Up and Down Through the Gravity Field

Abstract

The knowledge of the gravity field has widespread applications in geosciences, in particular in Geodesy and Geophysics. The point of view of the paper is to describe the properties of the propagation of the potential, or of its relevant functionals, while moving upward or downward. The upward propagation is always a properly posed problem, in fact a smoothing and somehow related to the Newton integral and to the solution of boundary value problems (BVP). The downward propagation is always improperly posed, not only due to its intrinsic numerical instability but also because of the non-uniqueness that is created as soon as we penetrate layers of unknown mass density.

So the paper focuses on recent results on the Geodetic Boundary Value Problems on the one side and on the inverse gravimetric problem on the other, trying to highlight the significance of mathematical theory to numerical applications. Hence, on the one hand we examine the application of the BVP theory to the construction of global gravity models, on the other hand the inverse gravimetric problem is studied for layers together with proper regularization techniques.

The level of the mathematics employed in the paper is willingly kept at medium level, often recursing to spherical examples in support to the theory. Most of the material is already present in literature but for a few parts concerning global models and the inverse gravimetric problem for layers.

Zusammenfassung

Die Kenntnis des Schwerefeldes hat weitreichende Anwendungen in den Geo- wissenschaften, insbesondere in Geodsie und Geophysik. Unser Anliegen in diesem Beitrag ist die Beschreibung von Eigenschaften zur Fortpflanzung des Potentials oder seiner relevanten Funktionale nach oben und nach unten. Die Fortpflanzung des Potentials nach oben (“upward continuation”) ist stets ein wohlgestelltes Problem. “Downward Continuation” ist stets ein schlechtgestelltes Problem, nicht nur wegen der numerischen Instabilitten, sondern vor allem wegen der Nichteindeutigkeit der Bestimmung von Massenschichtung aus Potentialwerten.

Als Konsequenz fokussiert sich der Beitrag auf neuere Resultate aus dem Bereich geodätischer Randwertprobleme und zum anderen auf das inverse Gravimetrieproblem. Dabei machen wir den Versuch, die Bedeutung von mathematischer Theorie für numerische Anwendungen heraus zu streichen.

Das Paper ist vom mathematischem Anspruch her schlicht gehalten. Dazu bedienen wir uns oftmals der Rückführung auf sphärische Beispiele. Der größte Teil des Materials ist bereits in der Literatur vorhanden, bis auf Teile für die globalen Modelle und das inverse Gravimetrieproblem für Schichtungen.

This chapter is part of the series Handbuch der Geodäsie, volume “Mathematical Geodesy/Mathematische Geodäsie”, edited by Willi Freeden, Kaiserslautern.

Appendices

Appendix A

On the relation between terrestrial and satellite global models

In this Appendix we aim at proving Eq. 88, giving on the same time the explicit form of F _nm(σ), G _nm(σ). We do that, not only adapting the second Green identity to an oblique derivative as opposed to a normal problem (in our case e _r ≠ n), but also taking advantage of the hypothesized star-shaped geometry of the boundary S. While the first step is perfectly known in literature (see [16], I.6 ), the second is not, at least to the knowledge of the authors, although of chief interest in Geodesy.

So let P = (x) ≡ (r, σ) be any point in Ω; likewise we will use Q =y ≡ (s, σ′) for any Q in Ω or on S. We start then from the known formula ( [18] 1–5)

$$\displaystyle \begin{aligned} u(P) = \frac{1}{4 \ \pi} \int_S \bigg\{u \ \frac{\partial}{\partial n_Q} \ \frac{1}{l_{PQ}} - \frac{1}{l_{PQ}} \ \frac{\partial u}{\partial n_Q}\bigg\} \ d S_Q. {} \end{aligned} $$

(184)

