\xi d\eta =\mathrm{O}\biggl ({1\over N^2}\biggr ),\qquad \int _{D_i}\biggl |{{\partial }F_m\over {\partial }\xi }\biggr |\,d\xi d\eta =\mathrm{O}\biggl ({1\over N}\biggr ); \end{aligned}$$
  • for \(i=2, 4\),
    $$\begin{aligned} \int _{D_i}\!\!|F_m|\,d\xi d\eta =\mathrm{O}({N^2}),\qquad \int _{D_i}\biggl |{{\partial }F_m\over {\partial }\eta }\biggr |\,d\xi d\eta =\mathrm{O}({N}); \end{aligned}$$
  • for \(i=3\),
    $$\begin{aligned} \int _{D_i}\!\!|F_m|\,d\xi d\eta =\mathrm{O}({N^2}),\qquad \int _{D_i}\biggl |{{\partial }F_m\over {\partial }\xi }\biggr |\,d\xi d\eta =\mathrm{O}({N}); \end{aligned}$$
  • for \(i=5,8\),
  • for \(i=6,7\),

Proof

By definition, for \(\mathrm{Re}\,m<1\),Therefore there is a constant \(C_1\) such that for all \((\xi ,\eta )\in \mathbb {H}^+\), we haveSimilarly, we haveand as before, as for all \(\theta \in [0,\pi ]\),there exists a constant \(C_2\) such that for all N large enough and all \((\xi ,\eta )\in \mathbb {H}^+\), we haveFinally, asthere exists a constant \(C_3\) such that for all N large enough and all \((\xi ,\eta )\in \mathbb {H}^+\), we haveUsing these inequalities, we estimate integrals on domains \(D_i\).
\(\underline{\hbox {On }D_1}\): Inequality (5) implies
Then, thanks to (6), we have
\(\underline{\hbox {On }D_2}\): Due to inequality (5), we have
Then, thanks to (7), we have
\(\underline{\hbox {On }D_3}\): Due to inequality (5), we have
Then, thanks to (6), we have

\(\underline{\hbox {On }D_4}\): This case is analogous to the case \(D_2\).

\(\underline{\hbox {On }D_5}\): Due to inequality (5), we have
$$\begin{aligned} \int _{D_5}|F_m|\,d\xi d\eta&=\mathrm{O}(1)\int _{{1/ 2N}}^{{1/ N}}d\xi \int _{{N/2}}^N {\xi \, d\eta \over \bigl [(x-\xi )^2+(\eta -y)^2\bigr ]^{1-{\mathrm{Re\,} m/ 2}}} \\&=\mathrm{O}\biggl (\frac{1}{N^2}\biggr )\int _{{N/2}}^N { d\eta \over (\eta -y)^{2-\mathrm{Re\,} m}} \\&=\mathrm{O}\biggl (\frac{1}{N^2}\biggr )\biggl [{1\over (N-y)^{1-\mathrm{Re\,}m}}-{1\over ({N/2}-y)^{1-\mathrm{Re\,}m}}\biggr ]=\mathrm{O}\biggl (\frac{1}{N^{3-\mathrm{Re\,}m}}\biggr ). \end{aligned}$$
Then, thanks to (6), we have
Estimate (7) gives
$$\begin{aligned} \int _{D_5}\biggl |{{\partial }F_m\over {\partial }\eta }\biggr |\,d\xi d\eta =\mathrm{O}(1)\int _{{1/ 2N}}^{{1/ N}}d\xi \int _{{N/2}}^{{N}} {\xi \,d\eta \over |\eta -y|^{3-\mathrm{Re\,}m}}=\mathrm{O}\biggl ({1\over N^2}\biggr ). \end{aligned}$$
\(\underline{\hbox {On }D_6}\): Due to (5), we have
$$\begin{aligned} \int _{D_6}\!\!|F_m|\,d\xi d\eta&=\mathrm{O}(1)\int _{{N/2}}^{{N}}d\xi \int _{{N/2}}^N {\xi \, d\eta \over \bigl [(x-\xi )^2+(\eta -y)^2\bigr ]^{1-{\mathrm{Re\,} m/ 2}}}\\&=\mathrm{O}(N^2)\int _{{N/2}}^N { d\eta \over (\eta -y)^{2-\mathrm{Re\,} m}}\\&=\mathrm{O}(N^2)\biggl [{1\over (N-y)^{1-\mathrm{Re\,}m}}-{1\over ({N/2}-y)^{1-\mathrm{Re\,}m}}\biggr ]=\mathrm{O}\bigl (N^{1+\mathrm{Re\,}m}\bigr ). \end{aligned}$$
Then, thanks to (6), we have
Estimate (7) gives
$$\begin{aligned} \int _{D_6}\biggl |{{\partial }F_m\over {\partial }\eta }\biggr |\,d\xi d\eta&=\mathrm{O}(1)\int _{{N/ 2}}^{{N}}d\xi \int _{{N/2}}^{{N}} {\xi \, d\eta \over |\eta -y|^{3-\mathrm{Re\,}m}}\\&=O(N^2)\int _{{N/2}}^{{N}} {d\eta \over |\eta -y|^{3-\mathrm{Re\,}m}} =\mathrm{O}\bigl (N^{\mathrm{Re\,}m}\bigr ). \end{aligned}$$
\(\underline{\hbox {On }D_7, D_8}\): These cases are analogous to the cases \(D_6\) and \(D_5\), respectively.\(\blacksquare \)

Lemma 5.6

Lemma 5.5 remains true for \(\mathrm{Re }\,m\ge 1\).

