1 Introduction

One of the most interesting problems in geophysics is the determination of the gravitational or magnetic field of the Earth from surface measurements of its intensity. It is in fact much more convenient to measure field intensities rather than field directions. In mathematical terms, the problem can be formulated as follows.

We shall represent the Earth’s surface by the unit sphere \({\mathcal {S}}\subset \mathbb {R}^3\) (centered at the origin). We will denote by \(\Omega \) the exterior of \({\mathcal {S}}\), i.e. the unbounded component of \(\mathbb {R}^3\setminus {\mathcal {S}}\). Also, we suppose a positive continuous function g is given on \({\mathcal {S}}\). If u represents the Earth’s external scalar potential associated to the gravitational or magnetic field, then \(|\nabla u|\) represents the field’s intensity. Backus’s problem then consists in finding solutions \(u\in C^2(\Omega )\cap C^1(\overline{\Omega })\) of the following nonlinear boundary value problem:

$$\begin{aligned} \Delta u =0 \ \text{ in } \ \Omega , \quad |\nabla u|=g \ \text{ on } \ {\mathcal {S}}, \quad u\rightarrow 0 \ \text{ as } \ |x|\rightarrow \infty . \end{aligned}$$
(1.1)

It is also important to specify sufficient conditions that ensure the uniqueness of a solution.

A large part of the results known about this problem is contained in the pioneering work of G. E. Backus [4]. There, the problem of finding solutions of the first two equations in (1.1) is first examined for the two-dimensional case in a bounded domain and, by conformal mappings, also in exterior domains. A complete analysis is carried out: existence of solutions is proved, a severe lack of uniqueness is pointed out, and conditions which restore uniqueness are stated. Related to this case, questions about the regularity of solutions are investigated in [12], together with an analysis of the case in which the data g may vanish at isolated points.

In [4], a condition for uniqueness of a solution for problem (1.1) is given in the gravitational case. Non-uniqueness for the geomagnetic case is noted in [5]; [10] contains a general uniqueness condition for (1.1), which includes both the gravitational and geomagnetic case. The solution of the linearized problem near the dipole is also constructed in [10] by means of an expansion in spherical harmonics. By a similar technique, the linearization near quadripoles is considered in [1] and solved.

For what concerns the existence of a solution of (1.1) in physical dimension, a first conclusive result is contained in [9]. There, it is proved local existence and uniqueness of a solution in the gravitational case. It is obtained by linearizing (1.1) at the so-called monopole,

$$\begin{aligned} \Phi (x)=\frac{1}{|x|}, \ x\in \overline{\Omega }, \end{aligned}$$

which is nothing else than a normalized version of the fundamental solution of Laplace’s equation. More precisely, one can prove existence and uniqueness of a solution of the form

$$\begin{aligned} u(x)=\Phi (x)+w(x), \ x\in \overline{\Omega }, \end{aligned}$$
(1.2)

provided g is sufficiently close to \(|\nabla \Phi |\equiv 1\) in the norm of \(C^{0,\alpha }({\mathcal {S}})\), \(0<\alpha <1\), the space of \(\alpha \)-Hölder continuous functions on \({\mathcal {S}}\). In fact, plugging the ansatz (1.2) into (1.1) gives that w must solve the problem:

$$\begin{aligned} \Delta w=0 \ \text{ in } \ \Omega , \ w_\nu +\frac{1}{2}\,|\nabla w|^2=\frac{g^2-1}{2} \ \text{ on } \ {\mathcal {S}}, \ w\rightarrow 0 \ \text{ as } \ |x|\rightarrow \infty . \end{aligned}$$
(1.3)

Here, \(\nu \) denotes the unit normal to \({\mathcal {S}}\), exterior to \(\Omega \).

In [9] (see also [13]), the nonlinear problem (1.3) is solved by a Neumann series, essentially based on the following fixed point argument. In fact, one can introduce a (nonlinear) operator \({\mathcal {T}}_\Phi \) by formally setting

$$\begin{aligned} {\mathcal {T}}_\Phi [f]=|\nabla v|^2, \end{aligned}$$

where v is the solution of the Neumann problem:

$$\begin{aligned} \Delta v=0 \ \text{ in } \ \Omega , \quad v_\nu =\frac{1}{2}\,f \ \text{ on } \ {\mathcal {S}}, \quad v\rightarrow 0 \ \text{ as } \ |x|\rightarrow \infty . \end{aligned}$$
(1.4)

Since we know that (1.4) always has a unique solution \(v\in C^{1,\alpha }(\overline{\Omega })\) for any \(f\in C^{0,\alpha }({\mathcal {S}})\), then \({\mathcal {T}}_\Phi \) turns out to be well-defined as an operator on \(C^{0,\alpha }({\mathcal {S}})\) into itself. The problem (1.3) is thus converted into the fixed-point equation:

$$\begin{aligned} f+{\mathcal {T}}_\Phi [f]=g^2-1 \ \text{ on } \ {\mathcal {S}}. \end{aligned}$$

This can be uniquely solved by a function \(f^*\in C^{0,\alpha }({\mathcal {S}})\), provided \(g^2-1\) is sufficiently small in the norm of \(C^{0,\alpha }({\mathcal {S}})\). A solution of (1.1) is therefore obtained by means of (1.2), where w is the solution of (1.4) corresponding to \(f^*\).

We conclude our review of known results with a couple of papers, [7, 8], which provide a genuinely nonlinear approach to problem (1.1). In [7], (1.1) is converted into a boundary value problem in the unit ball B:

$$\begin{aligned} \Delta U=0 \ \text{ in } \ B, \quad (U+U_\nu )^2+|\nabla _{\mathcal {S}}U|^2=g^2 \ \text{ on } \ {\mathcal {S}}, \end{aligned}$$
(1.5)

where \(\nabla _{\mathcal {S}}U\) denotes the tangential gradient of U on \({\mathcal {S}}\). Here, U is the Kelvin’s transformation of u, which is such that

$$\begin{aligned} u(x)=|x|^{-1} U(x/|x|^2) \ \text{ for } \ x\in \Omega . \end{aligned}$$

Then, a solution of (1.5) is obtained by solving the following boundary value problem:

$$\begin{aligned} \Delta U=0 \ \text{ in } \ B, \quad U+U_\nu =\sqrt{(g^2-|\nabla _{\mathcal {S}}U|^2)_+} \ \text{ on } \ {\mathcal {S}}. \end{aligned}$$
(1.6)

Since \(U+U_\nu =-u_\nu \) on \({\mathcal {S}}\), the corresponding solution u of (1.1) is such that \(u_\nu \le 0\) on \({\mathcal {S}}\). It turns out that harmonic functions in \(\Omega \), which vanish at infinity and are subject to the constraint \(u_\nu \le 0\) on \({\mathcal {S}}\), satisfy some sort of comparison principle. This property is then instrumental to the definition of maximal and minimal solutions of (1.1) such that \(u_\nu \le 0\) on \({\mathcal {S}}\). This fact allows the construction of a suitably defined viscosity solution of (1.6). In [8], a numerical scheme to construct a maximal solution is proposed.

The aim of this paper is to study the local resolution of the geomagnetic case, i.e. the (local) existence and uniqueness of solutions of (1.1) near the dipole defined by

$$\begin{aligned} d(x)=\frac{x_3}{|x|^3} \ \text{ for } \ x\in \overline{\Omega }. \end{aligned}$$

In spherical coordinates \(x=r\,(\cos \theta \cos \varphi , \cos \theta \sin \varphi , \sin \theta )\), d can be written as

$$\begin{aligned} d=\frac{\sin \theta }{r^2}. \end{aligned}$$

Here \(r=|x|\), \(-\pi /2\le \theta \le \pi /2\) is the latitude, and \(-\pi \le \varphi <\pi \) is the longitude on the Earth \({\mathcal {S}}\). To the best of our knowledge, existence results for this problem are not present in the literature.

Thus, similarly to the gravitational case, we linearize problem (1.1) by setting \(u=d+w\) and obtain the following problem for w:

$$\begin{aligned}&\Delta w=0 \ \text { in } \ \Omega , \nonumber \\ {}&\nabla d\cdot \nabla w+\frac{1}{2}\,|\nabla w|^2=\frac{g^2-|\nabla d|^2}{2} \ \text { on } \ {\mathcal {S}}, \\ {}&w\rightarrow 0 \ \text { as } \ |x|\rightarrow \infty . \nonumber \end{aligned}$$
(1.7)

Note that, being as \(\nu (x)=-x\), we have that

$$\begin{aligned} \nabla d(x)=\tau (x)+2 x_3\,\nu (x) \ \text{ for } \ x\in {\mathcal {S}}. \end{aligned}$$
(1.8)

The vector field

$$\begin{aligned} \tau (x)=e_3+x_3\,\nu (x) \ \text{ for } \ x\in {\mathcal {S}}, \end{aligned}$$

is tangential to S and is obtained by projecting \(e_3=(0,0, 1)\) on the tangent plane of \({\mathcal {S}}\) at \(x\in {\mathcal {S}}\). Notice that \(\nabla d(x)\) has intensity \(|\nabla d(x)|=\sqrt{1+3 x_3^2}\) for \(x\in {\mathcal {S}}\), and points outward to the Earth’s surface on the south hemisphere, becomes tangential on the equator \({\mathcal {E}}=\{ x\in {\mathcal {S}}: x_3=0\}\), and points inward on the north hemisphere. This behavior of \(\nabla d\) tells us that neither d nor any solution of (1.1) sufficiently close to d falls within the class of solutions studied in [7, 8].

Proceeding as in the monopole case gives the (irregular) oblique derivative problem:

$$\begin{aligned} \Delta v=0 \ \text{ in } \ \Omega , \quad \nabla d\cdot \nabla v=\frac{1}{2}\,f \ \text{ on } \ {\mathcal {S}}, \quad v\rightarrow 0 \ \text{ as } \ |x|\rightarrow \infty . \end{aligned}$$
(1.9)

Differently from regular oblique derivative problems, in which the relevant directional derivative is controlled by a vector field that points either inward or outward on the whole boundary, the irregular ones may present at least two setbacks. These are caused precisely by a change of direction, as described for \(\nabla d\).

The former is a non-trivial lack of uniqueness. In fact, uniqueness can be obtained only by prescribing Dirichlet boundary values on the subset of the boundary in which the relevant vector field becomes tangential (the equator \({\mathcal {E}}\) in the problem at stake). In other words, the homogeneous problem related to (1.9) has infinitely many solutions.

The latter is the so-called loss of derivatives. In fact, it may happen that suitably determined unique solutions of oblique boundary value problems with, say, \(C^{0,\alpha }\)-regular boundary data, do not gain \(C^{1,\alpha }\)-regularity up to the boundary, as it does happen for the Neumann problem or the regular oblique derivative problem (see for instance [2, 15, 16]). In other words, the linear operator on \(C^{0,\alpha }({\mathcal {S}})\) associating the oblique derivative data to the (trace on the boundary of the) solution of the problem may be unbounded. A similar behavior also occurs if we try to solve the oblique derivative problem in the scale of fractional Sobolev spaces \(H^s({\mathcal {S}})\) (this fact can be detected by an inspection of the solution obtained in [10]).

Besides causing a loss of surjectivity of the relevant operator, more importantly, the loss of derivatives disrupts the iterative scheme on which a classical contraction argument is based. Thus, the strategy of defining an operator \({\mathcal {T}}_{d,h}\) by first setting

$$\begin{aligned} {\mathcal {T}}_{d,h}[f]=|\nabla v|^2, \end{aligned}$$

where, for some fixed \(h:{\mathcal {E}}\rightarrow \mathbb {R}\), v is the solution of (1.9), subject to the Dirichlet-type condition

$$\begin{aligned} v=h \ \text{ on } \ {\mathcal {E}}, \end{aligned}$$
(1.10)

and then solving the equation

$$\begin{aligned} f+{\mathcal {T}}_{d,h}[f]=g^2-|\nabla d|^2, \end{aligned}$$

may miserably fail.

