Debye Sources, Beltrami Fields, and a Complex Structure on Maxwell Fields
The Debye source representation for solutions to the time harmonic Maxwell equations is extended to bounded domains with finitely many smooth boundary components. A strong uniqueness result is proved for this representation. Natural complex structures are identified on the vector spaces of time-harmonic Maxwell fields. It is shown that in terms of Debye source data, these complex structures are uniformized, that is, represented by a fixed linear map on a fixed vector space, independent of the frequency. This complex structure relates time-harmonic Maxwell fields to constant- Beltrami fields, i.e. solutions of the equation
A family of self-adjoint boundary conditions are defined for the
Beltrami operator. This leads to a proof of the existence of
zero-flux, constant-, force-free Beltrami fields for
any bounded region in , as well as a constructive method
to find them. The family of self-adjoint boundary value problems
defines a new spectral invariant for bounded domains in
Keywords: Maxwell’s equations, Debye sources, -Neumann fields, complex structure, Beltrami fields, constant- force-free Beltrami fields, self-adjoint boundary value problems for the curl operator.
In several previous papers [4, 5], we introduced a new representation for the time-harmonic Maxwell equations in based on two scalar densities defined on the surface of a smooth bounded region , which we refer to as generalized Debye sources. Recall that a pair of vector fields defined in an open subset of , satisfies the THME([)] if
If , then this implies the divergence equations
In the discussion that follows, unless otherwise noted, denotes a bounded region in with smooth boundary . When considering an exterior problem, the unbounded domain with smooth boundary is often referred to as though is sometimes used to refer to bounded components of as well.
As in [4, 5], we use exterior forms to represent Maxwell fields. We use a 1-form to represent the electric field and 2-form for the magnetic field. Faraday’s law and Ampere’s law (the curl equations) then take the form:
For , these equations imply the divergence equations, which take the form:
In our earlier work the emphasis was on applications to scattering theory. This previous analysis was performed largely in an exterior domain where the solution takes the form
The components are called the incoming field; the scattered field, , is assumed to satisfy the outgoing radiation condition in . For the electric field, this reads:
where . The same condition is satisfied by We use the notation for the Hodge star operator acting on forms defined on a -dimensional oriented, Riemannian manifold. In the present context is the Hodge star operator on with the Euclidean metric, and standard orientation. It is a classical result that if solves the THME([)] for , and one component is outgoing, then so is the other. When the equations for and decouple; the divergence equations, (1.4), are no longer a consequence of the curl equations, but are nonetheless assumed to hold.
In this paper our emphasis shifts to an application of the Debye source representation in bounded domains . We assume that is smooth, but allow it to have multiple connected components. The Debye source representation uses non-physical scalar sources along with harmonic 1-forms on to represent the space of solutions to (1.3) in a manner that is insensitive to the choice of wave number . The Debye sources are used to define a pair of pseudocurrents and on , which we call Debye currents. For an appropriate relationship between these pseudocurrents, e.g. (3.23), we show that the Debye source representation is injective. This representation leads to Fredholm equations of the second kind for solutions to (1.3) with specified tangential or components along ; this implies that the representation with the restricted source data is also surjective.
We then turn our attention to a very interesting feature of the space of solutions to the time-harmonic Maxwell system: it has a complex structure. If solves (1.3) in a domain , then so does
In terms of the more traditional vector field notation, this is just the statement that if solves the THME([)], then so does .
Since , we see that
In other words, defines a complex structure on the vector space, , of solutions to THME([)] in . Its eigenvalues are and this immediately implies that this space of solutions is a direct sum of the two eigenspaces, one in which and another in which . These subspaces are denoted by and , respectively. In these subspaces the electric field satisfies the equations
which are called Beltrami equations. The operator is the exterior form representation of the vector curl operator. In the classical vector notation, these equations are
For applications in fluid mechanics and plasma physics it is especially interesting to find solutions with vanishing normal components [2, 8, 9, 12, 13]. These are often referred to as constant-, force-free fields. A relationship of this sort was used in  to study the Beltrami equation, though without explicitly defining the complex structure on .
If are the Debye currents that represent a solution , then are the currents that represent . This shows that the Debye representation provides a uniformization of this complex structure on : for any value of , the structure is represented by a fixed linear transformation on a fixed vector space. This observation leads to effective numerical methods for solving the Beltrami equation in either a bounded or unbounded domain.