Proof

For \(\mathrm{Re}\,m\ge 1\), we have
Since for all \((\xi ,\eta )\in \mathbb {H}^+\), we havethenand there is a constant \(C'_1\) such that for all \((\xi ,\eta )\in \mathbb {H}^+\), we haveThis inequality does not suffice to estimate integrals over \(D_1\). We shall improve it as follows. Rewrite \(F_m\) aswhere
For \((x,y)\in \mathbb {H}^+\), \(\theta \in [0,\pi ]\) and \(\eta \in {{\mathbb {R}}}\) fixed, define a function \(g_m\) on \([-1/N,1/N]\), with \(1/N<x\), byThis function is well defined because
and the last term is greater than \((x-1/N)^2>0\).
We have
$$\begin{aligned} K_m(x,y,\xi ,\eta ,\theta )=g_m(\xi )-g_m(-\xi ) \end{aligned}$$
thus
$$\begin{aligned} \bigl |K_m(x,y,\xi ,\eta ,\theta )\bigr |\le 2\xi \!\!\sup _{[-\xi ,\xi ]}\!|g_m'|\le 2|m|\xi \, {|\xi -x|+2x\over \bigl [(x-\xi )^2+(y-\eta )^2\bigr ]^{1+\mathrm{Re\,}m/2}}, \end{aligned}$$
which implies that there exists a constant \(c'_1\) such that for all \((\xi ,\eta )\in D_1\),
$$\begin{aligned} |F_m|\le c'_1\,{\xi ^{\mathrm{Re\,}m+1}\over \bigl [(x-\xi )^2+(\eta -y)^2\bigr ]^{1+\mathrm{Re\,}m/2}}. \end{aligned}$$
(10)
Similarly, we haveand as before, for all \(\theta \in [0,\pi ]\),
and thanks to (8), for all \(\theta \in [0,\pi ]\),
These estimates, (11) and (9) show that there is a constant \(C'_2\) such that for large enough N and all \((\xi ,\eta )\in \mathbb {H}^+\), we haveWe can improve this inequality on \(D_1\), by using inequality (10) instead of (9), then there are two constants \(C''_2\) and \(C'''_2\) (which do not depend on N) such that for all \((\xi ,\eta )\in D_1\),Finally,
Similarly, there is a constant \(C'_3\) such that for all N large enough and all \((\xi ,\eta )\in \mathbb {H}^+\), we haveThanks to these inequalities, we can now estimate the corresponding integrals over domains \(D_i\).
\(\underline{\hbox {On }D_1}\): Due to (10), we have
Then thanks to (13),
\(\underline{\hbox {On }D_2}\): Due to (9), we have
because we integrate a bounded function (independent of N) on a domain with measure controlled by \(\mathrm{O}(N^2)\).
Then, inequality (14) implies
\(\underline{\hbox {On }D_3}\): Due to (9), we have
Then, thanks to (12), we have

\(\underline{\hbox {On }D_4}\): This case is analogous to the case \(D_2\).

\(\underline{\hbox {On }D_5}\): Due to (9), we have
Then, thanks to (12), we have
Applying inequality (14), we have
\(\underline{\hbox {On }D_6}\): Due to (9), we have
Then, thanks to (12), we obtain
Finally, inequality (14) implies

\(\underline{\hbox {On }D_7, D_8}\): These cases are analogous to the cases \(D_6\) and \(D_8\), respectively. \(\blacksquare \)

In the following table, we summarize results obtained on the previous lemmas:

i

Open image in new window

\(\int _{D_i}\!|F_m|\)

\((|{\partial }_{\xi }\phi _N|,|{\partial }_{\eta }\phi _N|)\)

\(\int _{D_i}\!|{\partial }_{\xi }F_m|\)

\(\int _{D_i}\!|{\partial }_{\eta }F_m|\)

1

\(\mathrm{O}(N^2)\)

\(\mathrm{O}(1/N^2)\)

\((\mathrm{O}(N),0) \)

\(\mathrm{O}({1/ N})\)

\(\times \)

2

\(\mathrm{O}(1/N^2)\)

\(\mathrm{O}(N^2)\)

\((0,\mathrm{O}({1/ N})) \)

\(\times \)

\(\mathrm{O}(N)\)

3

\(\mathrm{O}(1/N^2)\)

\(\mathrm{O}(N^2)\)

\((\mathrm{O}({1/ N}),0) \)

\(\mathrm{O}(N)\)

\(\times \)

4

\(\mathrm{O}(1/N^2)\)

\(\mathrm{O}(N^2)\)

\((0,\mathrm{O}({1/ N})) \)

\(\times \)

\(\mathrm{O}(N)\)

5

\(\mathrm{O}(N^2)\)

\(\mathrm{O}(1/N^2)\)

\((\mathrm{O}(N),\mathrm{O}({1/ N}))\)

\(\mathrm{O}({1/ N})\)