If the equation of S is written as

$$\displaystyle \begin{aligned} r = R_{\sigma}, {} \end{aligned} $$

(185)

it is easy to verify that the vector

$$\displaystyle \begin{aligned} \mathbf{n} = \frac{1}{p} \ (R_{\sigma} \ {\mathbf{e}}_r - \nabla_{\sigma} \ R_{\sigma}) \quad , \qquad (p = \sqrt{R_{\sigma}^2 + \vert \ \nabla_{\sigma} R_{\sigma} \ \vert^2}) {} \end{aligned} $$

(186)

is the outer normal, so that

$$\displaystyle \begin{aligned} cos I = \mathbf{n} \cdot {\mathbf{e}}_r = \frac{R_{\sigma}}{p} > 0. {} \end{aligned} $$

(187)

On the same time

$$\displaystyle \begin{aligned} dS = \frac{R_{\sigma}^2}{cos I} \ d \sigma = p \ R_{\sigma} \ d \sigma. {} \end{aligned} $$

(188)

We introduce now a slightly different notation for the gradient with respect to the spherical angular coordinates (σ), namely for any differentiable function f(r, σ) we write ∂ _σf(r, σ) instead of ∇_σf(r, σ); this is because we will need to restrict then ∂ _σf(r, σ) to the surface S, so obtaining

$$\displaystyle \begin{aligned} \partial_{\sigma} f (r, \sigma) \vert_S = \partial_{\sigma} f (R_{\sigma}, \sigma) {} \end{aligned} $$

(189)

but we want to avoid any confusion with ∇_σ[f(R _σ, σ)]. In fact we have

$$\displaystyle \begin{aligned} \nabla_{\sigma} [ f(r, \sigma) \vert_S] = \nabla_{\sigma} f(R_{\sigma}, \sigma) \equiv f'(R_{\sigma}, \sigma) \ \nabla_{\sigma} R_{\sigma} + \partial_{\sigma} f(R_{\sigma}, \sigma), {} \end{aligned} $$

(190)

where as usual we have denoted with f′ the radial derivative of f(r, σ).

Finally we recall that in spherical coordinates it holds

$$\displaystyle \begin{aligned} \nabla = {\mathbf{e}}_r \ \frac{\partial}{\partial r} + \frac{1}{r} \ \partial_{\sigma}. {} \end{aligned} $$

(191)

So for any regular function f(r, σ) we can write

$$\displaystyle \begin{aligned} \frac{\partial f}{\partial n} \bigg \vert_S = \mathbf{n} \cdot \nabla f \vert_S = \frac{R_{\sigma}}{p} \ f'(R_{\sigma}, \sigma) - \frac{\nabla_{\sigma} R_{\sigma} \cdot \partial_{\sigma} f(R_{\sigma}, \sigma)}{p^{R_\sigma}} {} \end{aligned} $$

(192)

and

$$\displaystyle \begin{aligned} \frac{\partial f}{\partial n} \bigg \vert_S dS = \bigg [ R_{\sigma}^2 \ f'(R_{\sigma}, \sigma) - \nabla_{\sigma} R_{\sigma} \cdot \partial_{\sigma} f(R_{\sigma}, \sigma) \bigg ] \ d \sigma. {} \end{aligned} $$

(193)

Note that we continue to use the notation ∇_σR _σ because R _σ is function of σ only and not of r, so that no ambiguity can rise.

Now we observe that if we take P ∈ S _e, namely r = R _e, and Q ∈ S, namely $Q = \textbf {y} \equiv (R_{\sigma '}, \sigma ')$, since $R_{e} > R_{\sigma '} (\forall \ \sigma ')$ the function $l_{PQ}^{-1}$ is analytic in Q ≡y, so that Eq. 193 applies and we can write Eq. 184 in the form

$$\displaystyle \begin{aligned} \begin{aligned} u(P) = u(R_e, \sigma) &= \frac{1}{4 \ \pi} \int \bigg \{ u(R_{\sigma'}, \sigma') \ \bigg [ R_{\sigma'}^2 \ \frac{\partial}{\partial s} \ \frac{1}{l_{PQ}} - \nabla_{\sigma'} R_{\sigma'} \cdot \partial_{\sigma'} \frac{1}{l_{PQ}}\bigg ] + \\ &- \int \frac{1}{l_{PQ}} \ \bigg [ R_{\sigma'}^2 \ u'(R_{\sigma'}, \sigma') - \nabla_{\sigma'} R_{\sigma'} \cdot \partial_{\sigma'} u(R_{\sigma'}, \sigma') \bigg] \bigg \} \ d \sigma' \end{aligned} {} \end{aligned} $$