As in our earlier work on solving the Maxwell system, we obtain Fredholm equations of second kind for the normal component of a Beltrami field. If is not simply connected, then these integral equations need to be augmented with algebraic conditions, where is the genus of . There are many possible choices for these conditions, which are effectively parametrized by the Lagrangian subspaces, relative to the canonical wedge product pairing, of the de Rham cohomology group . Each defines a self-adjoint boundary value problem for the curl-operator with a real spectrum
The intersection of all these spectra,
is an invariant of the embedding of into A priori one might expect this to be a finite set, but for round balls and tori of revolution we show that is infinite. We also show that, in all cases, if and only if there is a 1-form defined in with
with representing the trivial class in Here, and in the sequel:
These conditions have been defined and analyzed by several authors, see for example [7, 17, 18]. We provide a new, and somewhat simpler proof, of the self-adjointness of these unbounded operators. Unlike in the existing literature, we provide a straightforward way to reduce the solution of the Beltrami equation to well-conditioned, integral equations on the boundary. Using an alternative integral representation, Kress worked out something similar in the case of a torus, see . Additionally, the strong uniqueness result for the Debye source representation also allows us to use these boundary equations to find both the spectrum and the eigenvectors of these self-adjoint extensions of the curl operator. We give numerical results for the cases of the unit ball and torus-of-revolution. We close by briefly considering the analogous self-adjoint boundary conditions for the time-harmonic Maxwell equations in a bounded domain.
2 Debye Sources and Potentials
In this section, we quickly review the representation of solutions to the THME([)] in terms of both potentials and anti-potentials. For the moment, we will assume that we are working in either a bounded or an unbounded domain with smooth boundary, connected . The final assumption is just for ease of exposition; later in the paper we consider regions whose boundaries have several components.
where is a scalar function, a 1-form, a 2-form, and , a 3-form; is the usual vector potential and the corresponding scalar potential, while is the vector anti-potential and the corresponding scalar anti-potential.
We assume that all of the potentials solve the Helmholtz equation, in the correct form degree. Here denotes the (negative) Laplace operator. In order for to satisfy Maxwell’s equations in :
it suffices to check that (in the Lorenz gauge)
We let denote the fundamental solution for the scalar Helmholtz equation with Helmholtz parameter which satisfies the outgoing Sommerfeld radiation condition:
All of the potentials can be expressed in terms of a pair of 1-forms defined on , which define electric and magnetic currents. As is embedded in , these 1-forms can be expressed in terms of the ambient basis from , e.g.,
We normalize both with the requirement
which is analogous to the classical requirement that the current be tangent to the boundary. We call forms satisfying this condition tangential 1-forms on
For most applications of the Debye source representation we do not use the currents and as the fundamental parameters, though in the present context we will sometimes find this to be useful. In , we introduced the notion of generalized Debye sources, , , which are scalar functions defined on by the differential equations:
known as consistency conditions. From this definition, we see that and are exact and hence their mean values vanish on ,
This is necessary for the conditions in (2.7) to hold, and thus, for to satisfy the Maxwell equations. In terms of the generalized Debye sources:
It should be emphasized that any data that satisfy (2.7) and (2.8) define a solution to the THME([)k] in . We call the pairs scalar Debye source data. This flexibility allows us to easily construct elements of , which thereby leads to an efficient method for finding Beltrami fields.
2.1 Boundary equations
Following the convention in [4, 5], we use and , which correspond to and , respectively, to represent the tangential components. Here refers to the limit from the unbounded component of and the limit from the bounded component. Note that . The scalar functions and , which equal , and , represent the normal components. These limiting values satisfy the integral equations:
For non-zero wave numbers, we can use (2.7) to rewrite these boundary equations in terms of the currents and alone. For example:
where , on a 1-form. Through this formulation, any pair of 1-forms defines a solution to the THME([)] in In general these fields might vanish in one component or the other. By imposing relations between these 1-forms we can obtain injective, and therefore surjective representations. This is described in the next section.
3 Uniqueness Theorems
We now turn our attention to Maxwell fields in interior domains. As is always the case with representations yielding Fredholm equations of index zero, proving surjectivity is reduced to proving injectivity. In  and  this is established for outgoing solutions in an exterior domain , possibly with multiple boundary components. In those works, we produced a Fredholm system of second kind for the solution to the perfect electric conductor problem, wherein is specified on , the boundary of . By showing that this system of equations has trivial null-space it follows from the Fredholm alternative that the representation is surjective.