\(\mathrm{O}({1/ N^2})\)

6

\(\mathrm{O}(1/N^2)\)

\(\mathrm{O}(N^2)\)

\((\mathrm{O}({1/ N}),\mathrm{O}({1/ N}))\)

\(\mathrm{O}(N)\)

\(\mathrm{O}(N)\)

7

\(\mathrm{O}(1/N^2)\)

\(\mathrm{O}(N^2)\)

\((\mathrm{O}({1/ N}),\mathrm{O}({1/ N}))\)

\(\mathrm{O}(N)\)

\(\mathrm{O}(N)\)

8

\(\mathrm{O}(N^2)\)

\(\mathrm{O}(1/N^2)\)

\((\mathrm{O}(N),\mathrm{O}({1/ N}))\)

\(\mathrm{O}({1/ N})\)

\(\mathrm{O}({1/ N^2})\)

We can easily check that for each \(i\in \{1,\, \ldots ,\, 8\}\), the quantitiesare bounded. Therefore,
$$\begin{aligned} u(x,y)=\mathrm{o}(1)\qquad \text {as}\quad N\rightarrow +\infty . \end{aligned}$$
Thus \(u\equiv 0\) and this completes the proof of Proposition 5.3.\(\square \)

Proposition 5.7

Let \(u\in {\mathscr {D}}(\mathbb {H}^+)\) and let \((x,y)\in \mathbb {H}^+\), definethen \(\lim _{\Vert (x,y)\Vert \rightarrow +\infty } U=0\), and for all \(y\in {{\mathbb {R}}}\), \(\lim _{(0,y)} U=0\). Moreover, Open image in new window and for all \((x,y)\not \in { \mathrm supp }\,u\) we have \(L_{m,x,y}U(x,y)=0\).

Proof

Fix \((\xi ,\eta )\). For \({\mathrm{Re\,}}\,m<1\),hence \(F_m(x,y,\xi ,\eta )\rightarrow 0\) as \(\Vert (x,y)\Vert \rightarrow +\infty \). For \(\mathrm{Re}\,m\ge 1\),
hence \(F_m(x,y,\xi ,\eta )\rightarrow 0\) as \(\Vert (x,y)\Vert \rightarrow +\infty \). So the first statement of the proposition is shown.
For the second statement, for \(\mathrm{Re}\,m<1\), we haveas \((x,y)\rightarrow (0,y')\), which implies the desired result.
Now, assume that \(\mathrm{Re}\,m\ge 1\). Let \((\xi ,\eta )\) be fixed in the support of u, which is a compact set in \(\mathbb {H}^+\). In particular, there exist \(M>0\) and \(\alpha >0\) which do not depend on u such that \(\Vert (\xi ,\eta )\Vert \le M\) and \(\xi \ge 2\alpha \). Let y be in \({{\mathbb {R}}}\). DenoteBy the mean value inequality, for \(x>0\) near 0, we haveandthenIn particular,
$$\begin{aligned} \sup _{\mathop {(\xi ,\eta )\in \mathrm{supp\,} u}\limits _{ y\in {{\mathbb {R}}}}} \!\!\!|F_m(x,y)|=\mathrm{O}(x) \end{aligned}$$
as \(x\rightarrow 0+\). The second statement is proved.
The last statement can be deduced from the fact that if \((x,y)\not =(\xi ,\eta )\) are both in \(\mathbb {H}^+\), then

Remark 5.8

If \(U\in {\mathscr {D}}(\mathbb {H}^+)\), then \(L_{m,x,y}U=u\), but this identity is not necessarily true if \(U\not \in {\mathscr {D}}(\mathbb {H}^+)\). In particular, we cannot say that in Proposition 5.7 we have \(L_mU=u\).

Now, we will prove a decomposition theorem for axisymmetric potentials, it is interesting to compare it with the known result in [6, Section 4, Theorem 2]. The fundamental difference is that in this paper, the conductivity is not extended by reflection through the boundary \({\partial }{\Omega }\) to the whole domain.

Note that, due to our construction of fundamental solutions, the proof of this theorem is more or less the same as the proof of the decomposition theorem in [4, Chapter 9]. Note also that in our situation, the domain of our functions is \(\mathbb {H}^+\) not \({\mathbb {C}}\).

Theorem 5.9

Let \(m\in {\mathbb {C}}\). Let \({\Omega }\) be an open set in \(\mathbb {H}^+\) and let K be a compact set in \({\Omega }\). If Open image in new window satisfies \(L_mu=0\) in Open image in new window , then u has a unique decomposition as
$$\begin{aligned} u=v+w, \end{aligned}$$
where \(v\in {C}^2({\Omega })\) satisfies \(L_mv=0\) in \({\Omega }\) and Open image in new window satisfies \(L_mw=0\) in Open image in new window with \(\lim _{{\partial }\mathbb {H}^+} w=0\).

Proof

For \(E\subset {\mathbb {C}}\) and \(\rho >0\), define \(E_\rho =\{x\in {\mathbb {C}}: d(x,E)<\rho \}\), i.e. \(E_\rho \) is a neighborhood of E.