(194)

We multiply Eq. 194 by Y _nm(σ) and compute the integral over σ, recalling that

$$\displaystyle \begin{aligned} u_{nm} (R_e) = \frac{1}{4 \ \pi} \int u(R_e, \sigma) \ Y_{nm} (\sigma) \ d \sigma. {} \end{aligned} $$

(195)

On the other hand we also have

$$\displaystyle \begin{aligned} \begin{aligned} &\frac{1}{4 \ \pi} \int \frac{1}{l_{PQ}} \ Y_{nm} (\sigma) \ d \sigma = \frac{1}{4 \ \pi} \int \frac{1}{\vert \ \textbf{x} - \textbf{y} \ \vert } \ Y_{nm} (\sigma) \ d \sigma =\\ &= \frac{1}{4 \ \pi} \int \sum_{l = 0}^{+ \infty} \sum_{k = -l}^{l} \frac{Y_{lk} (\sigma) \ Y_{lk} (\sigma')}{2 \ l + 1} \ \frac{R_{\sigma'}^l}{R_{e}^{l+1}} \ Y_{nm} (\sigma) \ d \sigma = \frac{R_{\sigma'}^n}{R_{e}^{n+1}} \ \frac{Y_{nm}(\sigma')}{2n+1} \end{aligned} {} \end{aligned} $$

(196)

so that, using Eqs. 195 and 196 in Eq. 194, we get the identity

(197)

With Eq. 197 we are almost done, however we want to eliminate in it the dependence from $\partial _{\sigma '} u(R_{\sigma '}, \sigma ')$.

For this purpose we use Eq. 190 in the integral

$$\displaystyle \begin{aligned} \begin{aligned} J &= \int R_{\sigma'}^n \ Y_{nm} (\sigma') \ \nabla_{\sigma'} R_{\sigma'} \cdot \partial_{\sigma'} \ u \ d \sigma' = \\ &= \int R_{\sigma'}^n \ Y_{nm}(\sigma') \bigg [ \nabla_{\sigma'} R_{\sigma'} \cdot \nabla_{\sigma'} u \vert_S - u' \vert \ \nabla_{\sigma'} R_{\sigma'} \ \vert^2 \bigg ] \ d \sigma'. \end{aligned} {} \end{aligned} $$

(198)

Now it is enough to recall the Gauss theorem for the spherical surface, namely

$$\displaystyle \begin{aligned} \int \underline{h}(\sigma) \cdot \nabla_{\sigma} f(\sigma) \ d \sigma = - \int \bigg [ \nabla \cdot \underline{h}(\sigma) \bigg ] f(\sigma) \ d \sigma {} \end{aligned} $$

(199)

that applied to Eq. 198 gives

$$\displaystyle \begin{aligned} \begin{aligned} J = &- \int \bigg\{ u' \vert \ \nabla_{\sigma'} R_{\sigma'} \ \vert^2 \ Y_{nm}(\sigma') + \bigg[ n \ R_{\sigma'}^{n-1} \ \vert \ \nabla_{\sigma'} R_{\sigma'} \ \vert^2 \ Y_{nm}(\sigma') + \\ &+ R_{\sigma'}^n \ \varDelta_{\sigma'} R_{\sigma'} \ Y_{nm} (\sigma') + R_{\sigma'} \ \nabla_{\sigma'} R_{\sigma'} \cdot \nabla_{\sigma'} Y_{nm} (\sigma') \bigg] \ u\bigg\} \ d \sigma' \end{aligned} {} \end{aligned} $$

(200)

where $\varDelta _{\sigma '} = \nabla _{\sigma '} \cdot \nabla _{\sigma '}$ is the well known Laplace Beltrami operator.