There is an analogous system of Fredholm equations of second kind for the solution to the interior problem for equations (1.3) in with either or specified on . The interior problem is different from the exterior in that there exists a sequence of real frequencies for which there are non-trivial solutions to (1.3) that have a vanishing tangent component of the -field. On the other hand we can use the following lemma to obtain the desired result.
Let , and be a bounded domain with smooth connected boundary . If , then with
is injective. Here is the inverse of the (negative) Laplacian on scalar functions of mean zero and is a tangential harmonic 1-form on
We first observe that if is in the null-space of the map in (3.2), then the tangent components, both vanish. Theorem 4.1 in , or Theorem 5.5.1 in  show that is determined in by the pair of tangent components along , and therefore vanishes identically in
We first treat the case . Assume that we have data so that, with , the solution in defined by (2.1) is zero. The jump conditions then imply that
In the exterior , the outgoing condition and equation (6.14) from  state that:
Here . Using the PDE and (3.3) we see that
The case is quite similar to the argument in the proof of Theorem 5.2 in . Suppose that we have data so that the vector sources satisfy and the harmonic fields . We note that at zero frequency, . The jump conditions therefore show that and . On the other hand, and represent cohomology classes in , and therefore and belong to the complexification of a real Lagrangian subspace of with respect to the wedge product pairing, hence
The fact that and both vanish, along with (2.11), implies that
For later applications the following surjectivity result is very useful.
If is a bounded connected region in with smooth connected boundary , then for , all solutions to THME([)] in are represented by Debye source data Similarly, for an unbounded region, all outgoing solutions are represented by such data.
The unbounded case is done in . To prove this in the bounded case for non-zero frequencies, we use a hybrid system of Fredholm equations of second kind analogous to equations (5.5-5.7) in . Indeed, we use precisely the same set-up with the single change that instead of . We write this schematically as
If has genus then we append the cohomological conditions:
for . Here The -cycles bound chains, , contained in and the -cycles bound chains, , contained in . Together are a basis for which can be taken to be formal sums of smooth, simple closed curves on As these conditions entail restriction to codimension 1 submanifolds, they are given by bounded functionals provided that for an For bounded solutions to the THME([)], as tends to zero
For small , it is instead useful to apply Stokes theorem and (1.3) to replace the integrals over the -cycles with the conditions:
In  it is shown that the operator in the center of (3.8) is of the form where is a system of classical pseudodifferential operators of order The vector space and so for any the system of equations defined by (3.8)–(3.9) is of the form with compact, as a map from to itself, and hence Fredholm of index zero. Let denote pairs of functions of mean zero on and the -closure of
The functionals in (3.9) can be replaced by functionals that extend as bounded functionals on and are unchanged on closed 1-forms. The simplest way to do this is to use a basis for comprised of smooth embedded, simple closed curves, Each of the cycles is then homologous to a sum of the form
For , the equations in (3.9) can then be replaced with
Since is a smoothly embedded simple closed curve in the tubular neighborhood theorem implies that there is a smooth family of simple, closed curves so that:
an open subset of
We can replace the integrals over appearing in (3.14) with
Since the curves are all homologous it is clear that if is a closed 1-form then
From the properties of the family of curves it follows that there is a bounded 1-form supported in so that
and this therefore is an -bounded functional. Replacing the conditions in (3.9) with the conditions
As in our earlier work, if the non-trivial data belongs to the null-space of this system, then it follows that is a topologically trivially harmonic form and therefore On the other hand, Lemma 3.1 implies that the solution is non-trivial. Hence only arises if is in the spectrum of the Maxwell equations in with the Dirichlet conditions on , i.e. . The surjectivity for follows from the Fredholm alternative.
Let denote the non-zero exceptional wave numbers for which there exist non-trivial solutions to
with . Let denote the dimension of the eigenspace corresponding to and the dimension of the null-space of the hybrid system, (3.8)– (3.9). Clearly the injectivity of the Debye representation shows that . Suppose that is a basis for this eigenspace. If
then integrating by parts shows that
For details, see (7.128) later on. Since the tangent components , Theorem 4.1 in  again implies that the 1-forms are linearly independent. Thus (3.21) constitutes independent conditions that are necessary and sufficient for equation (3.19), with the boundary condition , to be solvable. On the other hand, the hybrid system is solvable if and only if satisfies linear conditions. This shows that , and that the Debye source representation is surjective in this case as well.