First, assume that \({\Omega }\) is a relatively compact open set in \(\mathbb {H}^+\). Choose \(\rho \) small enough so that \(K_\rho \) and \(({\partial }{\Omega })_\rho \) are disjoint. There is a function \(\varphi _\rho \in {\mathscr {D}}(\mathbb {H}^+)\) compactly supported on Open image in new window such that \(\varphi _\rho \equiv 1\) in a neighborhood of Open image in new window (Fig. 4).
Fig. 4

\(\varphi _{\rho }\equiv 1\) on the grey domain

For Open image in new window , denote
$$\begin{aligned} F_z(\zeta )=F_m(x,y,\xi ,\eta ),\qquad L_\zeta =L_{m,\xi ,\eta }\quad \qquad \mathrm{for}\quad \zeta =\xi +i\eta . \end{aligned}$$
Thanks to Proposition 4.7, we have
Then, the last result of Proposition 5.7 shows us that \(v_\rho \) satisfies \(L_mv_\rho =0\) on Open image in new window and \(w_\rho \) satisfies \(L_mw_\rho =0\) on Open image in new window . We also have \(\lim _{{\partial }\mathbb {H}^+}w_\rho =0\).

Now, assume that \(\sigma <\rho \). As previously, we obtain the decomposition \(u=v_\sigma + w_\sigma \) on Open image in new window . We claim that \(v_\rho =v_\sigma \) on Open image in new window and \(w_\rho =w_\sigma \) on Open image in new window . To see this, note that if Open image in new window , then \(v_\rho (z)+w_\rho (z)=v_\sigma (z)+w_\sigma (z)\).

The function \(w_\rho -w_\sigma \) satisfies (1) on \(\mathbb {H}^+{\setminus } K_\rho \), which is equal to \(v_\sigma -v_\rho \) on \({\Omega }{\setminus }(K_\rho \cup ({\partial }{\Omega })_\rho )\), therefore \(v_\sigma -v_\rho \) extends to a solution of (1) on \({\Omega }{\setminus }({\partial }{\Omega })_\rho \). Finally, \(w_\rho -w_\sigma \) extends to a solution of (1) on \(\mathbb {H}^+\), and \(\lim _{{\partial }\mathbb {H}^+}(w_\rho -w_\sigma )=0\). Due to Proposition 5.3, we have \(w_\rho =w_\sigma \), and hence \(v_\rho =v_\sigma \).

For \(z\in {\Omega }\), we can define \(v(z)=v_\rho (z)\) for \(\rho \) small enough so that Open image in new window . Similarly, for Open image in new window , we put \(w(z)=w_\rho (z)\) for small \(\rho \). Thus we have established the desired decomposition \(u=v+w\).

Now, assume that \({\Omega }\) is an arbitrary domain of \(\mathbb {H}^+\) and let u be a solution of \(L_mu=0\) on Open image in new window . Choose \(a\in \mathbb {H}^+\) and R large enough so that \(K\subset D(a,R)\) and D(aR) is a relatively compact set in \(\mathbb {H}^+\). Let \(\omega ={\Omega }\cap D(a,R)\). Note that K is a compact set in \(\omega \) which is a relatively compact open set in \(\mathbb {H}^+\) and u satisfies (1) on Open image in new window . Applying the results demonstrated for relatively compact open sets, we obtain
$$\begin{aligned} u(z)=\widetilde{v}(z)+\widetilde{w}(z) \end{aligned}$$
for Open image in new window , where \(\widetilde{v}\) satisfies (1) on the set \(\omega \) and \(\widetilde{w}\) satisfies (1) on Open image in new window with \(\lim _{{\partial }\mathbb {H}^+}\widetilde{w}=0\). Note that \(V=u-\widetilde{w}\) satisfies (1) on Open image in new window and V can be extended to a solution of (1) in a neighborhood of K because \(V=\widetilde{v}\) on \(\omega \). The sum \(u=V+\widetilde{w}\) provides with the desired decomposition of u.

If we have another decomposition \(u=v+w\) with \(v\in {C}^2({\Omega })\), \(L_mv=0\) and with Open image in new window , \(L_mw=0\) and \(\lim _{{\partial }\mathbb {H}^+}w=0\), then we have \(V-v=w-\widetilde{w}\) on Open image in new window . The function \(w-\widetilde{w}\) can be extended on \(\mathbb {H}^+\) to a solution of \(L_m(w-\widetilde{w})=0\) on \(\mathbb {H}^+\) with \(\lim _{{\partial }\mathbb {H}^+}(w-\widetilde{w})=0\). Thanks to Proposition 5.3, we obtain \(w=\widetilde{w}\), then \(V=v\), which completes the proof of the decomposition theorem. \(\square \)

The following proposition is a Poisson formula for axisymmetric potentials in \(\mathbb {H}^+\).