Introducing the notation

$$\displaystyle \begin{aligned} q_{\sigma} = \frac{R_{\sigma}}{R_e}, \end{aligned}$$

and wrapping up, we finally obtain

$$\displaystyle \begin{aligned} u_{nm} (R_e) = \frac{1}{4 \ \pi} \int F_{nm} (\sigma') \ u' (R_\sigma^{\prime}, \sigma') \ d \sigma' + \frac{1}{4 \ \pi} \int G_{nm}(\sigma') \ u(R_{\sigma^{\prime}}, \sigma') \ d \sigma' {} \end{aligned} $$

(201)

with

$$\displaystyle \begin{aligned} F_{nm}(\sigma') = - R_e \ q_{\sigma'}^n \ (q_{\sigma'}^2 + \vert \ \nabla_{\sigma'} q_{\sigma'} \ \vert^2) \ \frac{Y_{nm} (\sigma')}{2 \ n+1} {} \end{aligned} $$

(202)

$$\displaystyle \begin{aligned} \begin{aligned} G_{nm} (\sigma') = &\bigg [ n \ q_{\sigma'}^{n+1} \ Y_{nm} (\sigma') - 2 \ q_{\sigma'}^2 \ \nabla_{\sigma'} q_{\sigma'} \cdot \nabla Y_{nm}(\sigma') + \\ &- n \ q_{\sigma'}^{n+1} \ \vert \ \nabla_{\sigma'} q_{\sigma'} \ \vert^2 - q_{\sigma'}^2 \ \varDelta_{\sigma'} q_{\sigma'} \ Y_{nm}(\sigma') \bigg ] (2 \ n +1)^{-1}. \end{aligned} {} \end{aligned} $$

(203)

After such somewhat cumbersome calculations, it seems not unuseful to verify the correctness of Eq. 201 at least when R _σ = R ₀ and $q_{\sigma } = \frac {R_0}{R_e} = q$. In such a case ∇_σq _σ = 0 and Δ _σq _σ = 0, so that we have

$$\displaystyle \begin{aligned} F_{nm}(\sigma') = - R_e \ q^{n+2} \ \frac{Y_{nm}(\sigma')}{2 \ n+1} \quad , \quad G_{nm} (\sigma') = n \ q^{n+1} \ \frac{Y_{nm}(\sigma')}{2 \ n +1}. \end{aligned}$$

Therefore Eq. 201 reads

$$\displaystyle \begin{aligned} \begin{aligned} u_{nm}(R_e) = &- \frac{R_e \ q^{n+2}}{2 \ n +1} \ \frac{1}{4 \ \pi} \int u' (R_0, \sigma') \ Y_{nm} (\sigma') \ d \sigma' + \\ &+ n \ q^{n+1} \ \frac{1}{4 \ \pi} \int u (R_0, \sigma') \ Y_{nm} (\sigma') \ d \sigma'. \end{aligned} {} \end{aligned} $$

(204)

If we put

$$\displaystyle \begin{aligned} u(r, \sigma) = \sum_{n = 0}^{+ \infty} \sum_{m = -n}^{n} \bigg( \frac{R_0}{r} \bigg)^{n+1} u_{nm} (R_0) \ Y_{nm}(\sigma) \end{aligned}$$

so that

$$\displaystyle \begin{aligned} -u' (R_0, \sigma') = \sum_{n = 0}^{+ \infty} \sum_{m = -n}^n \frac{n+1}{R_0} \ u_{nm} (R_0) \ Y_{nm} (\sigma') \end{aligned}$$

we see that Eq. 204 reduces to

$$\displaystyle \begin{aligned} u_{nm}(R_e) = u_{nm}(R_0) \ q^{n+1} = u_{nm}(R_0) \bigg ( \frac{R_0}{R_e} \bigg)^{n+1}, \end{aligned}$$

which is nothing but Eq. 27 and is known to be correct.

