In the present chapter we solve some boundary value problems of physical interest in the asymmetric environment that is described by the ellipsoidal geometry. More precisely, we solve: (i) the problem of thermal equilibrium of an ellipsoidal body, which was the problem that gave rise to the theory of ellipsoidal harmonics [223] as well as to the introduction of the general curvilinear system [228]; (ii) the problem of gravitational attraction by a homogeneous ellipsoid, which was an important problem of Newtonian Mechanics for many years and was finally solved by Jacobi, Gauss, Rodrigues, and others in the early nineteenth century [51]; (iii) the problem of an ellipsoidal perfect conductor [329]; (iv) the problem of the polarization potential, in terms of which the polarization tensor and the electric polarizability tensor are expressed [200, 286]; (v) the problem of the virtual mass potential in terms of which the virtual mass tensor and the magnetic polarizability tensor are expressed [200, 286]; and (vi) the problem of the generalized polarization potentials, in terms of which the general polarizability tensor is defined [216]. We also include a short section on the reduction of these solutions to the case of prolate and oblate spheroids, their asymptotic forms, and the sphere. General results on polarization tensors can be found in [5–10] as well as in [11]. Further references on boundary value problems in ellipsoidal geometry are [34, 35, 38, 56, 69, 82, 87, 120, 122, 139-142, 144, 145, 154-156, 185, 196, 205, 207, 218, 221, 222, 242, 247, 251-253, 261, 275, 277, 283, 289-293, 295, 297, 302, 304, 324, 325, 336, 362].

The construction of inversion algorithms provide challenging mathematical problems which shape the direction of research in modern science and technology. Medical imaging, non-destructive evaluation and testing, RADAR and SONAR technology, oil exploration, and remote sensing are some areas where mathematical modelling leads to inverse problems of contemporary interest. Keller [199] gives the following definition, “two problems are inverse of one another if the formulation of each involves all or part of the solution of the other,” and continues, “historically, one of the two problems has been studied extensively for some time, while the other is newer and not so well understood. The former is called the direct problem, while the other is called the inverse problem.” Inverse problems are usually not well-posed, most of the time because of a lack of uniqueness and sometimes because of a lack of stability as well. Nevertheless, uniqueness can be secured if a-priori information is available, so that the possible set of solutions is severely restricted. This is the case, for example, when we know that the object we want to reconstruct is an ellipsoid.

In this chapter we will analyze a few inverse problems that are associated with ellipsoidal geometry. In the first three subsections we discuss the inverse problem of identifying an ellipsoid from low-frequency scattering data, from high-frequency time-dependent scattering, and from tomographic data. Following similar approaches, one can identify the thickness of a penetrable ellipsoidal shell surrounding a confocal ellipsoidal core [105].

Scattering theory investigates the interaction of a propagating wave, or incident wave, with an obstacle, or scatterer. The existence of the scatterer disturbs the incident wave in a way that depends on the physical and geometrical properties of the scatterer. In the forward scattering problem one knows the incident wave, as well as the physical and geometrical characteristics of the scatterer, and seeks the effect that the scatterer has on the propagation of the incident wave. More interesting and much more difficult is the inverse scattering problem, where one again knows the incident wave and has (usually partial) knowledge about the form of disturbance caused by the scatterer, and the goal is to identify as much information as possible about the physics and/or geometry of the scatterer. A large portion of modern science and technology is founded on inverse scattering problems. In this chapter, we state the appropriate boundary value problems for scattering of acoustic, electromagnetic, and elastic waves, then we focus on the special case where the wavelength of the incident wave is much larger than the characteristic dimension of the scatterer, and finally we solve some representative scattering problems when the scatterer is, or can be approximated by, an ellipsoid. The study of small scatterers, as they compare to the wavelength, is known as the theory of low-frequency scattering, and was initiated by Rayleigh [287], formally developed by Stevenson [326], and established into a rigorous mathematical theory by Kleinman [215].

In the present chapter we introduce vector ellipsoidal harmonics and discuss their peculiarities as well as the limitations that prevented their appearance for many years. In fact, it was not until 2009 that these functions were first introduced [121] and their complete understanding and effectiveness are still open for further investigation [82, 83].

Vector ellipsoidal harmonics

Without any knowledge of vectorial harmonics, all vector boundary value problems, governed by the Laplace equation in ellipsoidal domains, can be solved in the framework of a combined Cartesian–ellipsoidal treatment. That is, each Cartesian component of the vector solution we seek can be expanded in scalar ellipsoidal harmonics and then the boundary conditions can be used to calculate the coefficients of these expansions. However, this method is either very difficult or impossible, because the vectorial character of the field is cast into the Cartesian coefficients of the series expansions, and these are not compatible with the ellipsoidal geometry of the boundary. The correct approach is to use eigensolutions that carry the vectorial structure of the field, and leave the coefficients of the corresponding expansions in scalar form. This method best fits the needs of any vector problem, but it demands knowledge of a complete set of vector eigenfunctions. It was successfully applied to the spherical case by Hansen in 1935 [172], who introduced vector spherical harmonics in connection with an antenna radiation problem.