Proposition 5.10

Let \(m\in {\mathbb {C}}\) be such that \(\mathrm{Re}\,m<1\) and \(u:{{\mathbb {R}}}\rightarrow {{\mathbb {R}}}\) be a continuous and bounded function. Then there is a unique axisymmetric potential \(U\in {C}^2(\mathbb {H}^+)\) such that \(\lim _{\Vert (x,y)\Vert \rightarrow +\infty } U(x,y)=0\) and for all \(y\in {{\mathbb {R}}}\),
$$\begin{aligned} \lim _{(0,y)}U=u(y). \end{aligned}$$
Moreover, we have for all \((x,y)\in \mathbb {H}^+\),where

Proof

Let us first show that (15) is a solution of \(L_m U=0\). Let \(f(x,y)=x^{1-m}/ (x^2+(y-\eta )^2)^{1-{m/2}}\). Interchanging differentiation and integration, it suffices to prove that \(L_m f=0\). We have
Then,and we deduce that \(L_m f(x,y)=0\).
We have
By a change of variable \(t=(y-\eta )/ x\), we obtainThanks to the dominated convergence theorem, it suffices to show thatTo see this, according [1, p. 258], note thatwhere \(\mathrm{B}\) is the Euler beta function and
By the duplication formula for the \(\mathrm{\Gamma }\) function,and by the recurrence formula \(\mathrm{\Gamma }(z+1)=z\mathrm{\Gamma }(z)\), we obtain the desired result:The uniqueness follows from Proposition 5.3. The proof is complete. \(\square \)

Remark 5.11

One may ask if there is a reproducing formula for the case \(\mathrm{Re\,}m\ge 1\). Let \(m\in \mathbb {N}^*\) and let \(u\in C^2(\overline{\mathbb {H}^+})\) be such that \(L_mu=0\) on \(\mathbb {H}^+\), then the function v defined on \({{\mathbb {R}}}^{m+2}\) by
$$\begin{aligned} v(x_1,\ldots ,x_{m+2})=u\Bigl (\sqrt{x_1^2+\cdots +x_{m+1}^2},x_{m+2}\Bigr ) \end{aligned}$$
is harmonic on Open image in new window . In particular, if \(m\ge 2\), by [19, Proposition 18, p. 310], v can be extended to a harmonic function on \({{\mathbb {R}}}^{m+2}\), which tends to 0 at infinity. We then deduce that \(v\equiv 0\) hence \(u\equiv 0\). This shows that solving \(L_mu=0\) with u tending to 0 at infinity and with prescribed values of u on the y-axis is a problem which does not make sense. In this case, the fact that there is no solution to this Dirichlet problem is a consequence of the loss of ellipticity of \(L_m\) on the boundary of \(\mathbb {H}^+\). Therefore, we do not deal with the case \(\mathrm{Re\,}m\ge 1\).

6 Fourier–Legendre decomposition

First, we will introduce a specific system of coordinates \((\tau ,\theta )\) called bipolar coordinates, see [39]. The numerical applications on extremal bounded problems using this system of coordinates can be found in [25, 26, 27].

Let \(\alpha >0\). Suppose that there is a positive charge at Open image in new window and a negative charge at \(B=(\alpha ,0)\) (the absolute values of the two charges are identical). The potential generated by these charges at a point M is Open image in new window (modulo a multiplicative constant) (Fig. 5).
Fig. 5

Bipolar coordinates

Definition 6.1

The coordinatesare called bipolar coordinates.
The bipolar coordinates are related to the Cartesian coordinates by the following formulas:Let \(R>0\) and \(a=\sqrt{R^2+\alpha ^2}\), the disk with center (a, 0) and radius R is defined in terms of bipolar coordinates asThe right half-plane is defined as
$$\begin{aligned} \mathbb {H}^+=\bigl \{(\tau ,\theta ):\tau \in (0+\infty ], \theta \in [0,2\pi )\bigr \}. \end{aligned}$$
The level lines \(\tau =\tau _0\) are circles with center \((\alpha \coth \tau _0,0)\) and radii \(\alpha /{\text{ sh }}\,\tau _0\). This implies that for all \(\tau _0,\tau _1\) such that \(0<\tau _0<\tau _1\), the set \(\{(\tau ,\theta ): \tau \ge \tau _0\}\) is a closed disk and the set \(\{(\tau ,\theta ):0<\tau <\tau _1\}\) is the complement in \(\mathbb {H}^+\) of the closed disk \(\{\tau \ge \tau _1\}\) (Fig. 6).
Fig. 6

Level lines (with \(\alpha =1\))

The following theorem is well known in physics for \(m=-1\) [3, 15, 42, 45, 47, 48]. We extend it to \(m\in {\mathbb {C}}\).

Theorem 6.2

Let u be a solution to \(L_mu=0\) in an open set in \(\mathbb {H}^+\). Letwhere, by definition,Then
$$\begin{aligned} {{\partial }^2v_m\over {\partial }\tau ^2}+{{\partial }^2v_m\over {\partial }\theta ^2}+\coth \tau \,{{\partial }v_m\over {\partial }\tau }+\biggl ({1\over 4}-{(m-1)^2\over 4\,{\text{ sh }}^2\tau }\biggr ) v_m=0. \end{aligned}$$
(16)