To by-pass these difficulties, a standard idea would be to use the Nash-Moser implicit function theorem. This often works when a loss of derivatives occurs. The second author and M. C. Jorge have tried this pathway, but with no success. The main difficulty is the lack of sufficiently precise estimates for the relevant oblique derivative problems involved. In fact, in such approach, one would need to precisely control estimates not only for the solution of (1.9), but also for a class of oblique derivative problems obtained by perturbing \(\nabla d\).

In this paper, we turn back to a fixed-point approach and obtain local existence near the dipole for the nonlinear problem (1.1) in the case the boundary data g is axially symmetric around the Earth’s axis. This result is obtained in the framework of fractional Sobolev spaces and is made possible from the discovery that the relevant oblique derivative problem (1.9) no longer loses derivatives in presence of axially symmetric data. Hence, a fixed-point scheme still works for problem (1.7).

From a technical viewpoint, we construct solutions of (1.9)-(1.10) by means of series expansions of spherical harmonics as done in [10]. This time, we trace more carefully the dependence on the data f and h of the coefficients of the relevant expansion. The aim is to obtain precise estimates for the operator \({\mathcal {T}}_{d,h}\) in the scale of Sobolev spaces \(H^s({\mathcal {S}})\). It turns out that \({\mathcal {T}}_{d,h}\) is well-defined as an operator on the subspace \(H^s_\mathrm{ax}({\mathcal {S}})\) of axially symmetric functions on \({\mathcal {S}}\). (For a precise definition of \(H^s({\mathcal {S}})\) and \(H^s_\mathrm{ax}({\mathcal {S}})\), see Sect. 2.)

Our main result is then the following existence and uniqueness theorem.

Theorem 1.1

Suppose that \(g \in H^s_\mathrm{ax}({\mathcal {S}})\) for some \(s>1\) and that \(h\equiv h_0\) on \({\mathcal {S}}\) with \(h_0\in \mathbb {R}\). If \(\Vert g -|\nabla d|\Vert _{H^s({\mathcal {S}})}\) and \(|h_0|\) are sufficiently small, then problem (1.1) has a unique solution \(u \in C^2(\Omega ) \cap C^1(\overline{\Omega })\) satisfying (1.10). Moreover, we have that \(u|_{{\mathcal {S}}} \in H^{s+1}_\mathrm{ax}({\mathcal {S}})\) and

$$\begin{aligned} \Vert u-d\Vert _{H^{s+1}({\mathcal {S}})}\le C\left( \Vert g-|\nabla d|\Vert _{H^s({\mathcal {S}})}+|h_0|\right) \end{aligned}$$
(1.11)

for some constant \(C>0\).

From the continuous embedding of \(H^s({\mathcal {S}})\) into \(C({\mathcal {S}})\) (see Proposition 2.1 below) and the maximum principle for harmonic functions, we see that (1.11) holds if the left-hand side is replaced with \(\Vert u-d\Vert _{C(\overline{\Omega })}\).

We conclude this introduction with one more important technical remark about the proof of Theorem 1.1.

In fact, it should be noticed that the Hilbert-space structure we adopt for \(H^s({\mathcal {S}})\) and \(H^s_\mathrm{ax}({\mathcal {S}})\) is based on an inner product of a spectral-type. In other words, the relevant inner product is defined in terms of the coefficients in the spherical-harmonics series expansions of the functions at stake. In order to deal with problem (1.7), which shows a quadratic nonlinearity in the gradient, a Banach-algebra structure for \(H^s({\mathcal {S}})\) is desirable. As a matter of fact, Banach-algebra structures for Sobolev spaces on Riemannian manifolds are systematically studied in [6] (see [6, Theorem 24] for the most relevant result), in which the norm is defined by

$$\begin{aligned} |\!|\!|u|\!|\!|_{H^{s,p}({\mathcal {S}})}=\Vert u\Vert _{L^p({\mathcal {S}})}+\Vert (-\Delta )^{s/2}u\Vert _{L^p({\mathcal {S}})}, \end{aligned}$$

where the fractional Laplace-Beltrami operator \((-\Delta )^{s/2}\) is defined by the Dunford integral. It is shown that \((-\Delta )^{s/2}u\) is comparable to an integral representation of the fractional derivative of u, and hence an algebra structure is derived by Leibniz’ rule. However, to the best of our knowledge, there is no explicit comparison between our spectral norm \(\Vert u\Vert _{H^s({\mathcal {S}})}\) and \(|\!|\!|u|\!|\!|_{H^{s,2}({\mathcal {S}})}\) in the literature. Thus, in Theorem 2.4, we provide our own proof for the case of axially symmetric functions. This is based on the series expansions in spherical harmonics of products of spherical harmonics, and the so-called Wigner 3j-symbols. (In Proposition 2.5, we show that such a structure also holds in other instances.)

The paper is organized as follows. We begin with Sect. 2, in which we collect all the functional analytical results on the spaces \(H^s({\mathcal {S}})\) and \(H^s_\mathrm{ax}({\mathcal {S}})\) useful for our purposes. Then, in Sect. 3, we derive the appropriate estimates for the relevant oblique derivative problem. The proof of Theorem 1.1 is given in Sect. 4.

2 The fractional spectral Sobolev space \(H^s({\mathcal {S}})\) and its subspace \(H^s_\mathrm{ax}({\mathcal {S}})\)

In this section, we collect old and new results on the spectral Hilbert space \(H^s({\mathcal {S}})\).

2.1 Spherical harmonics and the space \(H^s({\mathcal {S}})\)

As mentioned in the introduction, we adopt a system of spherical coordinates by setting

$$\begin{aligned} x=r\,\left( \cos \theta \cos \varphi , \cos \theta \sin \varphi , \sin \theta \right) , \ r>0, \ -\frac{\pi }{2}\le \theta \le \frac{\pi }{2}, \ -\pi \le \varphi <\pi ; \end{aligned}$$

we call \(\theta \) the latitude and \(\varphi \) the longitude on the earth surface \({\mathcal {S}}\). With this parametrization, the surface element on \({\mathcal {S}}\) is given by \(dS_x=\cos \theta \,d\theta \,d\varphi \).

We denote by \(\{ Y_l^m(\theta ,\varphi )\}_{|m|\le l, l=0, 1,\dots }\) the spherical harmonics system of functions. We have that

$$\begin{aligned} Y_l^m(\theta ,\varphi )= \alpha _l^m P_l^{|m|} (\sin \theta )\, e^{im \varphi } \qquad \left( |m| \le l, \ l=0, 1, \dots \right) , \end{aligned}$$

where \(P_l^m(z)\) is the associated Legendre polynomial of degree l and order m, and \(\alpha _l^m\) is defined by

$$\begin{aligned} \alpha _l^m=(-1)^{\frac{m+|m|}{2}} \sqrt{\frac{(2l+1)(l-|m|)!}{4\pi (l+|m|)!}}. \end{aligned}$$

It is well-known that \(\{ Y_l^m(\theta ,\varphi )\}_{|m|\le l, l=0, 1,\dots }\) forms an orthonormal basis of \(L^2({\mathcal {S}})\), that is, the equality

$$\begin{aligned} \psi (\theta ,\varphi )= & {} \sum _{l=0}^\infty \sum _{m=-l}^l \widehat{\psi }_l^m Y_l^m(\theta ,\varphi ) \ \text{ with } \\ \widehat{\psi }_l^m= & {} \int _{-\pi }^\pi \int _{-\frac{\pi }{2}}^{\frac{\pi }{2}} \psi (\theta ,\varphi ) \, \overline{Y_l^m(\theta ,\varphi )} \cos \theta \, d\theta d\varphi \end{aligned}$$

holds in \(L^2({\mathcal {S}})\) for any function \(\psi \in L^2({\mathcal {S}})\).

For any non-negative real number s, we define the fractional Sobolev space:

$$\begin{aligned} H^s({\mathcal {S}})=\biggl \{ \psi \in L^2({\mathcal {S}}): \sum _{l=0}^\infty \sum _{m=-l}^l (l+1)^{2s} |\widehat{\psi }_l^m|^2 <+\infty \biggr \}. \end{aligned}$$

Thus, we have that \(H^0({\mathcal {S}})=L^2({\mathcal {S}})\) and we know that \(H^s({\mathcal {S}})\) is a Hilbert space endowed with the inner product

$$\begin{aligned} \langle \psi ,\phi \rangle _{H^s({\mathcal {S}})} =\sum _{l=0}^\infty \sum _{m=-l}^l (l+1)^{2s} \widehat{\psi }_l^m \overline{\widehat{\phi }_l^m}. \end{aligned}$$

Similarly, we set

$$\begin{aligned}&H^s({\mathcal {E}})=\biggl \{ \Psi \in L^2({\mathcal {E}}): \sum _{m=-\infty }^\infty (|m|+1)^{2s} |\widehat{\Psi }_m|^2 <+\infty \biggr \},\\&\quad \text{ where } \ \widehat{\Psi }_m=\frac{1}{2\pi } \int _{-\pi }^\pi \Psi (\varphi ) \, e^{-im\varphi } \, d\varphi , \end{aligned}$$

and write the norm on \(H^s({\mathcal {E}})\) as

$$\begin{aligned} \Vert \Psi \Vert _{H^s({\mathcal {E}})} =\sqrt{\sum _{m=-\infty }^\infty (|m|+1)^{2s} |\widehat{\Psi }_m|^2}. \end{aligned}$$

Then, any function \(\Psi \in H^s({\mathcal {E}})\) can be expanded in \(H^s({\mathcal {E}})\) as

$$\begin{aligned} \Psi (\varphi )=\sum _{m=-\infty }^\infty \widehat{\Psi }_m e^{im\varphi }. \end{aligned}$$

The following properties of \(H^s({\mathcal {S}})\) will be useful in the sequel.

Proposition 2.1

Set \(s>1\). Then, the Sobolev space \(H^s({\mathcal {S}})\) is continuously embedded into \(C({\mathcal {S}})\).

Proof

It is known that the identity

$$\begin{aligned}&\sum _{m=-l}^l Y_l^m(\theta _1,\varphi _1) \overline{Y_l^m\left( \theta _2,\varphi _2\right) }\nonumber \\&\quad =\frac{2l+1}{4\pi }P_l\left( \sin \theta _1 \sin \theta _2 +\cos \theta _1 \cos \theta _2 \cos \left( \varphi _2-\varphi _1\right) \right) \end{aligned}$$
(2.1)

holds for any \(l=0, 1, \dots \), \(-\pi /2\le \theta _1, \theta _2 \le \pi /2\), and \(-\pi \le \varphi _1, \varphi _2<\pi \) (see [3, (16.57), (16.59)]). In particular, this identity and the fact that \(P_l(1)=1\) give

$$\begin{aligned} \sum _{m=-l}^l|Y_l^m(\theta ,\varphi )|^2=\frac{2l+1}{4\pi }. \end{aligned}$$
(2.2)

Hence, by the Cauchy-Schwarz inequality, we have that

$$\begin{aligned} \sum _{m=-l}^l\left| \widehat{u}_l^mY_l^m(\theta ,\varphi )\right| \le \sqrt{\frac{2l+1}{4\pi }} \sqrt{\sum _{m=-l}^l\left| \widehat{u}_l^m\right| ^2}. \end{aligned}$$

Taking the sum in l and using the Cauchy-Schwarz inequality again, we obtain

$$\begin{aligned} \sum _{l=0}^\infty \sum _{m=-l}^l\left| \widehat{u}_l^mY_l^m(\theta ,\varphi )\right|&\le \sqrt{\sum _{l=0}^\infty \frac{(l+1)^{-2s}(2l+1)}{4\pi }} \sqrt{\sum _{l=0}^\infty \sum _{m=-l}^l (l+1)^{2s}\left| \widehat{u}_l^m\right| ^2}\\&\le \sqrt{\sum _{l=0}^\infty \frac{(l+1)^{1-2s}}{2\pi }} \Vert u\Vert _{H^s({\mathcal {S}})}. \end{aligned}$$

Thus, the series (of continuous functions) on the left-hand side defines a continuous function, since it converges uniformly and absolutely for \(s>1\). \(\square \)

Proposition 2.2

Let k be any natural number. Then, \(C^k({\mathcal {S}})\) is continuously embedded into \(H^k({\mathcal {S}})\).