Appendix B

On the application of Tikhonov regularization to the inverse problem of a layer with constant density

We first show the mathematical reasoning that proves that conditions (1) to (4), in Sect. 6, are sufficient to guarantee that:

$$\displaystyle \begin{aligned} m = \inf_{R_L \in C} T(R_L) {} \end{aligned} $$

(205)

is a true minimum, namely that $\exists \ \overline {R}_L \in C$ such that

$$\displaystyle \begin{aligned} m = T \left( \overline{R}_L \right). {} \end{aligned} $$

(206)

Then we will show that with $T\left (R_L\right )$ given by Eq. 163, $F\left [ R_L\right ]$ given by Eq. 162 and $J\left (R_L\right )=J_1\left (R_L\right )$, given by Eq. 165, such conditions are satisfied.

Given the definition of m, Eq. 205, and the fact that $T\left (R_L\right )\geq 0$, there must a sequence $R_{Ln}\in C \left ( \subset H \right )$, such that

$$\displaystyle \begin{aligned} m=\lim_{n \to\infty} T\left(R_{Ln}\right). {} \end{aligned} $$

(207)

Given Eq. 207, there must be a constant $\overline {T}$ such that

$$\displaystyle \begin{aligned} T\left(R_L\right)<\overline{T}; {} \end{aligned} $$

(208)

Eq. 208 implies that

$$\displaystyle \begin{aligned} J\left(R_L\right)<\frac{\overline{T}}{\lambda}. {} \end{aligned} $$

(209)

So

$$\displaystyle \begin{aligned} \left\lbrace R_{Ln}\right\rbrace \in C \cap \left\lbrace J\left(R_{Ln}\right)<\frac{\overline{T}}{\lambda} \right\rbrace\ {} \end{aligned} $$

(210)

and, according to the condition (3), there must be a subsequence, that we call again $\left \lbrace R_{Ln}\right \rbrace$, which is convergent in H,

$$\displaystyle \begin{aligned} R_{Ln} \underset{H}{\rightarrow} \overline{R}_L. {} \end{aligned} $$

(211)

Since C is closed, by (1), it has to be

$$\displaystyle \begin{aligned} \overline{R}_L \in C. {} \end{aligned} $$

(212)

On the other hand, $F\left (R_L\right )$ is continuous H → H, and therefore

$$\displaystyle \begin{aligned} F\left(R_{Ln}\right) \underset{H}{\rightarrow} F\left(\overline{R}_L\right), {} \end{aligned} $$

(213)

$$\displaystyle \begin{aligned} \Vert \ \overline{g}- F\left(R_{Ln}\right) \ \Vert^2_H \to \Vert \ \overline{g} - F\left(\overline{R}_{L}\right) \ \Vert^2_H. {} \end{aligned} $$

(214)

But then one has too

$$\displaystyle \begin{aligned} \begin{aligned} \lambda \lim J\left(R_{Ln} \right) &= \lim \left[ T\left(R_{Ln}\right) - \Vert \ \overline{g}- F\left(R_{Ln}\right) \ \Vert^2_H\right] = \\ &= m - \Vert \ \overline{g} - F\left(\overline{R}_{Ln}\right) \ \Vert^2_H. \end{aligned} {} \end{aligned} $$

(215)

On the same time

$$\displaystyle \begin{aligned} \lambda \lim J\left(R_{Ln} \right) \equiv \lambda \underline{\lim} J\left(R_{Ln}\right) \geq \lambda J\left(\overline{R}_{L}\right). {} \end{aligned} $$

(216)

Considering Eqs. 215 and 216 we see that

$$\displaystyle \begin{aligned} \lambda T \left(\overline{R}_{L}\right)+ \Vert \ \overline{g} - F\left(\overline{R}_{L}\right) \ \Vert^2_H \equiv J\left(\overline{R}_L\right)\leq m; {} \end{aligned} $$

(217)

since $\overline {R}_L \in C$ and m is the Inf of $T\left (R_L\right )$ on C, (217) can only be an equality and $\overline {R}_L$ is a minimum point of $J\left (R_L \right )$ in C _J, as it was to be proven.