Electromagnetic activity of the brain

The electromagnetic activity of the human brain is governed by the quasi-static theory of Maxwell's equations [229, 284] which, to some extent, decouples the electric from the magnetic behavior. The brain is modelled as a conductive medium with certain conductivity and a magnetic permeability which is equal to the permeability of the air. The brain is surrounded first by the cerebrospinal fluid, then by the skull, and finally by the scalp, all having different conductivities but the same magnetic permeability. In most cases though the brain–head system is assumed to be a single homogeneous medium. A primary neuronal current within the conductive brain tissue excites a secondary induction current and both currents generate an electric potential, which is measured on the surface of the head, and a magnetic flux, which is measured outside, but close to, the head. Recordings of the electric potential concern the theory of electroencephalography (EEG) and recordings of the magnetic flux density concern the theory of magnetoencephalography (MEG). The forward problem of EEG or MEG consists of the calculation of the electric potential or the magnetic induction, respectively, from a complete knowledge of the primary neuronal current. The inverse problem of EEG or MEG consists of the identification of the location and intensity of the neuronal current from the electric potential or the surface of the head, or of the magnetic induction outside the head, respectively.

Eigensolutions of the ellipsoidal biharmonic equation

A function is called biharmonic if it is annihilated by two successive applications of the Laplacian. That is, if we apply the Laplace operator on a biharmonic function we end up with a harmonic function. The most amazing result concerning biharmonic functions was proved by Almansi in 1899 [4]. Almansi proved that, if U is a biharmonic function, then there exist harmonic functions u1 and u2 such that

where r denotes the ordinary Euclidean distance.

The Almansi formula (10.1) provides an algebraic representation of a biharmonic function in terms of harmonic functions, and at a first glance it seems that, with this formula, we can solve boundary value problems for the biharmonic operator in ellipsoidal geometry. Indeed, replacing the functions u1 and u2 in (10.1) with ellipsoidal harmonics, we end up with ellipsoidal biharmonic functions. Since the ellipsoidal harmonics form a complete set of harmonic eigenfunctions, the corresponding biharmonics form a complete set of biharmonic eigenfunctions. Nevertheless, the effectiveness of these biharmonic eigenfunctions depends on their orthogonality properties, and these properties are not inherited from the orthogonality of the ellipsoidal harmonics, since the Euclidean distance is a function of ρ, μ, and ν. Note that since r is a spherical coordinate, the Almansi representation is tailor-made for problems in spherical geometry. In order to deal with the orthogonality problem we need a cumbersome algebraic analysis which is based on the exact form of the particular ellipsoidal harmonic. Therefore, only biharmonics of the few first degrees can be calculated in closed form.

The theory of harmonic functions was initiated in 1782 by Laplace, when he derived the partial differential equation that is known today as Laplace's equation. The same year Legendre developed the theory of zonal spherical harmonics, which is a solution of the Laplace equation with axial symmetry, while Laplace himself solved his equation in spherical geometry without any symmetry, introducing the concept of tesseral spherical harmonics. Both papers were published in 1785 [230, 233].

The sphere is invariant under rotation and therefore provides the geometrical visualization of isotropy. In an anisotropic space however, where only a finite number of symmetries are possible, the sphere is transformed into an ellipsoid. The study of harmonic functions in the presence of anisotropic structure, which is undertaken in the present book, is more complicated by far than the corresponding study of harmonic functions in the presence of isotropy. The ellipsoidal shape appears naturally in many different forms. For example, Rayleigh has proved that the ultimate shape of pebbles, as they are worn down by attrition, is a generic ellipsoid, see [36, 124, 126, 128, 129, 288]. It is also known that the RGB points, which determine the color of objects in our visual neuronal system, exhibit color insensitivity whenever they vary in a small ellipsoid [184]. Many more cases appear in physics, such as the inertia ellipsoid in mechanics, the directivity ellipsoid, the reciprocal ellipsoid in wave propagation within crystalographic structures, and so on.

The ellipsoidal coordinate system replaces the spherical system whenever the geometrical standards of the space depend on direction. In other words, when the space exhibits some kind of anisotropy. This anisotropy is characterized by three orthogonal directions, specifying the principal directions, and the unit lengths along these directions, specifying the semi-axes of the reference ellipsoid. Hence, the reference ellipsoid encodes the complete structure of the anisotropic behavior of the space and defines the appropriate coordinate system. One of the variables of the ellipsoidal system, denoted by ρ, specifies a family of ellipsoids and therefore corresponds to the radial variable of the spherical system. The other two variables, denoted by μ and ν, specify a point on the ellipsoid and therefore they correspond to the spherical angular variables. Since the variables vary in successive intervals of the real line in the order (ρ, μ, ν), it is customary to refer to them in this particular order. We should keep in mind, however, that this order corresponds to a sinistral system. The order that leads to a dextral system is (ρ, μ, ν). The ellipsoidal system stems out of three couples of foci, two of which lie along the longest semi-axis and one lies along the intermediate semi-axis of the reference ellipsoid. These six foci define the focal ellipse, which has the two focal distances as its axes and the third one as its own focal distance.