Proof

Assume \(u \in C^k({\mathcal {S}})\). Let \(\nabla _{{\mathcal {S}}}\) and \(\Delta _{{\mathcal {S}}}\) denote the gradient and the Laplace-Beltrami operator on \({\mathcal {S}}\), respectively. In order to prove the proposition, we verify that the equalities

$$\begin{aligned} \sum _{l=0}^\infty \sum _{m=-l}^l \left[ l(l+1)\right] ^{2j} |\widehat{u}_l^m|^2= & {} \left\| (-\Delta _{{\mathcal {S}}})^j u\right\| _{L^2({\mathcal {S}})}^2, \end{aligned}$$
(2.3)
$$\begin{aligned} \sum _{l=0}^\infty \sum _{m=-l}^l \left[ l(l+1)\right] ^{2j+1} |\widehat{u}_l^m|^2= & {} \left\| \nabla _{{\mathcal {S}}} \left[ (-\Delta _{{\mathcal {S}}})^ju\right] \right\| _{L^2({\mathcal {S}})}^2 \end{aligned}$$
(2.4)

hold for any nonnegative integer j with \(2j \le k\), \(2j+1 \le k\), respectively.

By repeated integration by parts and the fact that

$$\begin{aligned} -\Delta _{{\mathcal {S}}} Y_l^m=l(l+1)\,Y_l^m, \end{aligned}$$

the Fourier-Laplace coefficient of \((-\Delta _{{\mathcal {S}}})^j u\) is computed as

$$\begin{aligned} \begin{aligned} \left[ \widehat{(-\Delta _{{\mathcal {S}}})^j u}\right] _l^m&=\int _{{\mathcal {S}}}[(-\Delta _{{\mathcal {S}}})^ju]\, \overline{Y_l^m}\,dS\\&=\int _{{\mathcal {S}}}u\,[(-\Delta _{{\mathcal {S}}})^j\overline{Y_l^m}]\,dS =\left[ l(l+1)\right] ^j\widehat{u}_l^m. \end{aligned} \end{aligned}$$
(2.5)

Hence (2.3) follows. To derive (2.4), we set \(w=(-\Delta _{{\mathcal {S}}})^ju\). For the moment, we suppose that \(w \in C^2({\mathcal {S}})\). Then, we have that

$$\begin{aligned} \left\| \nabla _{{\mathcal {S}}} w\right\| _{L^2({\mathcal {S}})}^2 =-\int _{{\mathcal {S}}} w \Delta _{{\mathcal {S}}} w dS =-\sum _{l=0}^\infty \sum _{m=-l}^l \overline{\widehat{w}_l^m} \int _{{\mathcal {S}}} \overline{Y_l^m} \Delta _{{\mathcal {S}}} w dS, \end{aligned}$$

where we used integration by parts and the expansion

$$\begin{aligned} w=\sum _{l=0}^\infty \sum _{m=-l}^l \widehat{w}_l^m Y_l^m =\sum _{l=0}^\infty \sum _{m=-l}^l \overline{\widehat{w}_l^m} \overline{Y_l^m}. \end{aligned}$$

From the definition of w and (2.5), we see that

$$\begin{aligned} -\overline{\widehat{w}_l^m} \int _{{\mathcal {S}}} \overline{Y_l^m} \Delta _{{\mathcal {S}}} w dS =\overline{\left[ \widehat{(-\Delta _{{\mathcal {S}}})^j u}\right] _l^m} \left[ \widehat{(-\Delta _{{\mathcal {S}}})^{j+1} u}\right] _l^m =\left[ l(l+1)\right] ^{2j+1} |\widehat{u}_l^m|^2, \end{aligned}$$

and therefore

$$\begin{aligned} \left\| \nabla _{{\mathcal {S}}} w\right\| _{L^2({\mathcal {S}})}^2 =\sum _{l=0}^\infty \sum _{m=-l}^l \left[ l(l+1)\right] ^{2j+1} |\widehat{u}_l^m|^2. \end{aligned}$$

By approximation, this equality also holds if \(w \in C^1({\mathcal {S}})\). We thus obtain (2.4).

Using (2.3) and (2.4), we deduce that

$$\begin{aligned} \Vert u\Vert _{H^{k}({\mathcal {S}})}^2&\le C\sum _{l=0}^\infty \sum _{m=-l}^l\left( \left[ l(l+1)\right] ^k+1\right) |\widehat{u}_l^m|^2\\&=\left\{ \begin{aligned}&C\left( \left\| (-\Delta _{{\mathcal {S}}})^{\frac{k}{2}}u\right\| _{L^2({\mathcal {S}})}^2+\left\| u\right\| _{L^2({\mathcal {S}})}^2\right) \quad \text{ if } \ k \ \text{ is } \text{ even },\\&C\left( \left\| \nabla _{{\mathcal {S}}} \left[ (-\Delta _{{\mathcal {S}}})^{\frac{k-1}{2}} u\right] \right\| _{L^2({\mathcal {S}})}^2+\left\| u\right\| _{L^2({\mathcal {S}})}^2\right) \quad \text{ if } \ k \ \text{ is } \text{ odd } \end{aligned} \right. \\&\le C\Vert u\Vert _{C^k({\mathcal {S}})}^2, \end{aligned}$$

where \(C>0\) is some constant. This proves the proposition. \(\square \)

2.2 The subspace \(H^s_\mathrm{ax}({\mathcal {S}})\) and its Banach-algebra structure

In our analysis, the subspace of axially symmetric functions defined by

$$\begin{aligned} H^s_\mathrm{ax}({\mathcal {S}})=\bigl \{\psi \in H^s({\mathcal {S}}): \psi \text{ does } \text{ not } \text{ depend } \text{ on }\, \varphi \bigr \} \end{aligned}$$

will play a crucial role. It is clear that

$$\begin{aligned} H^s_\mathrm{ax}({\mathcal {S}})=\bigl \{\psi \in H^s({\mathcal {S}}): \widehat{\psi }^m_l=0, \, 1\le |m|\le l, \, l=1, 2, \dots \bigr \}. \end{aligned}$$

This section is the technical core of this paper. In fact, observe that, for the solvability of the nonlinear problem (1.7), we need to deal with the quadratic term \(|\nabla w|^2\). In other words, in the functional framework adopted, we must be sure that the product of two functions in the relevant space still belongs to the same space. We shall show that the subspace \(H^s_\mathrm{ax}({\mathcal {S}})\) enjoys this property, i.e. it is a Banach algebra with respect to the pointwise product.

To this aim, we recollect some notations and results about products of spherical harmonics. We recall (see [14, Appendix III]) that the product of two spherical harmonics \(Y_{l_1}^{m_1}, Y_{l_2}^{m_2}\) is represented by the formula:

$$\begin{aligned}&Y_{l_1}^{m_1}(\theta ,\varphi )Y_{l_2}^{m_2}(\theta ,\varphi )\\&\quad =\sum _{l=0}^\infty \!\sum _{m=-l}^{l}(-1)^m\sqrt{\frac{(2l_1\!+\!1)(2l_2\!+\!1)(2l\!+\!1)}{4\pi }} \!\begin{pmatrix} l_1 &{}\!\!\!l_2 &{}\! \!\!\!l\\ m_1 &{}\!\!\! m_2 &{}\!\!\!\! -m \end{pmatrix} \!\begin{pmatrix} l_1 &{}\! l_2 &{}\! l\\ 0 &{}\! 0 &{}\! 0 \end{pmatrix} Y_{l}^{m}(\theta ,\varphi ). \end{aligned}$$

In this formula, the so-called Wigner 3-j symbol is defined by

$$\begin{aligned} \begin{pmatrix} l_1 &{}\!\! l_2 &{}\!\! l_3\\ m_1 &{}\!\! m_2 &{}\!\! m_3 \end{pmatrix}= & {} \sqrt{\frac{(-l_1\!+\!l_2\!+\!l_3)!(l_1\!-\!l_2\!+\!l_3)!(l_1\!+\!l_2\!-\!l_3)!(l_3\!-\!m_3)!(l_3\!+\!m_3)!}{(l_1\!+\!l_2\!+\!l_3\!+\!1)!(l_1\!-\!m_1)!(l_1\!+\!m_1)!(l_2\!-\!m_2)!(l_2\!+\!m_2)!}} \\&\times \sum _k\frac{(-1)^{k+l_1+m_2-m_3}(l_2\!+\!l_3\!+\!m_1\!-\!k)!(l_1\!-\!m_1\!+\!k)!}{k!(-l_1\!+\!l_2\!+\!l_3\!-\!k)!(l_3\!-\!m_3\!-\!k)!(l_1\!-\!l_2\!+\!m_3\!+\!k)!}, \end{aligned}$$

if \(m_1+m_2+m_3=0\), \(|l_1-l_2|\le l_3\le l_1+l_2\), \(|m_1|\le l_1\), \(|m_2|\le l_2\) and \(|m_3|\le l_3\); the symbol is set to be zero otherwise. The summation in the formula is taken over all integers k for which all the factorials in the sum have nonnegative arguments.

By using the product formula, we find that the product uv of \(u,v\in H^s({\mathcal {S}})\) has the following Fourier-Laplace coefficients:

$$\begin{aligned} \widehat{uv}_l^m=\sum _{l_1=0}^\infty \sum _{m_1=-l_1}^{l_1}\sum _{l_2=0}^\infty \sum _{m_2=-l_2}^{l_2}\sqrt{(2l_1\!+\!1)(2l_2\!+\!1)(2l\!+\!1)}\,W^{l_1,l_2,l}_{m_1,m_2,m}\widehat{u}_{l_1}^{m_1}\widehat{v}_{l_2}^{m_2}, \end{aligned}$$

where

$$\begin{aligned} W^{l_1,l_2,l}_{m_1,m_2,m}:=\frac{(-1)^m}{\sqrt{4\pi }} \begin{pmatrix} l_1 &{}\!\! l_2 &{}\!\! l\\ m_1 &{}\!\! m_2 &{}\!\! -m \end{pmatrix} \!\begin{pmatrix} l_1 &{}\! l_2 &{}\! l\\ 0 &{} 0 &{} 0 \end{pmatrix}. \end{aligned}$$
(2.6)

Our proof of the Banach-algebra property of \(H^s_\mathrm{ax}({\mathcal {S}})\) is based on an l-sum relation (see [14, (7.61)]) satisfied by the Wigner 3-j symbols, that is

$$\begin{aligned} \sum _{l_j=0}^\infty (2l_j+1) \begin{pmatrix} l_1 &{}\!\! l_2 &{}\!\! l_3\\ m_1 &{}\!\! m_2 &{}\!\! m_3 \end{pmatrix} ^2=1 \quad (j=1,2,3), \end{aligned}$$

if \(m_1+m_2+m_3=0\) and \(|m_i|\le l_i\) for \(i\ne j\). This yields that

$$\begin{aligned} \begin{aligned} \sup _{l_2,m_2,l,m}\sum _{l_1=0}^\infty \sum _{m_1=-l_1}^{l_1}(2l_1+1)|W^{l_1,l_2,l}_{m_1,m_2,m}|&\le \frac{1}{\sqrt{4\pi }},\\ \sup _{l_1,m_1,l,m}\sum _{l_2=0}^\infty \sum _{m_2=-l_2}^{l_2}(2l_2+1)|W^{l_1,l_2,l}_{m_1,m_2,m}|&\le \frac{1}{\sqrt{4\pi }},\\ \sup _{l_1,m_1,l_2,m_2}\sum _{l=0}^\infty \sum _{m=-l}^{l}(2l+1)|W^{l_1,l_2,l}_{m_1,m_2,m}|&\le \frac{1}{\sqrt{4\pi }}, \end{aligned} \end{aligned}$$
(2.7)

by the Cauchy-Schwarz inequality.

