## Abstract

The objective of this paper is to present a modern and concise new derivation for the explicit expression of the interior and exterior Newtonian potential generated by homogeneous ellipsoidal domains in *N*≥3). The very short argument is essentially based on the application of Reynold's transport theorem in connection with the Green–Stokes integral representation formula for smooth functions on bounded domains of *N*-dimensional problem to a one-dimensional one. Owing to its physical relevance, a separate section is devoted to the the derivation of the *demagnetizing factors* of the general ellipsoid which are one of the most fundamental quantities in ferromagnetism.

## 1. Historical introduction, motivations

The computation of the gravitational potential induced by a homogeneous ellipsoid was one of the most important problems in mathematics for more than two centuries after Newton enunciated his universal law of gravitation [1–8]. Once an ellipsoidal domain *Ω* of *explicit* expression for the Newtonian potential induced by a constant mass/charge density on *Ω* [9]
*Ω* (interior problem) and in the exterior points of *Ω* (exterior problem).

In the case of a homogeneous *spherically symmetric* region, Newton (in 1687) proved what nowadays is known as Newton's shell theorem [10,6]: *if* *Ω* *is a homogeneous spherical region centred at the origin, then for all* *t*>1, *is constant in* *Ω*, *i.e.* *tΩ*∖*Ω* (the so-called *hollow ball*) *induces no gravitational force inside* *Ω*. *Moreover, a spherically symmetric body affects external objects gravitationally as though all of its mass were concentrated at a point at its centre*.

For what concerns an ellipsoidal domain *Ω* of *ellipsoidal* *interior* problem was for the first time solved by Gauss (in 1813) by means of what is nowadays known as the Gauss divergence theorem [11]. Later, in 1839, Dirichlet proposed a solution of the interior problem based on the theory of Fourier integrals [12].

The results of Gauss and Dirichlet can be summarized by saying that if *Ω* is an ellipsoidal domain centred at the origin, then the gravitational potential induced by a homogeneous ellipsoid in its internal points is a second-order polynomial. In other terms

### Remark 1.1

For the sake of completeness we recall that the converse statement (the *inverse homogeneous ellipsoid problem*) is also true [9,15–17], namely *if* *Ω* *is a bounded domain of* *such that* *is connected and* (1.2) *holds, then* *Ω* *is an ellipsoid*. Historically speaking, the inverse homogeneous ellipsoid problem was for the first time solved by Dive [18] in 1931 for *N*=3 and in 1932 by Hölder [19] for *N*=2. A modern proof of this result can be found in DiBenedetto & Friedman [15], who, in 1985, extended it to all *N*≥2. In 1994, Karp [17], by the means of certain topological methods, obtained an alternative proof of the inverse homogeneous ellipsoid problem.

Despite the well-knowness of (1.2) in the mathematical and physical community, and its importance in theoretical and applied studies [20–29], rigorous proofs of that result are not readily available in the literature: to the best knowledge of the author, relative modern treatments of the interior problem can be found in [13,14], and more recently in [30,31] where also the the exterior problem is investigated. However, in all the cited references, the solution of the problem is always based on the use of ellipsoidal coordinates which tends to focus the attention on the computational details of the question rather than on its geometric counterpart. Modern proofs of Newton's theorem and relation (1.2), as well as far reaching beautiful generalizations, can be found in [32], where nevertheless the problem of finding an analytic expression for the coefficients *c* and *P* is not touched.

The aim of this paper is to give a modern and concise derivation for the expression of the *interior* and *exterior* Newtonian potential (induced by homogeneous ellipsoids). The very short argument is essentially based on the application of Reynold's transport theorem in connection with the Green–Stokes integral representation formula for smooth functions on bounded domains of *N*-dimensional problem to a one-dimensional one, providing (in particular) at once a proof of (1.2) together with an explicit expression of the coefficients *P* and *c* in terms of one-dimensional integrals. More precisely, the paper is organized as follows.

Section 2 is devoted to the main result of the paper. We give a concise proof of the homogeneous ellipsoid problem. For completeness, we then derive Newton's shell theorem as a corollary.

In §3, we focus attention to the three-dimensional case. An expression in terms of the elliptic integrals is given for the coefficients of *P*. Owing to its physical relevance, particular attention is paid to the eigenvalues of *P*. Indeed, when *N*=3, the matrix *P* and its eigenvalues, known in the theory of ferromagnetism, respectively, as *the demagnetization tensor* and *the demagnetizing factors*, are one of the most important and well-studied quantities of ferromagnetism [20,21,23,28,29,33]. In fact, the following magnetostatic counterpart of the homogeneous ellipsoid problem holds: *given a uniformly magnetized ellipsoid, the induced magnetic field is also uniform inside the ellipsoid*. This result was for the first time showed by Poisson [34], while an explicit expression for the demagnetizing factors was for the first time obtained by Maxwell [35]. Their importance is in that they encapsulate the self-interaction of magnetized bodies: their knowledge being equivalent to the one of the corresponding demagnetizing (stray) fields [21]. Moreover, a similar result also occurs in linearized elasticity, where the Eshelby solution asserts that a uniform eigenstress on an ellipsoidal inclusion in an infinite elastic medium induces uniform strain inside the ellipsoid [36,37].