Proof

In particular, we haveTherefore, we obtain
According to definition of \(v_m\),Denotethen
and
Hence the equation Open image in new window can be rewritten as
with
and
This completes the proof. \(\square \)
Let us apply the method of separation of variables, i.e. assume \(v_m\) has the form \(v_m(\tau ,\theta )=A_m(\tau )B_m(\theta )\). Then (16) becomesThe term on the right depends only on \(\theta \) and the left-hand side depends only on \(\tau \), thus we deduce that both are constant. Let \(n\in {\mathbb {C}}\) be such that this constant is equal to \(n^2\). Then we have
$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle {A_m''+\coth \tau \, A_m'+\biggl (\frac{1}{4} -\frac{(m-1)^2}{4\,{\text{ sh }}^2\tau }-n^2\biggr )\,A_m=0,}\\ \displaystyle {B_m''+n^2B_m=0.} \end{array}\right. } \end{aligned}$$
The function \(B_m\) is a \(2\pi \)-periodic function as \(\theta \) represents an angle, therefore n should necessarily be an integer.
To examine the equation satisfied by \(A_m\), we carry out the following change of function
$$\begin{aligned} A_m(\tau )=C_m({\text{ ch }}\,\tau ). \end{aligned}$$
Then, \(C_m\) satisfieswhich can be rewritten as
This equation is called the hyperbolic associated Legendre equation. Note that if we put \(z={\text{ ch }}\,\tau \) and \(u(z)=C_m({\text{ ch }}\,\tau )\), then
where
This equation is called the associated Legendre equation, and it can be reduced to the Legendre equation if \(\mu =0\):
Two independent solutions of (LA) are given in Appendix, where they are denoted by \(P_\nu ^\mu ({\text{ ch }}\,\tau )\) and \(Q_\nu ^\mu ({\text{ ch }}\,\tau )\).

Theorem 6.3

Let \(m\in {\mathbb {C}}\) and \(0<\tau _0\). Let u be a smooth solution to \(L_mu=0\) on the disk \(\tau \ge \tau _0\) and let v be a smooth solution to \(L_mv=0\) on \(\mathbb {H}^+{\setminus }\{\tau >\tau _0\}\), which is the complement in \(\mathbb {H}^+\) of the disk \(\{\tau >\tau _0\}\), and assume that \(\lim _{{\partial }\mathbb {H}^+}v=0\). Then there are two sequences \((a_n)_{n\in \mathbb {Z}}\) and \((b_n)_{n\in \mathbb {Z}}\) of \(\ell ^2(\mathbb {Z})\) (rapidly decreasing) such that
The sequence \((a_n)\) is unique. In addition, the convergence of the first series is uniform on every compact set \([\tau _1,\tau _2]\) of the disk \(\tau >\tau _0\) with \(\tau _0\le \tau _1<\tau _2\). And the convergence of the second one is uniform on every compact set \([\tau _3,\tau _4]\) of the complement of the disk \(\tau >\tau _0\) in \(\mathbb {H}^+\) with \(0<\tau _3<\tau _4\le \tau _0\).

If \(\mathrm{Re}\,m<1\), then the sequence \((b_n)\) is unique.

Proof

Indeed, decomposing the functionas Fourier series with respect to the variable \(\theta \), yields the following Fourier expansion for \(u(\tau _0,{\cdot })\):where \(a_n\in \ell ^2(\mathbb {Z})\) satisfiesThe function Open image in new window is a smooth function of the variable \(\theta \), therefore the sequence \((a_n)_n\) is rapidly decreasing as \(|n|\rightarrow +\infty \).
The functioncoincides with u on the circle \(\tau =\tau _0\). Let us see that \(\widetilde{u}\) is well defined on the disk \(\tau \ge \tau _0\). Indeed, thanks to Proposition 8.1, as \(|n|\rightarrow +\infty \),and this equivalence is uniform on all compact sets \([\tau _1,\tau _2]\) with \(0<\tau _0\le \tau _1<\tau _2\). It follows that the series which defines \(\widetilde{u}\) is norm convergent on any compact set \([\tau _1,\tau _2]\) of the disk \(\tau \ge \tau _0\). The same is true for derivatives with respect to \(\tau \) and \(\theta \) (which are also expressed through the associated Legendre functions, see Appendix).

Due to the fact that the solution of an elliptic equation is uniquely determined by its boundary values (this follows from the maximum principle), we deduce that \(\widetilde{u}\) is the unique axisymmetric potential on the disk \(\tau \ge \tau _0\) which coincides with u on the circle \(\tau =\tau _0\).

For v, the proof is similar. Indeed, decomposing the functionas Fourier series with respect to the variable \(\theta \), yields the following Fourier expansion for Open image in new window :where \(b_n\in \ell ^2(\mathbb {Z})\) satisfiesThe function Open image in new window is a smooth function of the variable \(\theta \), therefore the sequence \((b_n)_n\) is rapidly decreasing as \(|n|\rightarrow +\infty \).
The functioncoincides with v on the circle \(\tau =\tau _0\). Let us see that \(\widetilde{v}\) is well defined on the complement of the disk \(\tau >\tau _0\). Indeed, thanks to Proposition 8.1, as \(|n|\rightarrow +\infty \),and this equivalence is uniform on all compact sets \([\tau _1,\tau _2]\) with \(0<\tau _1<\tau _2\le \tau _0\). It follows that the series which defines \(\widetilde{v}\) is norm convergent on any compact set \([\tau _1,\tau _2]\) of the complement of the disc \(\tau >\tau _0\). The same is true for derivatives with respect to \(\tau \) and \(\theta \).
We will show that
$$\begin{aligned} \lim _{\tau \rightarrow 0+}\!\widetilde{v}=0. \end{aligned}$$
If \(\mathrm{Re}\,m<1\), by formula (18), for \(n\in \mathbb {N}\) we have
hence
$$\begin{aligned} \lim _{\tau \rightarrow 0+}\!P_{n-{1/2}}^{(m-1)/2}({\text{ ch }}\,\tau )=0. \end{aligned}$$
In addition, for \(n>1-{\mathrm{Re\,}m/2}\), we have
thusBy Proposition 8.1, we obtainSo, \(\lim _{\tau \rightarrow 0+}\widetilde{v}=0\).
The uniqueness of \((b_n)\) for \(\mathrm{Re}\,m<1\) follows from the following fact established in the next section: the families
form a Riesz basis. \(\square \)