We conclude our preliminaries with the following simple result.

Lemma 2.3

For any \(\sigma <s-1/2\), there is a constant \(C>0\) such that

$$\begin{aligned} \sum _{l=0}^\infty (2l+1)^\sigma |\widehat{u}_l^0|\le C\Vert u\Vert _{H^s({\mathcal {S}})} \end{aligned}$$

for any \(u\in H^s_\mathrm{ax}({\mathcal {S}})\).

Proof

The Cauchy-Schwarz inequality implies that

$$\begin{aligned} \sum _{l=0}^\infty (2l+1)^\sigma |\widehat{u}_l^0|\le \sqrt{\sum _{l=0}^\infty (2l+1)^{2(\sigma -s)}}\sqrt{\sum _{l=0}^\infty (2l+1)^{2s}|\widehat{u}_l^0|^2}, \end{aligned}$$

where the first series on the right hand side converges if \(2(\sigma -s)<-1\). \(\square \)

We are now ready to state and prove the main result of this section.

Theorem 2.4

(\(H^s_\mathrm{ax}({\mathcal {S}})\) is a Banach algebra). Let \(s>1\). If \(u,v\in H^s_\mathrm{ax}({\mathcal {S}})\), then \(uv\in H^s_\mathrm{ax}({\mathcal {S}})\) and

$$\begin{aligned} \Vert uv\Vert _{H^s({\mathcal {S}})}\le C\Vert u\Vert _{H^s({\mathcal {S}})}\Vert v\Vert _{H^s({\mathcal {S}})} \end{aligned}$$

for some constant \(C>0\) independent of u, v.

Proof

Let us simply write \(\widehat{u}_l, \widehat{v}_l, \widehat{uv}_l\) for \(\widehat{u}_l^0, \widehat{v}_l^0, \widehat{uv}_l^0\), respectively. Since \(W^{l_1,l_2,l}_{0,0,0}\) is nonzero only when \(l\le l_1+l_2\), and in this situation \(l_1\le l_2\) implies \(l\le 2l_2\), while \(l_1\ge l_2\) implies \(l\le 2l_1\), we have

$$\begin{aligned} (l+1)^s|\widehat{uv}_l|&\le \sqrt{2}\sum _{l_1=0}^\infty \sum _{l_2=l_1}^\infty \sqrt{2l_1+1}(2l_2+1)^{s}(l+1)|W^{l_1,l_2,l}_{0,0,0}||\widehat{u}_{l_1}||\widehat{v}_{l_2}|\\&\quad +\sqrt{2}\sum _{l_2=0}^\infty \sum _{l_1=l_2}^\infty (2l_1+1)^{s}\sqrt{2l_2+1}(l+1)|W^{l_1,l_2,l}_{0,0,0}||\widehat{u}_{l_1}||\widehat{v}_{l_2}|. \end{aligned}$$

In what follows, we denote the two summands in this formula by \(I_l\) and \(J_l\), respectively.

Now, Lemma 2.3 shows that

$$\begin{aligned} \sum _{l_1=0}^\infty \sqrt{2l_1+1}|\,\widehat{u}_{l_1}|\le C\Vert u\Vert _{H^s({\mathcal {S}})}, \quad \sum _{l_2=0}^\infty \sqrt{2l_2+1}|\,\widehat{v}_{l_2}|\le C\Vert v\Vert _{H^s({\mathcal {S}})}. \end{aligned}$$
(2.8)

Thus, we use Lemma A.3 with the settings

$$\begin{aligned} p_k= & {} \sqrt{2k+1}\,|\widehat{u}_k|, \quad q_k=(2k+1)^{s}|\widehat{v}_k|, \ \text{ and } \\ r_{i, j,k}= & {} {\left\{ \begin{array}{ll} (k+1)|W_{0,0,0}^{i, j ,k}| \ \text{ for } \ i\le j, \\ 0 \ \text{ for } \ i>j, \end{array}\right. } \end{aligned}$$

and, by recalling the l-sum relation (2.7), we see that

$$\begin{aligned} \sqrt{\sum _{l=0}^\infty |I_l|^2}\le \frac{C}{\sqrt{2\pi }}\Vert u\Vert _{H^s({\mathcal {S}})}\Vert v\Vert _{H^s({\mathcal {S}})}. \end{aligned}$$

A similar formula can be obtained for the terms \(J_l\). Therefore, the proof is completed. \(\square \)

For future reference, we conclude this section by showing that the algebra structure still holds for the entire space \(H^s({\mathcal {S}})\) if \(s>3/2\).

Proposition 2.5

If \(u,v\in H^s({\mathcal {S}})\) with \(s>3/2\), then \(uv\in H^s({\mathcal {S}})\) and

$$\begin{aligned} \Vert uv\Vert _{H^s({\mathcal {S}})}\le C\Vert u\Vert _{H^s({\mathcal {S}})}\Vert v\Vert _{H^s({\mathcal {S}})}. \end{aligned}$$

Proof

The proof runs similarly to that of Theorem 2.4. All what is needed is an extension of (2.8) to the case of two independent variables l and m. Indeed, we simply have that the inequality

$$\begin{aligned} \sum _{l=0}^\infty \sum _{m=-l}^l\sqrt{2l+1}|\,\widehat{u}_l^m|\le C\Vert u\Vert _{H^s({\mathcal {S}})} \end{aligned}$$

holds for \(u\in H^s({\mathcal {S}})\) if \(s>3/2\), as in the proof of Lemma 2.3. \(\square \)

2.3 The square of the gradient of a function in \(H^s_\mathrm{ax}({\mathcal {S}})\)

In the case of axially symmetric functions, we have that

$$\begin{aligned} |\nabla w|^2=w_r^2+w_\theta ^2. \end{aligned}$$

The following two lemmas will be decisive for the proof of existence for problem (1.1) of Sect. 4.

Lemma 2.6

Let u, v be harmonic functions in \(\Omega \), continuous up to the boundary \({\mathcal {S}}\), and such that \(u, v\rightarrow 0\) as \(|x|\rightarrow \infty \). If \(u,v\in H^{s+1}_\mathrm{ax}({\mathcal {S}})\) for \(s>1\), then \(u_r v_r\in H^s_\mathrm{ax}({\mathcal {S}})\) and

$$\begin{aligned} \Vert u_rv_r\Vert _{H^s({\mathcal {S}})}\le C\Vert u\Vert _{H^{s+1}({\mathcal {S}})}\Vert v\Vert _{H^{s+1}({\mathcal {S}})} \end{aligned}$$

for some \(C>0\) independent of u, v.

Proof

If \(u=\sum \limits _{l=0}^\infty r^{-l-1}\widehat{u}_l\,Y_l^0\), then \(u_r=-\sum \limits _{l=0}^\infty r^{-l-2} (l+1)\widehat{u}_l\,Y_l^0\), and similarly for v. Hence, Theorem 2.4 gives

$$\begin{aligned} \Vert u_rv_r\Vert _{H^s({\mathcal {S}})}&\le C\Vert u_r\Vert _{H^s({\mathcal {S}})}\Vert v_r\Vert _{H^s({\mathcal {S}})}\\&=C\sqrt{\sum _{l=0}^\infty (l+1)^{2s+2}|\widehat{u}_l|^2}\,\sqrt{\sum _{l=0}^\infty (l+1)^{2s+2}|\widehat{v}_l|^2}\\&=C\Vert u\Vert _{H^{s+1}({\mathcal {S}})}\Vert v\Vert _{H^{s+1}({\mathcal {S}})}, \end{aligned}$$

as desired. \(\square \)

We now turn to the estimate for \(v_\theta ^2\) in the \(H^s({\mathcal {S}})\)-norm. Unlike the case of \(v_r\), we do not have a simple expression of the spherical harmonics expansion of \(v_\theta \). Nevertheless, we will show that there is one for \(v_\theta ^2\), by using the fact that the Legendre polynomial \(P_l^0(z)\) of degree l, which we will denote here by \(P_l(z)\), solves the differential equation:

$$\begin{aligned} \left[ \left( 1-z^2\right) P_l'\right] '+l(l+1)P_l=0. \end{aligned}$$
(2.9)

Lemma 2.7

Let u, v be harmonic functions in \(\Omega \), continuous up to the boundary \({\mathcal {S}}\), and such that \(u, v\rightarrow 0\) as \(|x|\rightarrow \infty \). If \(u,v\in H^{s+1}_\mathrm{ax}({\mathcal {S}})\) for \(s>1\), then \(u_\theta v_\theta \in H^s_\mathrm{ax}({\mathcal {S}})\) and

$$\begin{aligned} \Vert u_\theta v_\theta \Vert _{H^s({\mathcal {S}})}\le C\Vert u\Vert _{H^{s+1}({\mathcal {S}})}\Vert v\Vert _{H^{s+1}({\mathcal {S}})} \end{aligned}$$

for some \(C>0\) independent of u, v.

Proof

Recall that

$$\begin{aligned} Y_l^0(\theta ,\varphi )=\sqrt{\frac{2l+1}{4\pi }}P_l(\sin \theta ). \end{aligned}$$

Hence, if \(u(\theta )=\sum \limits _{i=0}^\infty \widehat{u}_iY_i^0(\theta ,\varphi )\) and \(v(\theta )=\sum \limits _{j=0}^\infty \widehat{v}_jY_j^0(\theta ,\varphi )\), we infer that

$$\begin{aligned} u_\theta (\theta ) v_\theta (\theta )&=\sum _{i=0}^\infty \sum _{j=0}^\infty \widehat{u}_i\widehat{v}_j\sqrt{\frac{(2i+1)(2j+1)}{(4\pi )^2}}\cos ^2\theta \, P_i'(\sin \theta )\,P_j'(\sin \theta )\\&= \sum _{l=0}^\infty c_lY_l^0(\theta ,\varphi ), \end{aligned}$$

where

$$\begin{aligned} c_l&=2\pi \int _{-\frac{\pi }{2}}^{\frac{\pi }{2}} u_\theta (\theta ) v_\theta (\theta ) Y_l^0(\theta ,\varphi )\,\cos \theta \,d\theta \\&=2\pi \sum _{i=0}^\infty \sum _{j=0}^\infty \widehat{u}_i\widehat{v}_j\sqrt{\frac{(2i+1)(2j+1)(2l+1)}{(4\pi )^3}}\int _{-1}^1(1-z^2)P_i'(z)P_j'(z)P_l(z)\,dz. \end{aligned}$$

The last integral can be represented by the symbols \(W_{0,0,0}^{i,j,l}\) defined in (2.6). Indeed, (2.9) gives that

$$\begin{aligned} \left\{ \left( 1-z^2\right) \left[ P_i'(z)P_l(z)-P_i(z)P_l'(z)\right] \right] \}'P_j=\left[ l(l+1)-i(i+1)\right] P_i(z)P_j(z)P_l(z). \end{aligned}$$

An integration by parts then gives:

$$\begin{aligned} \left[ i(i+1)-l(l+1)\right] \int _{-1}^1P_iP_jP_l\,dz&=\int _{-1}^1\left( 1-z^2\right) \left( P_i'P_l-P_iP_l'\right) P_j'\,dz\\&=2\int _{-1}^1\left( 1-z^2\right) P_i'P_j'P_l\,dz\\&\quad \,-j(j+1)\int _{-1}^1P_iP_jP_l\,dz. \end{aligned}$$

Hence, we compute:

$$\begin{aligned} c_l&=\sum _{i, j=0}^\infty \widehat{u}_i\widehat{v}_j\left[ i(i\!+\!1)+j(j\!+\!1)\!-\!l(l\!+\!1)\right] \sqrt{\frac{(2i\!+\!1)(2j\!+\!1)(2l\!+\!1)}{64\,\pi }}\int _{-1}^1\!P_iP_jP_l\,dz\\&=\sum _{i, j=0}^\infty \frac{i(i\!+\!1)+j(j\!+\!1)\!-\!l(l\!+\!1)}{2}\sqrt{(2i\!+\!1)(2j\!+\!1)(2l\!+\!1)} \, \,W_{0,0,0}^{i,j,l}\,\widehat{u}_i\,\widehat{v}_j. \end{aligned}$$

Note that \(W_{0,0,0}^{i,j,l}\) is nonzero only when \(|i-j|\le l\le i+j\), so that we have:

$$\begin{aligned} -ij\le \frac{i(i+1)+j(j+1)-l(l+1)}{2}\le ij. \end{aligned}$$

Therefore, as in the proof of Theorem 2.4, we can split up the sum into two summands and obtain the inequality:

$$\begin{aligned} (l+1)^s|c_l|&\le \sum _{i=0}^\infty \sum _{j=i}^\infty (2i+1)^{3/2}(2j+1)^{s+1}(l+1)|W^{i,j,l}_{0,0,0}||\widehat{u}_{i}||\widehat{v}_{j}|\\&\quad +\sum _{j=0}^\infty \sum _{i=j}^\infty (2i+1)^{s+1}(2j+1)^{3/2}(l+1)|W^{i,j,l}_{0,0,0}||\widehat{u}_{i}||\widehat{v}_{j}|. \end{aligned}$$

The desired estimate then follows thanks to the same arguments as those in the proof of Theorem 2.4. \(\square \)

3 Series solution for the linearized problem

In this section, we collect the results on the linearized problem (1.9)–(1.10), which will be instrumental for the proof of our main theorem in Sect. 4.