## 2. The interior and exterior potential of a homogeneous ellipsoid

In what follows, we denote by *Ω* the ellipsoidal domain of *N*≥3) having *a*_{1},*a*_{2},…,*a*_{N} as semi-axes lengths. We then denote by *Ω*≡*Ω*_{0}. Finally, we denote by *Ω*_{t}:
*c*_{N}:=[(*N*−2)*ω*_{N}]^{−1} and *ω*_{N} the surface measure of the unit sphere in

The main result of the paper is stated in the following.

### Theorem 2.1

*Let* *be the ellipsoidal domain of* *having* *as semi-axes lengths. For every* *where we have denoted by τ the non-negative real-valued function
*

### Remark 2.2

The function *τ* defined by (2.4) can be computed by solving, for every *x*∈*Ω*^{c}, the equation in the *λ* variable given by |*ϕ*_{λ}(*x*)|^{2}=1. Indeed, for every *x*∈*Ω*^{c} there exists a unique *ϕ*_{λ}(*x*)|^{2}=1 (cf. figure 1). In particular, when *Ω* is a spherical region of radius *a* (centred around the origin), the function *τ* reduces to the function *τ*(*x*):=(|*x*|^{2}−*a*^{2})1_{Ωc}(*x*) and the integral in (2.3) can be readily computed. For general ellipsoidal domains the integral in (2.3) can be evaluated by the means of the theory of elliptic integrals (cf. [39,40]).

### Proof.

For every *ϕ*_{t}|^{2}−1 is the unique solution of the homogeneous Dirichlet problem for the Poisson equation Δ*u*=2 *tr*(*A*_{t}) in *Ω*_{t}, *u*≡0 on ∂*Ω*_{t}. Thus, according to the Green–Stokes representation formula [9], for every _{n}|*ϕ*_{t}|^{2} concentrated on ∂*Ω*_{t} (cf. [9,38]). Next, we observe that the *N*-dimensional Newtonian kernel |*x*|^{2−N} is in

### Corollary 2.3 (Newton's shell theorem)

*Let* *be a homogeneous spherical region (centred around the origin) of radius a and of total mass M. For every* *the induced gravitational potential is the same as though all of its mass were concentrated at a point at its centre. Moreover, for all t>1,* *is constant in Ω, i.e. the hollow ball induces no gravitational force inside Ω.*

### Proof.

We denote by *ρ*1_{Ω}(*x*), (*ρ*:=*M*/|*Ω*|) the uniform density of mass in *Ω*. The gravitational potential induced by *ρ* in *G* the *gravitational constant*. In this geometrical setting the function *τ* defined by (2.4) reduces to the function *Ω*. The gravitational field is given by ** g**:=−∇

*u*

_{ρ}and the fact that the hollow ball

*tΩ*∖

*Ω*,

*t*>1, induces no gravitational force inside

*Ω*can be immediately seen by splitting the uniform density of mass in

*tΩ*∖

*Ω*in the form

*ρ*=(

*M*/|

*tΩ*∖

*Ω*|)1

_{tΩ}−(

*M*/|

*tΩ*∖

*Ω*|)1

_{Ω}. Indeed by linearity we get that

*u*

_{ρ}is constant in

*Ω*and therefore

**=**

*g***0**. ▪

## 3. The demagnetizing factors of the general ellipsoid

We know focus on the three-dimensional framework (*N*=3) and, in particular, on the so-called demagnetizing factors of the general ellipsoid [29]. To this end, we recall that the demagnetizing (stray) field ** h**[

**] associated with a magnetization**

*m**φ*

_{m}[42,43]. Precisely,

**[**

*h***]:=−∇**

*m**φ*

_{m}in

**is constant in**

*m**Ω*, then

*demagnetizing tensor*, whose diagonal

*i*-entry (the

*i*th

*demagnetizing factor*), by virtue of (2.3), is given by

### Proposition 3.1

*We have* *P*_{i}≥0 *for every* *and if* *a*_{1}≥*a*_{2}≥*a*_{3} *then* *P*_{1}≤*P*_{2}≤*P*_{3}. *The trace of* *P* *satisfies the relation* tr(*P*)=1.

### Proof.

The first statement is obvious. The relation tr(*P*)=1 can of course be verified by a direct evaluation of the integrals in (3.2), but it is also possible to observe that as the Newtonian potential

Assuming *a*_{1}≥*a*_{2}≥*a*_{3}, from (3.2) and the theory of elliptic integrals, we get
*p*<1, we have denoted by
*parameter form* [39,40]. In particular, in the case of a *prolate spheroid* (*a*_{1}≥*a*_{2}=*a*_{3}), we get
*oblate spheroid* (*a*_{1}=*a*_{2}≥*a*_{3})
*sphere* (*a*_{1}=*a*_{2}=*a*_{3}), one finish with

## Competing interests

I have no competing interests.

## Funding

This work was partially supported by the labex LMH through the grant no. ANR-11-LABX-0056-LMH in the *Programme des Investissements d'Avenir*.

## Acknowledgements

The author gratefully acknowledge helpful discussions with Prof. François Alouges and Prof. Claudio Serpico.

- Received March 21, 2016.
- Accepted May 27, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.