Corollary 6.4

The solution of the Dirichlet problem for \(L_mu=0\) on D((a, 0), R), with \(u=\varphi \) on \({\partial }D((a,0),R)\), is given bywhere \(\{\tau =\tau _0\}\) corresponds to the circle with center (a, 0) and radius R andSimilarly, the functionis a solution to \(L_mv=0\) on \(\mathbb {H}^+{\setminus } D((a,0),R)\), which is equal to \(\varphi \) on \({\partial }D((a,0),R)\).

Moreover, if \(\mathrm{Re}\,m<1\), then v satisfies \(\lim _{{\partial }\mathbb {H}^+}\!v\!=\!0\), and (17) is the unique solution of the Dirichlet problem \(L_mv=0\) on \(\mathbb {H}^+{\setminus } D((a,0),R)\) which vanishes on \({\partial }\mathbb {H}^+\).

7 Riesz basis

We will prove that the first group of functions of the following family:is a basis of solutions on the disk \(\tau \ge \tau _1\) and the second one is a basis of solutions on \(\tau \le \tau _0\), which is the complement on \(\mathbb {H}^+\) of some disk, with \(0<\tau _0<\tau _1\). This fact is known for \(m=-1\), namely, for \(\mu =1\). We extend this result for complex m.
Let us recall the definition of a Riesz basis. The sequence \((x_n)_{n\in \mathbb {N}}\) is called a quasi-orthogonal or Riesz sequence of a Hilbert space X if there are two constants \(c,C>0\) such that for all sequences \((a_n)_{n\in \mathbb {Z}}\) in \(\ell ^2\), we have
$$\begin{aligned} c^2\sum _n|a_n|^2\le \biggl \Vert \sum _n a_n x_n\biggr \Vert ^2 \le C^2\sum _n|a_n|^2. \end{aligned}$$
If the family \((x_n)_{n\in \mathbb {Z}}\) is complete, it is called a Riesz basis. The matrix of scalar products \(\{\langle x_i,x_j\rangle \}_{i,j}\) is called the Gram matrix associated to \(\{x_i\}_i\). Let us recall the following characterization of a Riesz basis by the Gram matrix.

Property

([43, p. 170]) A family \(\{x_i\}_i\) is a Riesz basis for some Hilbert space X if \(\{x_i\}_i\) is complete in X and the Gram matrix associated to \(\{x_i\}_i\) defines a bounded and invertible operator on \(\ell ^2(\mathbb {N})\).

Let \({\mathscr {A}}\) and \({\mathscr {B}}\) be the families of solutions to \(L_mu=0\), respectively, inside the disk \(\tau >\tau _0\) and outside the disk \(\tau >\tau _1\), with \(0<\tau _0<\tau _1\):
Let \({\mathscr {C}}\) be the union of these two families,Denote the annulus defined in terms of bipolar coordinates \(\{0<\tau _0<\tau <\tau _1\}\) by \({\mathbb {A}}\). The space \(L^2({\partial }{\mathbb {A}})\) is equipped with the following inner product: for \(f,g\in L^2({\partial }{\mathbb {A}})\),

Proposition 7.1

\({\mathscr {C}}\) is a Riesz basis in the Hilbert space \(L^2({\partial }{\mathbb {A}})\).