In the spherical system of coordinates, we can compute that

$$\begin{aligned} d=\frac{\sin \theta }{r^2} \ \text{ and } \ \nabla u\cdot \nabla v=u_r \,v_r+\frac{u_\theta \, v_\theta }{r^2} +\frac{u_\varphi \, v_\varphi }{r^2 \cos ^2\theta }. \end{aligned}$$

Thus, problem (1.9)–(1.10) reads as

$$\begin{aligned}&\frac{1}{r^2}(r^2 v_r)_r +\frac{1}{r^2\cos \theta }(\cos \theta \, v_\theta )_\theta +\frac{1}{r^2\cos ^2 \theta }\,v_{\varphi \varphi }=0 \nonumber \\&\quad \text{ for } \ r\ge 1, \ -\pi /2\le \theta \le \pi /2, \ -\pi \le \varphi <\pi , \end{aligned}$$
(3.1a)
$$\begin{aligned}&-2\,\sin \theta \,v_r+\cos \theta \,v_\theta =\frac{1}{2}\,f \ \text{ for } \ r=1, \ -\pi /2\le \theta \le \pi /2, \ -\pi \le \varphi <\pi , \end{aligned}$$
(3.1b)
$$\begin{aligned}&v\rightarrow 0 \ \ \text{ uniformly } \text{ in } \ -\pi /2\le \theta \le \pi /2, \ -\pi \le \varphi <\pi \ \text{ as } \ r\rightarrow \infty , \end{aligned}$$
(3.1c)
$$\begin{aligned}&v=h \ \text{ for } \ r=1, \ \theta =0, \ -\pi \le \varphi <\pi . \end{aligned}$$
(3.1d)

The main result of this section is the following.

Theorem 3.1

(Unique existence and fractional Sobolev estimates). Suppose that \(f \in H^s({\mathcal {S}})\), with \(f_\varphi \in H^{s-1/2}({\mathcal {S}})\), and \(h \in H^{s+3/4}({\mathcal {E}})\) for some \(s>1\). Then (3.1) has a unique classical solution \(v \in C^2(\Omega ) \cap C^1(\overline{\Omega })\). Furthermore, the solution satisfies \(v|_{{\mathcal {S}}} \in H^{s+1}({\mathcal {S}})\) and

$$\begin{aligned} \Vert v\Vert _{H^{s+1}({\mathcal {S}})} \le C\left( \Vert f\Vert _{H^s({\mathcal {S}})} +\Vert f_\varphi \Vert _{H^{s-1/2}({\mathcal {S}})} +\Vert h\Vert _{H^{s+3/4}({\mathcal {E}})}\right) , \end{aligned}$$
(3.2)

where \(C>0\) is a constant independent of f and h.

In particular, if \(f \in H^s_\mathrm{ax}({\mathcal {S}})\) and h is a constant, then \(v|_{{\mathcal {S}}} \in H^{s+1}_\mathrm{ax}({\mathcal {S}})\) and

$$\begin{aligned} \Vert v\Vert _{H^{s+1}({\mathcal {S}})} \le C\left( \Vert f\Vert _{H^s({\mathcal {S}})} +|h|\right) . \end{aligned}$$

The proof of Theorem 3.1 is given in Sect. 3.2.

3.1 Formal derivation of a series solution

We start by formally deriving a representation formula of a solution v of problem (3.1) (computations here will be verified in Proposition 3.7 below). The formula is given by

$$\begin{aligned} v(r,\theta ,\varphi )=\sum _{l=0}^\infty \sum _{m=-l}^l (b_l^m +c_l^m) r^{-l-1} Y_l^m(\theta ,\varphi ), \end{aligned}$$
(3.3)

where \(b_l^m\) and \(c_l^m\) are defined as follows. We set

$$\begin{aligned} \beta _l^m&=\sqrt{\frac{(l-|m|)(l+|m|)}{(2l-1)(2l+1)}}\, ,\\ \gamma _l^m&=-\frac{(l+1)\,\beta _l^m}{3(l+2)\,\beta _{l+1}^m} =-\frac{l+1}{3(l+2)} \sqrt{\frac{(2l+3)(l-|m|)(l+|m|)}{(2l-1)(l+1-|m|)(l+1+|m|)}}\,\\&\text{ for } \ m=0,\pm 1,\dots , \ l=|m|,|m|+1,\dots , \end{aligned}$$

and put

$$\begin{aligned} \Gamma _0^m=1, \qquad \Gamma _k^m=\prod _{j=1}^k \gamma _{|m|+2j-1}^m \ \text{ for } \ m=0,\pm 1,\dots , \ k=1, 2, \dots . \end{aligned}$$

For \(f \in L^2({\mathcal {S}})\) and \(h \in L^2({\mathcal {E}})\), we write

$$\begin{aligned} a_l^m=\frac{1}{6(l+2)\beta _{l+1}^m}\,\widehat{f}_l^m, \quad {\tilde{a}}^m=\frac{\widehat{h} _m-\sum \limits _{k=0}^\infty b_{|m|+2k}^m \alpha _{|m|+2k}^m P_{|m|+2k}^{|m|} (0)}{\sum \limits _{k=0}^\infty \Gamma _k^m \alpha _{|m|+2k}^m P_{|m|+2k}^{|m|} (0)} \end{aligned}$$

(note that the denominator in the definition of \({\tilde{a}}^m\) is nonzero due to the inequality (3.8) of Lemma 3.3 below). \(b_l^m\) is then defined by the recurrence relation

$$\begin{aligned}&b_{|m|-1}^m=b_{|m|}^m=0&\quad&\text{ for } \ m=0,\pm 1,\dots ,\\&b_{l+1}^m=\gamma _l^m b_{l-1}^m+a_l^m&\quad&\text{ for } \ m=0,\pm 1,\dots , \ l=|m|,|m|+1,\dots , \end{aligned}$$

and \(c_l^m\) is given by

$$\begin{aligned} c_{|m|+2k-1}^m=0, \quad c_{|m|+2k}^m=\Gamma _k^m \,{\tilde{a}}^m \ \text{ for } \ m=0,\pm 1,\dots , \ k=0,1,\dots . \end{aligned}$$

Let us derive (3.3). Let \(\widehat{v}_l^m(r)\) denote the Fourier-Laplace coefficients of \(v(r,\cdot ,\cdot )\), that is,

$$\begin{aligned} \widehat{v}_l^m(r) =\left\langle v(r,\cdot ,\cdot ), Y_l^m \right\rangle _{L^2({\mathcal {S}})} =\int _{-\pi }^\pi \int _{-\frac{\pi }{2}}^{\frac{\pi }{2}} v(r,\theta ,\varphi ) \, \overline{Y_l^m(\theta ,\varphi )}\cos \theta \, d\theta d\varphi . \end{aligned}$$

For abbreviation, we write \(\widehat{v}_l^m\) instead of \(\widehat{v}_l^m(1)\) and, for convenience, we set \(\widehat{v}_l^{m}=0\) if \(l<|m|\). First, we observe that v has the form

$$\begin{aligned} v(r,\theta ,\varphi )=\sum _{l=0}^\infty \sum _{m=-l}^l \widehat{v}_l^m r^{-l-1} Y_l^m(\theta ,\varphi ). \end{aligned}$$
(3.4)

This follows from (3.1a) and (3.1c). Indeed, multiplying (3.1a) by \(\overline{Y_l^m}\), integrating over \({\mathcal {S}}\) and using the fact that \(-\Delta _{{\mathcal {S}}} Y_l^m=l(l+1)Y_l^m\), we see that \(\widehat{v}_l^m(r)\) satisfies

$$\begin{aligned} \frac{d^2\widehat{v}_l^m}{dr^2}(r)+\frac{2}{r}\frac{d\widehat{v}_l^m}{dr}(r) -\frac{l(l+1)}{r^2}\widehat{v}_l^m(r)=0 \quad \text{ for } r>1. \end{aligned}$$

This together with the condition (3.1c) gives \(\widehat{v}_l^m(r)=\widehat{v}_l^m r^{-l-1}\), and hence we obtain (3.4).

Next, we derive a recurrence relation for \(\widehat{v}_l^m\) from (3.1b). It is known that the following recurrence relations hold (see [3, (15.88)] for the first equality and [3, (15.88), (15.89), (15.92)] for the second equality):

$$\begin{aligned} zP_l^{|m|} (z)= & {} \frac{l-|m|+1}{2l+1}P_{l+1}^{|m|} (z) +\frac{l+|m|}{2l+1}P_{l-1}^{|m|} (z),\\ \left( 1-z^2\right) \frac{d P_l^{|m|}}{dz} (z)= & {} -\frac{l(l-|m|+1)}{2l+1}P_{l+1}^{|m|} (z) +\frac{(l+1)(l+|m|)}{2l+1}P_{l-1}^{|m|} (z). \end{aligned}$$

Hence we have that

$$\begin{aligned} \sin \theta \, Y_l^m= & {} \beta _{l+1}^mY_{l+1}^m+\beta _l^mY_{l-1}^m,\\ \cos \theta \, \frac{\partial Y_l^m}{\partial \theta }= & {} -l\beta _{l+1}^mY_{l+1}^m+(l+1)\beta _l^mY_{l-1}^m. \end{aligned}$$

Here, \(Y_l^m=0\) if \(l<|m|\). From these identities, we see that

$$\begin{aligned} \langle \sin \theta \,v_r,Y_l^m \rangle _{L^2({\mathcal {S}})}&=\int _{-\pi }^\pi \int _{-\frac{\pi }{2}}^{\frac{\pi }{2}} v_r(r,\theta ,\varphi ) \, \sin \theta \, \overline{Y_l^m(\theta ,\varphi )} \cos \theta \, d\theta d\varphi \\&=\frac{d}{dr} \left( \beta _{l+1}^m \widehat{v}_{l+1}^m(r) +\beta _l^m \widehat{v}_{l-1}^m(r) \right) \\&=-(l+2)\beta _{l+1}^m \widehat{v}_{l+1}^m r^{-l-3} -l \beta _l^m \widehat{v}_{l-1}^m r^{-l-1}, \end{aligned}$$

and

$$\begin{aligned}&\langle \cos \theta \,v_\theta ,Y_l^m \rangle _{L^2({\mathcal {S}})}\\&\quad =\int _{-\pi }^\pi \int _{-\frac{\pi }{2}}^{\frac{\pi }{2}} v_\theta (r,\theta ,\varphi ) \, \overline{Y_l^m(\theta ,\varphi )} \cos ^2 \theta \, d\theta d\varphi \\&\quad =\int _{-\pi }^\pi \int _{-\frac{\pi }{2}}^{\frac{\pi }{2}} v(r,\theta ,\varphi ) \, \left( \cos \theta \, \overline{\frac{\partial Y_l^m}{\partial \theta }(\theta ,\varphi )} -2\sin \theta \overline{Y_l^m(\theta ,\varphi )} \right) \cos \theta \, d\theta d\varphi \\&\quad =(l+2)\beta _{l+1}^m \widehat{v}_{l+1}^m r^{-l-2} -(l-1) \beta _l^m \widehat{v}_{l-1}^m r^{-l}, \end{aligned}$$

where we used integration by parts. Thus, multiplying (3.1b) by \(\overline{Y_l^m}\) and integrating over \({\mathcal {S}}\), we find the recurrence relation

$$\begin{aligned} \widehat{v}_{l+1}^m=\gamma _l^m \widehat{v}_{l-1}^m+a_l^m \ \text{ for } \ m=0,\pm 1,\dots , \ l=|m|,|m|+1,\dots . \end{aligned}$$