In our specific case, $H \equiv L^2\left (\sigma \right )$ and C is given by

$$\displaystyle \begin{aligned} R_L \in C: \quad 0\leq R_{L\sigma} \leq R_\sigma. {} \end{aligned} $$

(218)

Since B is bounded and $\left \lbrace r = R_\sigma \right \rbrace$ is ∂B, one has for some $\overline {R} < +\infty$

$$\displaystyle \begin{aligned} R_\sigma \leq \overline{R} {} \end{aligned} $$

(219)

and therefore it is

$$\displaystyle \begin{aligned} R_{L\sigma} \leq \overline{R} {} \end{aligned} $$

(220)

too, so that

$$\displaystyle \begin{aligned} R_L \in C \Rightarrow \Vert \ R_{L\sigma} \ \Vert_{L^2} \leq \overline{R} \sqrt{\mu \left( B \right)} < + \infty, {} \end{aligned} $$

(221)

i.e., C is bounded.

Furthermore, let $\left \lbrace R_{Ln}\right \rbrace$ be a Cauchy sequence in $C\subset L^2\left (\sigma \right )$, converging to $\overline {R}_L$; then we know there is a subsequence, $\left \lbrace R_{Ln_{k}}\right \rbrace$, that is pointwise converging to $\overline {R}_L \ a.e.$.

But then

$$\displaystyle \begin{aligned} 0 \leq \overline{R}_L = \lim R_{Ln_{k}} \leq R_\sigma \ , \ \sigma \ a. \ e., {} \end{aligned} $$

(222)

i.e., $\overline {R}_L \in C$. So C is closed.

Now we want to prove that $F\left ( R_L\right )$ is continuous $L^2\left ( \sigma \right ) \to L^2\left ( \sigma \right )$. We will adopt the notation of Fig. 9, and of Eqs. 144, 145, and 147. So we consider

$$\displaystyle \begin{aligned} \lvert \ F\left(R_{L2} \right) - F\left(R_{L1} \right) \ \rvert^2 =\bigg \lvert \ G \int_{\mathscr{L}_1 \div \mathscr{L}_2} \frac{\rho_0\left(Q\right)}{\ell_{PQ}} dB_Q \ \bigg \rvert^2; {} \end{aligned} $$

(223)

we have already observed (see Eq. 106 and Fig. 6) that for whatever layer $\mathscr {L}\subset B$ one has

$$\displaystyle \begin{aligned} \int_{\mathscr{L}}\frac{1}{\ell^2} \ dB \leq 4 \ \pi \ \varDelta. {} \end{aligned} $$

(224)

So, by applying the Schwartz inequality to Eq. 223, one gets

$$\displaystyle \begin{aligned} \lvert \ F \left( R_{L2}\right) - F \left( R_{L1}\right) \ \rvert^2 \leq C_0^2 \ 4 \ \pi \ \varDelta \int_{\mathscr{L}_1 \div \mathscr{L}_2} \rho_0^2 \ dB. {} \end{aligned} $$

(225)

But $\rho _0^2 \equiv \rho ^2$ (constant) by definition of ρ ₀ (see Eq. 147) and so

$$\displaystyle \begin{aligned} \lvert \ F\left( R_{L2}\right) - F\left( R_{L1}\right) \ \rvert^2 \leq C_0^2 \ 4 \ \pi \ \varDelta \rho^2 \mu \left(\mathscr{L}_1 \div \mathscr{L}_2\right) {} \end{aligned} $$

(226)

