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

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## 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

*m*. The Weinstein equation reads

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.

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 \(\mathrm{supp}\,\varphi \subset K\).

*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

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

*L*be a linear differential operator on \({\Omega }\),

*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,

*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

*a*on \({\Omega }\) and if \(g\in {\mathscr {D}}({\Omega })\) is such that \(g=L(\varphi )\) with \(\varphi \in {\mathscr {D}}({\Omega })\), then

*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 \(L_m\).

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.

**Proposition 2.1**

*D*conjugates \(L_{-m}^\star \) and \(L_m\), which means that

*Remark 2.2*

- 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.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 haveDefine \(S_\sigma \) as$$\begin{aligned} \langle P_{\sigma }u,v\rangle&= \int _{{\Omega }}{1\over \sigma (x,y)}\,\mathrm{div}\bigl (\sigma (x,y)\nabla u(x,y)\bigr )v(x,y)\,dxdy\\&= -\int _{{\Omega }}\sigma \nabla u{\cdot }\nabla \biggl ({v\over \sigma }\biggr )\,dxdy= \int _{{\Omega }}u\mathrm{div}\biggl (\sigma \nabla \biggl ({v\over \sigma }\biggr )\biggr )dxdy\\&= \langle u,P_{\sigma }^\star v\rangle . \end{aligned}$$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.$$\begin{aligned} (S_\sigma u)(x,y)={1\over \sigma (x,y)}\,u(x,y), \qquad u\in C^2({\Omega }). \end{aligned}$$

*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

*m*real and complex, so we omit the proofs.

**Proposition 2.3**

**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*

**Proposition 3.1**

*Proof*

This proposition and the Weinstein principle imply the following result.

**Proposition 3.2**

*Proof*

## 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 \(m\in {\mathbb {C}}\).

**Proposition 4.1**

*Proof*

**Proposition 4.2**

*Proof*

*k*tends to \(+\infty \).

**Claim 4.3**

*Proof*

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**

*Proof*

*x*,

*y*) 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.

*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 \(\mathrm{I}_{\varepsilon }\).

**Claim 4.5**

*Proof*

**Claim 4.6**

*Proof*

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**

*ds*is the arc length element on \({\partial }{\Omega }\) (positively oriented).

*Proof*

*x*,

*y*) 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.

*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 \(\mathrm{Re}\,m\ge 1\),

**Definition 5.1**

**Proposition 5.2**

*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*

*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):

*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**

*Proof*

**Lemma 5.5**

- 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*

*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 \(D_i\).

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

**Lemma 5.6**

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

*Proof*

*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 \(D_i\).

*N*) on a domain with measure controlled by \(\mathrm{O}(N^2)\).

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

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

| \(\int _{D_i}\!|F_m|\) | \((|{\partial }_{\xi }\phi _N|,|{\partial }_{\eta }\phi _N|)\) | \(\int _{D_i}\!|{\partial }_{\xi }F_m|\) | \(\int _{D_i}\!|{\partial }_{\eta }F_m|\) | |
---|---|---|---|---|---|

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

**Proposition 5.7**

*Proof*

*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,

*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**

*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

*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*.

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 \(u=v+w\).

*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*(

*a*,

*R*) 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

*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**

*Proof*

*Remark 5.11*

*v*defined on \({{\mathbb {R}}}^{m+2}\) by

*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].

*M*is Open image in new window (modulo a multiplicative constant) (Fig. 5).

**Definition 6.1**

*bipolar coordinates*.

*a*, 0) and radius

*R*is defined in terms of bipolar coordinates asThe right half-plane is defined as

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**

*u*be a solution to \(L_mu=0\) in an open set in \(\mathbb {H}^+\). Letwhere, by definition,Then

*Proof*

*n*should necessarily be an integer.

*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**

*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*

*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 \(\tau =\tau _0\).

*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 \(|n|\rightarrow +\infty \).

*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 \(\theta \).

**Corollary 6.4**

*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

*m*.

*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

*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 \(\ell ^2(\mathbb {N})\).

**Proposition 7.1**

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

*Proof*

*G*has the form

## 8 Appendix: Associated Legendre functions of first and second kind

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**

*Proof*

## 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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