Finally, we consider the condition (3.1d). Interchanging the sum in (3.4), we have that

$$\begin{aligned} v(1,\theta ,\varphi )=\sum _{m=-\infty }^\infty \widehat{v}_m(\theta ) e^{im\varphi }, \quad \text{ where } \quad \widehat{v}_m(\theta )=\sum _{l=|m|}^\infty \widehat{v}_l^m \alpha _l^m P_l^m(\sin \theta ). \end{aligned}$$

Hence, by (3.1d), we deduce that

$$\begin{aligned} \widehat{h}_m=\widehat{v}_m(0)=\sum _{l=|m|}^\infty \widehat{v}_l^m \alpha _l^m P_l^m(0). \end{aligned}$$
(3.5)

The recurrence relations for \(b_l^m\) and \(\widehat{v}_l^m\) show that \(d_l^m=\widehat{v}_l^m-b_l^m\) satisfies

$$\begin{aligned}&d_{|m|-1}^m=0, \ d_{|m|}^m=\widehat{v}_{|m|}^m&\quad&\text{ for } \ m=0,\pm 1,\dots , \\&d_{l+1}^m=\gamma _l^m d_{l-1}^m&\quad&\text{ for } \ m=0,\pm 1,\dots , \ l=|m|,|m|+1,\dots , \end{aligned}$$

and therefore \(d_{|m|+2k-1}^m=0\) and \(d_{|m|+2k}^m=\Gamma _k^m \,\widehat{v\,}_{|m|}^m\). Plugging \(\widehat{v}_l^m=b_l^m+d_l^m\) into (3.5) and using the fact that \(P_l^{|m|} (0)=0\) if \(l-|m|\) is odd (see [3, (15.96)]), we find that

$$\begin{aligned} \widehat{h}_m=\sum _{k=0}^\infty b_{|m|+2k}^m \alpha _{|m|+2k}^m P_{|m|+2k}^{|m|} (0) +\widehat{v}_{|m|}^m \sum _{k=0}^\infty \Gamma _k^m \alpha _{|m|+2k}^m P_{|m|+2k}^{|m|} (0). \end{aligned}$$

This gives \(d_l^m=c_l^m\), and thus we obtain (3.3).

Remark 3.2

Suppose that f and h are independent of \(\varphi \), that is, \(\widehat{f}_l^m=\widehat{h}_m=0\) if \(m \ne 0\). Then, by definition, we see that \(b_l^m=c_l^m=0\) unless \(m=0\). This shows that the function v defined by (3.3) is also independent of \(\varphi \).

3.2 Proof of Theorem 3.1

We divide the proof of Theorem 3.1 into a sequence of lemmas. First we show some estimates to be mainly used in deriving (3.2).

Lemma 3.3

The inequalities

$$\begin{aligned}&c\,\root 4 \of {\frac{|m|+2k+1}{2k+1}}\nonumber \\&\quad \le (-1)^{\frac{|m|-m}{2}+k} \alpha _{|m|+2k}^m P_{|m|+2k}^{|m|} (0) \le C\,\root 4 \of {\frac{|m|+2k+1}{2k+1}}, \end{aligned}$$
(3.6)
$$\begin{aligned}&c\,\root 4 \of {\frac{|m|+1}{(2k+1)(|m|+2k+1)}} \le (-3)^k\Gamma _k^m \le C\,\root 4 \of {\frac{|m|+1}{(2k+1)(|m|+2k+1)}}, \qquad \end{aligned}$$
(3.7)
$$\begin{aligned}&c\,\root 4 \of {|m|+1} \le \left| \sum _{k=0}^\infty \Gamma _k^m \alpha _{|m|+2k}^m P_{|m|+2k}^{|m|} (0)\right| \le C\,\root 4 \of {|m|+1} \end{aligned}$$
(3.8)

hold with some positive constants c and C.

Proof

It is known (see [3, (15.96)]) that

$$\begin{aligned} P_{|m|+2k}^{|m|} (0)=(-1)^{|m|+k}\frac{(2|m|+2k-1)!!}{(2k)!!}, \end{aligned}$$

which gives:

$$\begin{aligned}&(-1)^{\frac{|m|-m}{2}+k} \alpha _{|m|+2k}^m P_{|m|+2k}^{|m|} (0)\\&\quad =\frac{\sqrt{(2k)!}}{(2k)!!} \frac{(2|m|+2k+1)!!}{\sqrt{(}2|m|+2k+1)!}\, \sqrt{\frac{2|m|+4k+1}{4\pi (2|m|+2k+1)}}. \end{aligned}$$

Also, by definition, we have that

$$\begin{aligned} (-3)^k\Gamma _k^m= & {} \sqrt{\frac{2|m|+4k+1}{2|m|+1}}\, \prod _{j=1}^k \frac{|m|+2j}{|m|+2j+1}\, \sqrt{\frac{2j-1}{2j} \frac{2|m|+2j-1}{2|m|+2j}}\\= & {} \frac{(|m|\!+\!1)!! \,(|m|\!+\!2k)!!}{|m|!! \, (|m|\!+\!2k\!+\!1)!!}\,\sqrt{\frac{2|m|\!+\!4k\!+\!1}{2|m|\!+\!1}}\\&\times \sqrt{\frac{(2k\!-\!1)!!}{(2k)!!} \frac{(2|m|)!!}{(2|m|\!-\!1)!!} \frac{(2|m|\!+\!2k\!-\!1)!!}{(2|m|\!+\!2k)!!}}. \end{aligned}$$

Therefore (3.6) and (3.7) follow from Lemma A.1 and some simple estimates. The inequality (3.8) is derived easily by (3.6) and (3.7). \(\square \)

Next, we derive an estimate of the \(H^s({\mathcal {S}})\)-norm of the formal solution (3.3). For \(a \in \mathbb {R}\), we write \(a_+=\max \{ a,0\}\).

Lemma 3.4

(Fractional Sobolev estimates for v). Suppose that \(f \in H^s({\mathcal {S}})\), \(f_\varphi \in H^{(s-1/2)_+}({\mathcal {S}})\) and \(h \in H^{s+3/4}({\mathcal {E}})\) for some \(s \ge 0\). Then the function v defined by (3.3) satisfies \(v|_{{\mathcal {S}}} \in H^{s+1}({\mathcal {S}})\) and

$$\begin{aligned} \Vert v\Vert _{H^{s+1}({\mathcal {S}})} \le C\left( \Vert f\Vert _{H^s({\mathcal {S}})} +\Vert f_\varphi \Vert _{H^{(s-1/2)_+}({\mathcal {S}})} +\Vert h\Vert _{H^{s+3/4}({\mathcal {E}})}\right) \end{aligned}$$
(3.9)

for some constant \(C>0\) independent of f and h.

Proof

Throughout the proof, c and C denote generic positive constants depending only on s, which may change from formula to formula.

Notice that, since \((Y_l^m)_\varphi =im Y_l^m\), we have that \( \widehat{(f_\varphi )}_l^m =im \widehat{f}_l^m, \) and hence

$$\begin{aligned} \Vert f_\varphi \Vert _{H^s({\mathcal {S}})}^2=\sum _{l=0}^\infty \sum _{m=-l}^l m^2 (l+1)^{2s} |{\widehat{f}}_l^m|^2. \end{aligned}$$
(3.10)

Next, we write \(v|_{{\mathcal {S}}}\) as

$$\begin{aligned} v|_{{\mathcal {S}}}=v_1+v_2, \quad v_1=\sum _{l=0}^\infty \sum _{m=-l}^l b_l^m Y_l^m, \quad v_2=\sum _{l=0}^\infty \sum _{m=-l}^l c_l^m Y_l^m, \end{aligned}$$

and estimate the norms of \(v_1\) and \(v_2\) separately. We first consider \(v_1\). It is elementary to show that \(|\gamma _l^m| \le 2/3\), and hence \(|b_{l+1}^m| \le 2|b_{l-1}^m|/3 +|a_l^m|\). Thus, applying Lemma A.2 with \(p_k=|b_{|m|+2k}^m|\), \(q_k=|a_{|m|+2k+1}^m|\), \(\sigma =2/3\), \(\tau _1=2(s+1)\), \(\tau _2=0\), and \(\chi =|m|\) gives:

$$\begin{aligned} \sum _{k=0}^\infty (|m|+2k+1)^{2(s+1)} \left| b_{|m|+2k}^m\right| ^2 \le C\sum _{k=0}^\infty (|m|+2k+1)^{2(s+1)} \left| a_{|m|+2k+1}^m\right| ^2. \end{aligned}$$

We use Lemma A.2 again with \(p_k=|b_{|m|+2k-1}^m|\) and \(q_k=|a_{|m|+2k}^m|\), and combine the resulting inequality with the above inequality to obtain that

$$\begin{aligned} \sum _{l=|m|}^\infty (l+1)^{2(s+1)} |b_l^m|^2 \le C\sum _{l=|m|}^\infty (l+1)^{2(s+1)} |a_l^m|^2. \end{aligned}$$

Since

$$\begin{aligned} (l+1)^{2(s+1)} |a_l^m|^2&\le \frac{(l+1)^{2(s+1)}}{(l+1-|m|)(l+1+|m|)} |\widehat{f}_l^m|^2\\&=(l+1)^{2s} |\widehat{f}_l^m|^2 +\frac{m^2 (l+1)^{2s}}{(l+1-|m|)(l+1+|m|)} |\widehat{f}_l^m|^2\\&\le (l+1)^{2s} |\widehat{f}_l^m|^2 +m^2 (l+1)^{(2s-1)_+} |\widehat{f}_l^m|^2, \end{aligned}$$

we find

$$\begin{aligned} \sum _{l=|m|}^\infty (l+1)^{2(s+1)} |b_l^m|^2 \le C\sum _{l=|m|}^\infty (l+1)^{2s} |\widehat{f}_l^m|^2 +C\sum _{l=|m|}^\infty m^2 (l+1)^{(2s-1)_+} |\widehat{f}_l^m|^2. \end{aligned}$$

By taking the sum over m and applying (3.10), we conclude that

$$\begin{aligned} \Vert v_1\Vert _{H^{s+1}({\mathcal {S}})} \le C\left( \Vert f\Vert _{H^s({\mathcal {S}})} +\Vert f_\varphi \Vert _{H^{(s-1/2)_+}({\mathcal {S}})}\right) . \end{aligned}$$
(3.11)

In order to examine \(v_2\), we estimate \({\tilde{a}}^m\). We see from (3.6) and the Cauchy-Schwarz inequality that

$$\begin{aligned}&\left| \sum _{k=0}^\infty b_{|m|+2k}^m \alpha _{|m|+2k}^m P_{|m|+2k}^{|m|} (0)\right| \\&\quad \le C \sum _{k=0}^\infty \root 4 \of {\frac{|m|+2k+1}{2k+1}} \left| b_{|m|+2k}^m \right| \\&\quad \le C \sqrt{\sum _{k=0}^\infty \frac{1}{(2k+1)^{3/2}}} \sqrt{\sum _{k=0}^\infty (2k+1) \sqrt{|m|+2k+1} \left| b_{|m|+2k}^m \right| ^2 }. \end{aligned}$$