On the other hand, again by Schwarz inequality,

$$\displaystyle \begin{aligned} \begin{aligned} \mu \left(\mathscr{L}_1 \div \mathscr{L}_2\right) &= \int d\sigma \int_{R_{-\sigma}}^{R_{+\sigma}}s^2 \ ds \leq \overline{R}^2\int d\sigma \lvert R_{+\sigma} - R_{-\sigma}\rvert\\ &\leq\overline{R}^2\sqrt{4 \ \pi}\left[\int d \sigma \ \lvert \ R_{+\sigma} - R_{-\sigma} \ \rvert^2 \right]^{\frac{1}{2}}. \end{aligned} {} \end{aligned} $$

(227)

By its definition it is $\lvert \ R_{+\sigma } - R_{-\sigma } \ \rvert \equiv \lvert \ R_{L2} - R_{L1} \ \rvert$, so that collecting Eqs. 226 and 227, one gets

$$\displaystyle \begin{aligned} \lvert \ F \left(R_{L2} \right) - F \left(R_{L1} \right) \ \rvert^2 \leq const. \ \Vert \ R_{L2}-R_{L1} \ \Vert_{L^2 \left(\sigma \right)}. {} \end{aligned} $$

(228)

As we see Eq. 228 implies that $F\left ( R_L\right )$ is continuous from $L^2\left (\sigma \right )$ to the space of continuous functions, $C\left (\sigma \right )$, and therefore also $L^2\left ( \sigma \right ) \to L^2\left ( \sigma \right )$, since

$$\displaystyle \begin{aligned} \Vert \ f \ \Vert_{L^2 \left( \sigma \right)} = \left[\int f^2(\sigma) d \sigma \right]^{\frac{3}{2}} \leq \sup \lvert \ f\left(\sigma\right) \ \rvert \sqrt{4 \ \pi} \equiv \Vert \ f \ \Vert_{C\left(\sigma\right)} \ \sqrt{4 \ \pi} . {} \end{aligned} $$

(229)

Finally we have to verify that the set

$$\displaystyle \begin{aligned} \varGamma_a = C \cap \left\lbrace J\left(R_L\right) < a\right\rbrace {} \end{aligned} $$

(230)

is compact in $L^2\left (\sigma \right )$, ∀ a.

But we have already observed that C is bounded, so that, for some positive constant a,

$$\displaystyle \begin{aligned} R_L \in C \ \Rightarrow \Vert \ R_L \ \Vert^2_{L^2\left(\sigma\right)} \leq a. {} \end{aligned} $$

(231)

Then we have

$$\displaystyle \begin{aligned} R_L \in \varGamma_a \Rightarrow \Vert \ R_L \ \Vert^2_{H^{1,2}\left(\sigma\right)}= \Vert \ R_L \ \Vert^2_{L^2\left(\sigma\right)}+J_1\left(R_L\right) \leq c+a, {} \end{aligned} $$

(232)

namely Γ _a is a bounded set in $H^{1,2}\left (\sigma \right )$. We have already observed that the embedding of $H^{1,2}\left (\sigma \right )$ into $L^2\left (\sigma \right )$ is compact, and in any way this is quite obvious because

$$\displaystyle \begin{aligned} f \in H^{1,2}; \quad \ f = \sum f_{nm}\ Y_{nm}\left(\sigma\right) {} \end{aligned} $$

(233)

implies

$$\displaystyle \begin{aligned} \begin{aligned} \Vert \ f \ \Vert^2_{H^{1,2}} &= \int f^2 d \sigma + \int \lvert \ \nabla_\sigma f \ \rvert^2d \sigma\\ &\equiv \int f^2 d \sigma - \int f \ \varDelta_\sigma f \ d \sigma \\ &\equiv 4 \ \pi \sum f_{nm}^2 \left[ 1+n \ \left(n+1\right)\right]<+\infty. \end{aligned} {} \end{aligned} $$

(234)

So Γ _ais compact in $L^2\left (\sigma \right )$ as it was to be proved.