Proof

To find the Gram matrix for \({\mathscr {C}}\), first calculate all its elements. For \(n\in \mathbb {Z}\),
In all other cases, the inner product is zero, hence the Gram matrix is diagonal by blocks and each block is the following Open image in new window matrix:The Gram matrix G has the form
$$\begin{aligned} \displaystyle G= \left( \begin{matrix} M_0 &{}\quad 0 &{}\quad \cdots &{}\quad \cdots &{}\quad \cdots &{}\quad \cdots &{}\quad \cdots &{}\quad \cdots \\ 0 &{}\quad M_{-1} &{}\quad 0 &{}\quad \cdots &{}\quad \cdots &{}\quad \cdots &{}\quad \cdots &{}\quad \cdots \\ \vdots &{}\quad 0 &{}\quad M_1 &{}\quad 0 &{}\quad \cdots &{}\quad \cdots &{}\quad \cdots &{}\quad \cdots \\ \vdots &{}\quad \ddots &{}\quad 0 &{}\quad M_{-2}&{}\quad 0 &{}\quad \cdots &{}\quad \cdots &{}\quad \cdots \\ \vdots &{}\quad \ddots &{}\quad \ddots &{}\quad 0 &{}\quad \ddots &{}\quad \ddots &{}\quad \cdots &{}\quad \cdots \\ \vdots &{}\quad \ddots &{}\quad \ddots &{}\quad \ddots &{}\quad \ddots &{}\quad M_{-n}&{}\quad \ddots &{}\quad \cdots \\ \vdots &{}\quad \ddots &{}\quad \ddots &{}\quad \ddots &{}\quad \ddots &{}\quad \ddots &{}\quad M_n &{}\quad \ddots \\ \vdots &{}\quad \ddots &{}\quad \ddots &{}\quad \ddots &{}\quad \ddots &{}\quad \ddots &{}\quad \ddots &{}\quad \ddots \end{matrix}\right) \!. \end{aligned}$$
The determinant of \(M_n\) isLet us show that \(M_n\) is invertible. Suppose the contrary, then \(\det M_n=0\), which is equivalent toIt can be rewritten aswith \(P_{n-1/2}^{(m-1)/2}({\text{ ch }}\,\tau _0), Q_{n-1/2}^{(m-1)/2}({\text{ ch }}\,\tau _1)\ne 0\). Therefore, there is \(\lambda \in {\mathbb {C}}{\setminus }\{0\}\) (which depends on \(m,n, \tau _0\) and \(\tau _1\)) such thatThen, by the asymptotic behavior of associated Legendre functions (see Proposition 8.1 in Appendix), on the one hand, we haveand on the other hand, we havewhat implies that \(\tau _0=\tau _1\), whereas it is not possible. Hence the matrix \(M_n\) is invertible and this completes the proof. \(\square \)

8 Appendix: Associated Legendre functions of first and second kind

In this section, we provide formulas of integral representation for the associated Legendre functions of the first and the second kind with \(z={\text{ ch }}\,\tau >1\), see [2, 39, 50].with \(\mathrm{Re}\,\nu >-1\) and \(\mathrm{Re}\,(\mu +\nu )<0\).with \(\mathrm{Re}\,\mu <{1/2}\).with \(\mathrm{Re}\,\mu <{1/2}\).with \(\mathrm{Re}\,\mu >-{1/2}\), \(\mathrm{Re}\,(\nu -\mu +1)<0\) and \(\mathrm{Re}\,(\nu +\mu +1)>0\).with \(\mathrm{Re}\,\mu <{1/2}\) and \(\mathrm{Re}\,(\mu +\nu +1)>0\).with \(\mathrm{Re}\,\nu >-1\) and \(\mathrm{Re}\,(\mu +\nu +1)>0\), see [50, pp. 4–6].
There are the following relations between the Legendre functions, see [50, p. 6] and [2, Formula 8.2.2]:
for \(\nu -\mu \not \in \mathbb {Z}\). In particular, for \(\nu =n-{1/2}\) and \(n\in \mathbb {Z}\), we haveThere hold the following Whipple formulas relating the associated Legendre functions of first and second kind, see [50, p. 6]:
There hold the following recursion formulas, see [50, pp. 6–7]:
All of these formulas are used to calculate the values \(P_\nu ^\mu ({\text{ ch }}\,\tau )\) and \(Q_\nu ^\mu ({\text{ ch }}\,\tau )\) for all \(\tau >0\) and \((\mu ,\nu )\in {\mathbb {C}}^2\).

If \(\mu \) and \(\tau \) are fixed, the following proposition describes the behavior of associated Legendre functions of the first and second kind when \(\nu =n-{1/2}\), \(n\in \mathbb {Z}\), as \(|n|\rightarrow +\infty \).

Proposition 8.1

Fix \(\tau >0\) and \(\mu \in {\mathbb {C}}\). Then if \(\nu =n-{1/2}\) and \(n\in \mathbb {Z}\), we have:
These equivalences are locally uniform with respect to the variable \(\tau \), i.e. uniform on the whole interval \([\tau _0,\tau _1]\) with \(0<\tau _0<\tau _1\).

Proof

If \(\nu =n-{1/2}\) and \(n\in \mathbb {N}\), see [50, p. 48], we have
By the Stirling formula as \(\nu \rightarrow +\infty \)
consequently,which gives us the first estimate.
The second estimate is obtained thanks to the relation \(P_\nu ^\mu =P_{-\nu -1}^\mu \). The third estimate follows from [50, Formula (8.3)]:and the last estimate follows from the fact that for \(\nu =n-{1/2}\) and \(n\in \mathbb {Z}\), we have
$$\begin{aligned} Q_{-\nu -1}^\mu =Q_\nu ^\mu . \end{aligned}$$
The local uniformity of these equivalences implies from explicit expressions of \(P_\nu ^\mu \) and \(Q_\nu ^\mu \) in terms of hypergeometric functions available in [21, pp. 124–138] and estimates of these hypergeometric functions which are locally uniform [50, pp. 178–182]. \(\square \)

Notes

Acknowledgments

Both authors thank Laurent Baratchart and Alexander Borichev for very useful discussions and remarks on the preliminary version of this paper.

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© Springer International Publishing AG 2015

Authors and Affiliations

  1. 1.I2M, UMR 7373, Aix-Marseille Université, CMIMarseilleFrance

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