Applying Lemma A.2 with \(p_k=|b_{|m|+2k}^m|\), \(q_k=|a_{|m|+2k+1}^m|\), \(\sigma =2/3\), \(\tau _1=1/2\), \(\tau _2=1\), and \(\chi =|m|\) yields that

$$\begin{aligned} \sum _{l=0}^\infty (2k\!+\!1) \sqrt{|m|\!+\!2k\!+\!1}\left| b_{|m|\!+\!2k}^m \right| ^2&\le C\sum _{k=0}^\infty (2k+1) \sqrt{|m|+2k+1} \left| a_{|m|+2k+1}^m \right| ^2\\&\le C\sum _{l=|m|}^\infty \sqrt{l+1}\, (l+1-|m|) |a_l^m|^2\\&\le C\sum _{l=|m|}^\infty \frac{1}{\sqrt{l+1}}\, |\widehat{f}_l^m|^2. \end{aligned}$$

Hence

$$\begin{aligned} \left| \sum _{k=0}^\infty b_{|m|+2k}^m \alpha _{|m|+2k}^m P_{|m|+2k}^{|m|} (0)\right| ^2 \le C\sum _{l=|m|}^\infty \frac{1}{(l+1)^{1/2}} |\widehat{f}_l^m|^2. \end{aligned}$$

From this and (3.8), we obtain that

$$\begin{aligned} |{\tilde{a}}^m|^2 \le \frac{C}{\sqrt{|m|+1}} \left[ |\widehat{h}_m|^2 +\sum _{l=|m|}^\infty \frac{1}{\sqrt{l+1}}\, |\widehat{f}_l^m|^2 \right] . \end{aligned}$$
(3.12)

Now, we are ready to estimate \(v_2\). Note that

$$\begin{aligned} \Vert v_2\Vert _{H^{s+1}({\mathcal {S}})}^2&=\sum _{l=0}^\infty \sum _{m=-l}^l (l+1)^{2(s+1)} |c_l^m|^2\\&=\sum _{m=-\infty }^\infty \left[ \sum _{k=0}^\infty (|m|+2k+1)^{2(s+1)} (\Gamma _k^m)^2 \right] |{\tilde{a}}^m|^2. \end{aligned}$$

By (3.7) and the fact that \(|m|+1 \le |m|+2k+1 \le (|m|+1)(2k+1)\), we have

$$\begin{aligned} \sum _{k=0}^\infty (|m|+2k+1)^{2(s+1)} (\Gamma _k^m)^2&\le C\sum _{k=0}^\infty \frac{[(|m|+1)(2k+1)]^{2(s+1)}}{3^{2k}}\\&\le C(|m|+1)^{2(s+1)}. \end{aligned}$$

This, together with (3.12), gives

$$\begin{aligned} \Vert v_2\Vert _{H^{s+1}({\mathcal {S}})}^2\le & {} C\sum _{m=-\infty }^\infty (|m|+1)^{2s+\frac{3}{2}} |\widehat{h}_m|^2 \\&\quad +C\sum _{m=-\infty }^\infty \sum _{l=|m|}^\infty \frac{(|m|+1)^{2s+3/2}}{\sqrt{l+1}} |\widehat{f}_l^m|^2. \end{aligned}$$

Since

$$\begin{aligned} (|m|+1)^{2s+\frac{3}{2}} \le 2(m^2+1) (l+1)^{2s-\frac{1}{2}} \le 2m^2(l+1)^{(2s-1)_+ +\frac{1}{2}} +2(l+1)^{2s+\frac{1}{2}}, \end{aligned}$$

we deduce that

$$\begin{aligned} \Vert v_2\Vert _{H^{s+1}({\mathcal {S}})}^2&\le C\sum _{m=-\infty }^\infty (|m|\!+\!1)^{2s+\frac{3}{2}} |\widehat{h}_m|^2\\&\quad +C\sum _{m=-\infty }^\infty \sum _{l=|m|}^\infty \left[ m^2(l\!+\!1)^{(2s-1)_+} +(l\!+\!1)^{2s} \right] |\widehat{f}_l^m|^2\\&=C\left( \Vert h\Vert _{H^{s+3/4}({\mathcal {E}})} +\Vert f_\varphi \Vert _{H^{(s-1/2)_+}({\mathcal {S}})} +\Vert f\Vert _{H^s({\mathcal {S}})}\right) . \end{aligned}$$

Combining this and (3.11), we obtain (3.9). Thus the proof is completed. \(\square \)

In the next lemma we check that, under the assumptions of Theorem 3.1, the function v given by (3.3) is indeed a classical solution of (3.1).

Lemma 3.5

(Regularity of v). Suppose that \(f \in H^s({\mathcal {S}})\), \(f_\varphi \in H^{s-1/2}({\mathcal {S}})\) and \(h \in H^{s+3/4}({\mathcal {E}})\) for some \(s>1\), and let v be defined by (3.3). Then v belongs to \(C^2 (\Omega ) \cap C^1(\overline{\Omega })\) and satisfies (3.1) in the classical sense.

Proof

We know that the Fourier-Laplace coefficients \(\widehat{v}_l^m\) of \(v|_{{\mathcal {S}}}\) is given by \(\widehat{v}_l^m=b_l^m+c_l^m\). From (2.2) and the Cauchy-Schwarz inequality, we have that

$$\begin{aligned}&\sum _{l=0}^\infty \sum _{m=-l}^l |\widehat{v}_l^m r^{-l-1} Y_l^m(\theta ,\varphi )|\nonumber \\&\quad \le r^{-1}\sum _{l=0}^\infty \sum _{m=-l}^l |\widehat{v}_l^m Y_l^m(\theta ,\varphi )| \nonumber \\&\quad \le r^{-1}\Vert v\Vert _{H^{s+1}({\mathcal {S}})} \sqrt{\sum _{l=0}^\infty \sum _{m=-l}^l(l+1)^{-2(s+1)}|Y_l^m(\theta ,\varphi )|^2} \nonumber \\&\quad \le r^{-1}\Vert v\Vert _{H^{s+1}({\mathcal {S}})} \sqrt{\frac{1}{2\pi }\sum _{l=0}^\infty (l+1)^{-1-2s}}. \end{aligned}$$
(3.13)

Lemma 3.4 and the fact that \(s>0\) show that the right-hand side is bounded by a constant independent of \((r,\theta ,\varphi )\). Hence the series on the left-hand side converges uniformly on \(\overline{\Omega }\). Since it is well-known that a uniform limit of a sequence of harmonic functions is smooth and harmonic, we deduce that v is of class \(C^2\) in \(\Omega \) and satisfies (3.1a). In addition, from (3.13), we see at once that (3.1c) is satisfied.

Next, we show that \(v \in C^1(\overline{\Omega })\) and that (3.1b) and (3.1d) hold. It suffices to show that the series

$$\begin{aligned} \begin{aligned}&\sum _{l=0}^\infty \sum _{m=-l}^l (l+1)\widehat{v}_l^m r^{-l-2} Y_l^m(\theta ,\varphi ) \left( =\sum _{l=0}^\infty \sum _{m=-l}^l \frac{\partial }{\partial r}\left( \widehat{v}_l^m r^{-l-2} Y_l^m(\theta ,\varphi )\right) \right) ,\\&\quad \sum _{l=0}^\infty \sum _{m=-l}^l \widehat{v}_l^m r^{-l-1} \frac{\partial Y_l^m}{\partial \theta }(\theta ,\varphi ), \qquad \sum _{l=0}^\infty \sum _{m=-l}^l \widehat{v}_l^m r^{-l-1} \frac{1}{\cos \theta } \frac{\partial Y_l^m}{\partial \varphi }(\theta ,\varphi ) \end{aligned}\nonumber \\ \end{aligned}$$
(3.14)

are uniformly convergent on \(\overline{\Omega }\). A computation similar to that in the derivation of (3.13) gives

$$\begin{aligned} \sum _{l=0}^\infty \sum _{m=-l}^l |(l+1)\widehat{v}_l^m r^{-l-2} Y_l^m(\theta ,\varphi )| \le \Vert v\Vert _{H^{s+1}({\mathcal {S}})} \sqrt{\frac{1}{2\pi }\sum _{l=0}^\infty (l+1)^{1-2s}}. \end{aligned}$$

Since the right-hand side is finite if \(s>1\), we deduce that the first series of (3.14) converges uniformly. For the second and third series, we use the identities

$$\begin{aligned} \sum _{m=-l}^l \left| \frac{\partial Y_l^m}{\partial \theta }(\theta ,\varphi ) \right| ^2 =\sum _{m=-l}^l \frac{1}{\cos ^2 \theta } \left| \frac{\partial Y_l^m}{\partial \varphi }(\theta ,\varphi ) \right| ^2 =\frac{l(l+1)(2l+1)}{8\pi }, \end{aligned}$$

which are derived by operating \(\partial ^2/\partial \theta _1 \partial \theta _2\) or \(\partial ^2/\partial \varphi _1 \partial \varphi _2\) to the equality (2.1) and then taking \((\theta _1,\varphi _1)=(\theta _2,\varphi _2)=(\theta ,\varphi )\). From these identities and the Cauchy-Schwarz inequality, we have that

$$\begin{aligned}&\sum _{l=0}^\infty \sum _{m=-l}^l \left| \widehat{v}_l^m r^{-l-1} \frac{\partial Y_l^m}{\partial \theta }(\theta ,\varphi )\right| +\sum _{l=0}^\infty \sum _{m=-l}^l \left| \widehat{v}_l^m r^{-l-1} \frac{1}{\cos \theta } \frac{\partial Y_l^m}{\partial \varphi }(\theta ,\varphi )\right| \\&\quad \le \Vert v\Vert _{H^{s+1}({\mathcal {S}})} \sqrt{\sum _{l=0}^\infty \sum _{m=-l}^l (l+1)^{-2(s+1)} \left| \frac{\partial Y_l^m}{\partial \theta }(\theta ,\varphi ) \right| ^2}\\&\qquad +|v\Vert _{H^{s+1}({\mathcal {S}})} \sqrt{\sum _{l=0}^\infty \sum _{m=-l}^l (l+1)^{-2(s+1)} \frac{1}{\cos ^2 \theta } \left| \frac{\partial Y_l^m}{\partial \varphi }(\theta ,\varphi ) \right| ^2}\\&\quad \le \Vert v\Vert _{H^{s+1}({\mathcal {S}})} \sqrt{\frac{1}{\pi }\sum _{l=0}^\infty (l+1)^{1-2s}}. \end{aligned}$$

This shows that the second and third series of (3.14) are uniformly convergent, and thus the assertion follows. \(\square \)

Finally, we prove the uniqueness of a solution of (3.1).

Lemma 3.6

Let \(v_1,v_2 \in C^2(\Omega ) \cap C^1(\overline{\Omega })\) satisfy (3.1). Then \(v_1=v_2\) on \(\overline{\Omega }\).

Proof

Although the lemma can be shown in the same way as [10, Theorem], we give a proof for readers’ convenience.

We know that the function \(w=v_1-v_2\) satisfies

$$\begin{aligned} \Delta w=0 \ \text { in } \ \Omega , \quad \nabla d\cdot \nabla w=0 \ \text { on } \ {\mathcal {S}}, \quad w\rightarrow 0 \ \text { as } \ |x|\rightarrow \infty , \quad w=0 \ \text { on } \ {\mathcal {E}}. \nonumber \\ \end{aligned}$$
(3.15)

We show that \(w \le 0\) on \(\overline{\Omega }\). On the contrary, suppose that w is positive somewhere. Then, we can take a point \(x_0 \in \overline{\Omega }\) such that \(w(x_0)=\sup _{\overline{\Omega }} w>0\), since w decays at infinity. By the last condition of (3.15), we have either \(x_0 \in \Omega \) or \(x_0 \in {\mathcal {S}}\setminus {\mathcal {E}}\). Assume \(x_0 \in \Omega \). Then, since w is harmonic in \(\Omega \) and vanishes on \({\mathcal {E}}\), we see from the strong maximum principle that \(w(x_0)=0\), a contradiction. Assume \(x_0 \in {\mathcal {S}}\setminus {\mathcal {E}}\). In this case, we note that the tangential derivative of w on \({\mathcal {S}}\) vanishes at \(x_0\). From (1.8) and the second condition of (3.15), it follows that

$$\begin{aligned} \left. \frac{\partial w}{\partial \nu }\right| _{x=x_0} =\left. \frac{1}{2x_3}\left( \nabla d \cdot \nabla w -\tau \cdot \nabla w\right) \right| _{x=x_0}=0. \end{aligned}$$

Hence the Hopf lemma and the last condition of (3.15) give \(w(x_0)=0\), which is impossible. Consequently w is nonpositive everywhere. The fact that \(w \ge 0\) can be shown in the same way, and therefore we obtain \(w=0\). \(\square \)

We can now prove Theorem 3.1.

Proof of Theorem 3.1

The unique existence of a solution of (3.1) follows from Lemmas 3.5 and 3.6 . The inequality (3.2) is a direct consequence of Lemma 3.4. From Remark 3.2, we see that the solution v satisfies \(v|_{{\mathcal {S}}} \in H^{s+1}_\mathrm{ax}({\mathcal {S}})\) if \(f \in H^s_\mathrm{ax}({\mathcal {S}})\) and h is a constant. Therefore the proof is completed. \(\square \)

3.3 Validity of the formula (3.3)

We have proved in Theorem 3.1 that any classical solution of (3.1) is given by the formula (3.3), provided that f and h are in certain Sobolev spaces. At the end of this section, we prove that this is still true without assuming extra regularity conditions on f and h.

Proposition 3.7

Any solution \(v \in C^2(\Omega ) \cap C^1(\overline{\Omega })\) of (3.1) is of the form (3.3).

Proof

Most of computations in Sect. 3.1 are valid, since the assumption \(v \in C^2(\Omega ) \cap C^1(\overline{\Omega })\) implies that \(\widehat{v}_l^m(r) \in C^2((1,\infty )) \cap C^1([1,\infty ))\), \(f \in C({\mathcal {S}})\) and \(h \in C^1({\mathcal {E}})\). The only point where we have to verify is (3.5). In order to ensure (3.5), we need to show that the equality (3.1) holds in \(L^2({\mathcal {E}})\) for \(\theta =0\). For this purpose, we use the inequality (see [11, Corollary 1])

$$\begin{aligned} \big | P_l^{|m|}(z)\big | \le \root 4 \of {\frac{64}{\pi ^3 (2l+1)\sqrt{1-z^2}}} \sqrt{\frac{(l+|m|)!}{(l-|m|)!}}, \end{aligned}$$

which gives

$$\begin{aligned} \big | \alpha _l^m P_l^{|m|} (\sin \theta )\big | \le \root 4 \of {\frac{4(2l+1)}{\pi ^5 \cos \theta }}. \end{aligned}$$

From this and the Cauchy-Schwarz inequality, we have that

$$\begin{aligned} |\widehat{v}_m(\theta )|^2&\le \sqrt{\frac{8}{\pi ^5 \cos \theta }}\left( \sum _{l=|m|}^\infty \root 4 \of {l+1} |\widehat{v}_l^m| \right) ^2\\&\le \sqrt{\frac{8}{\pi ^5 \cos \theta }}\left[ \sum _{l=0}^\infty \frac{1}{(l+1)^{3/2}} \right] \left[ \sum _{l=|m|}^\infty (l+1)^2|\widehat{v}_l^m|^2\right] . \end{aligned}$$

Since Proposition 2.2 and the assumption \(v|_{{\mathcal {S}}} \in C^1({\mathcal {S}})\) show that

$$\begin{aligned} \sum _{m=-\infty }^\infty \sum _{l=|m|}^\infty (l+1)^2|\widehat{v}_l^m|^2 =\Vert v\Vert _{H^1({\mathcal {S}})}<\infty , \end{aligned}$$

we see from the Weierstrass M-test that the series \(\sum _{m=-\infty }^\infty |\widehat{v}_m(\theta )|^2\) converges locally uniformly in \(\theta \in (-\pi /2,\pi /2)\). This means that the right-hand side of (3.1) is convergent in \(L^2({\mathcal {E}})\) locally uniformly in \(\theta \in (-\pi /2,\pi /2)\). Since we know that (3.1) holds in \(L^2({\mathcal {S}})\), we conclude that (3.1) is valid in \(L^2({\mathcal {E}})\) for every \(\theta \in (-\pi /2,\pi /2)\). Thus (3.5) is verified, and the proof is completed. \(\square \)

4 Axially symmetric solutions of Backus problem

In this section, we finally prove the existence of axially symmetric solutions of Backus problem (1.1) near the dipole.

Remark 4.1

Notice that, since \(|\nabla d(x)|=\sqrt{1+3\,x_3^2}\) for \(x\in {\mathcal {S}}\), then \(|\nabla d| \in H^s({\mathcal {S}})\) for any s, thanks to Proposition 2.2, being as \(|\nabla d|\in C^\infty ({\mathcal {S}})\).

Lemma 4.2

Let u, v be harmonic functions in \(\mathbb {R}^N\setminus \overline{B}\), continuous up to the boundary \({\mathcal {S}}\), and such that \(u, v\rightarrow 0\) as \(|x|\rightarrow \infty \). If \(u,v\in H^{s+1}_\mathrm{ax}({\mathcal {S}})\) for \(s>1\). Then, \(|\nabla u|^2 ,|\nabla v|^2\in H^s_\mathrm{ax}({\mathcal {S}})\) and

$$\begin{aligned} \Vert |\nabla u|^2\Vert _{H^s({\mathcal {S}})}&\le C\Vert u\Vert _{H^{s+1}({\mathcal {S}})}^2,\\ \Vert |\nabla u|^2-|\nabla v|^2\Vert _{H^s({\mathcal {S}})}&\le C\left( \Vert u\Vert _{H^{s+1}({\mathcal {S}})}+\Vert v\Vert _{H^{s+1}({\mathcal {S}})}\right) \Vert u-v\Vert _{H^{s+1}({\mathcal {S}})}, \end{aligned}$$

for some constant \(C>0\) independent of u, v.

Proof

The assertion follows from the decompositions

$$\begin{aligned} |\nabla u|^2&=u_r^2+u_\theta ^2,\\ |\nabla u|^2-|\nabla v|^2&=(u+v)_r(u-v)_r+(u+v)_\theta (u-v)_\theta , \end{aligned}$$

Lemmas 2.6 and 2.7 . \(\square \)

We are now in position to prove our main result.

Proof of Theorem 1.1

In the proof C denotes a generic positive constant depending only on s. We define operators \({\mathcal {T}}\) and \(\Psi \) by

$$\begin{aligned} {\mathcal {T}}[f]= |\nabla v|^2, \qquad \Psi [f]=\frac{1}{2} \left( g^2 -|\nabla d|^2 -{\mathcal {T}}[f] \right) , \end{aligned}$$

where v is a unique solution of (3.1). Due to Theorem 3.1, Remark 4.1 and Lemma 4.2, we see that \({\mathcal {T}}\) and \(\Psi \) are defined as mappings from \(H^s_\mathrm{ax}({\mathcal {S}})\) to \(H^s_\mathrm{ax}({\mathcal {S}})\). Put \(\delta =\Vert g-|\nabla d|\Vert _{H^s({\mathcal {S}})}+|h|\) and define a closed subset \(X_\delta \) of \(H^s({\mathcal {S}})\) by

$$\begin{aligned} X_\delta =\{ f \in H^s_\mathrm{ax}({\mathcal {S}}): \Vert f\Vert _{H^s({\mathcal {S}})} \le M\delta \}. \end{aligned}$$

We shall prove that \(\Psi \) has a unique fixed point, by showing that it is a contraction mapping on \(X_\delta \), for some number \(M>0\).

To this end, we observe that Theorems 2.4 and 3.1, Remark 4.1 and Lemma 4.2 give that

$$\begin{aligned} \Vert g^2 -|\nabla d|^2\Vert _{H^s({\mathcal {S}})}\le & {} C\Vert g+|\nabla d|\Vert _{H^s({\mathcal {S}})} \Vert g-|\nabla d|\Vert _{H^s({\mathcal {S}})} \le C(\delta +1)\, \delta ,\\ \Vert {\mathcal {T}}[f]\Vert _{H^s({\mathcal {S}})}\le & {} C\Vert v\Vert _{H^{s+1}({\mathcal {S}})}^2 \le C(\Vert f\Vert _{H^s({\mathcal {S}})}^2 +|h_0|^2) \le C(M^2+1)\,\delta ^2, \end{aligned}$$

for any \(f \in X_\delta \). Hence, the inequality

$$\begin{aligned} \Vert \Psi [f]\Vert _{H^s({\mathcal {S}})} \le C_1 \left[ (M^2 +1)\delta +1\right] \delta \end{aligned}$$
(4.1)

holds for some other positive constant \(C_1\) only depending on s.

Next, let \(f_j \in X_\delta \) and let \(v_j\) be a unique solution of (3.1) for \(f=f_j\) (\(j=1,2\)). We see from Theorem 3.1 and Lemma 4.2 that

$$\begin{aligned} \Vert {\mathcal {T}}[f_1] -{\mathcal {T}}[f_2]\Vert _{H^s({\mathcal {S}})}&\le C\left( \Vert v_1\Vert _{H^{s+1}({\mathcal {S}})} +\Vert v_2\Vert _{H^{s+1}({\mathcal {S}})} \right) \Vert v_1-v_2\Vert _{H^{s+1}({\mathcal {S}})}\\&\le C(\Vert f\Vert _{H^s({\mathcal {S}})} +|h_0|) \Vert f_1-f_2\Vert _{H^s({\mathcal {S}})}\\&\le C(M+1)\,\delta \, \Vert f_1-f_2\Vert _{H^s({\mathcal {S}})}. \end{aligned}$$

Therefore we have

$$\begin{aligned} \Vert \Psi [f_1] -\Psi [f_2]\Vert _{H^s({\mathcal {S}})} =\frac{1}{2} \Vert {\mathcal {T}}[f_1] -{\mathcal {T}}[f_2]\Vert _{H^s({\mathcal {S}})} \le C_2\,(M+1)\,\delta \, \Vert f_1-f_2\Vert _{H^s({\mathcal {S}})},\nonumber \\ \end{aligned}$$
(4.2)

for a constant \(C_2>0\), which only depends on s.

Now, in order to show that \(\Psi \) is a contraction mapping on \(X_\delta \), we must choose the positive parameters M and \(\delta \) such that

$$\begin{aligned} C_1 \left[ (M^2 +1)\delta +1\right] <M, \end{aligned}$$

so that \(\Psi (X_\delta )\subset X_\delta \) thanks to (4.1), and

$$\begin{aligned} C_2\,(M+1)\,\delta <1, \end{aligned}$$

from (4.2). The last two inequalities are certainly satisfied if we take \(M=2 C_1\) and

$$\begin{aligned} \delta < \min \left\{ \frac{1}{1+4\,C_1^2}, \frac{1}{C_2\,(1+2\,C_1)} \right\} . \end{aligned}$$

Thus, by the Banach fixed-point theorem, \(\Psi \) has a unique fixed point \(f_*\) in \(X_\delta \). Therefore, we can easily see that the solution \(v_*\) of (3.1) with \(f=f_*\) is such that \(u=d+v_*\) satisfies (1.1) with \(u=h\) on \({\mathcal {E}}\). Thus, the proof is completed. \(\square \)

Remark 4.3

The constant \(h_0\) can be chosen as the average on \({\mathcal {E}}\) of a function h. Thus, loosely speaking, Theorem 1.1 can be interpreted from a geophysics point of view as: for any field intensity of dipolar character given on the Earth’s surface, there exists a unique geomagnetic potential outside the Earth, with that field intensity on its surface, and with given average potential on the equator.