Suppose that for we have non-trivial data in the null-space of the operator in (3.8) for which the integrals over the -cycles in (3.9), along with those in (3.11), vanish. In this case would be a topologically trivial harmonic 1-form on , which must therefore vanish. Hence as well. The harmonic 2-form would vanish on the boundary and represent the trivial class in , which implies that it too must vanish. Thus the field defined by would vanish in and this contradicts Lemma 3.1. ∎
These results easily extend to bounded, connected domains whose boundary has more than one component. There is a unique component that is also the boundary of the unbounded component of . We let denote the other components of , each of which is the boundary of a connected component of . We let denote the real exceptional frequencies for which there are -Neumann fields for which the restriction of to each component of is topologically trivial. By -Neumann fields, we mean fields that satisfy the THME([)] with vanishing normal components on the boundary, see .
To represent solutions to THME([)] in we use Debye sources
on the boundary components. The scalar sources are assumed to have mean zero on each component of . As usual, we use these sources to define 1-forms via equation (2.7), enforcing the relations
for . Hence, for ,
Using such data it is easy to extend the uniqueness result from Lemma 3.1 to cover the case where has multiple components; surjectivity follows directly as well.
If is a bounded connected region in with smooth boundary , then, for , all solutions to THME([)] in are uniquely represented by data in .
Briefly, for any , the Debye representation of using the data in and the relations (3.23) is one-to-one and onto.
The uniqueness result is all that is really needed as we can give Fredholm equations of second kind on for both and in terms of data in , and use the argument from the proof of Lemma 3.1. This argument is essentially identical to that given to prove Theorem 3.4 in . We suppose that there is data in so that the solution, , specified in by this data vanishes in . We let
with the unbounded component and for .
We let denote the solution defined by this data in and the solution in . The tangential boundary data for these solutions in the components of are determined by jump conditions, and therefore:
for . Using the proof of Lemma 3.1 we deduce that , and therefore . Using the argument used to prove Theorem 7.1 in , we also conclude that for . This completes the proof of the uniqueness statement. Surjectivity is proved using the same argument as above to establish this property when is connected. The case follows by combining the argument at the end of the proof of Lemma 3.1 with the proof of Theorem 5.2 in . ∎
4 The Complex Structure
Let denote a domain in (bounded or unbounded, for now) with boundary , and let be a function defined in taking values in . For most applications we take to be locally constant in the connected components of , but the first results in this section do not require any such assumption.
Let and be non-zero complex constants and denote the set of all solutions, defined in , to the system of equations
We define a map on by setting
It is an easy calculation to check that if solves (4.1), then does as well. Since the Hodge star-operator satisfies , a further calculation shows that
We summarize this as follows:
The map defines a complex structure on the complex vector space .
This observation appears, in passing, in Section 4.4 of .
It is clear from its definition that the map is complex linear. In light of that, it defines two projection operators
We denote the image of under these projections by and respectively. Another easy calculation demonstrates the following proposition.
The subspace of is eigenspace of and is the eigenspace of .
If , then , and therefore satisfies the Beltrami equation:
If , then we have that
hence the space can be decomposed as a direct sum of solutions to these two Beltrami equations.
4.1 The Tangent Map
The space is an infinite dimensional vector space whose dependence on the function is complicated. It is often the case that an element of this vector space is uniquely determined by either or . Moreover, this data can be freely specified to be any tangent 1-form on . In this case, the operator defines a complex structure on the fixed vector space of -forms on , , which depends on the coefficient function . On the tangential boundary data, this operator, , is given by
This operator is the analogue for the time-harmonic Maxwell equations of the Dirichlet-to-Neumann map for a scalar elliptic equation. The remaining Cauchy data consists of the normal components of and , which in this formulation are just restrictions of the 2-forms to the boundary:
Using equation (4.1) these quantities can be determined from and :
From these relations it is clear why should be understood as the analogue of the Dirichlet-to-Neumann map.
As the dependence of on is now quite important we denote this operator on by . That is far from obvious, though nonetheless true. The dependence of on is quite complicated. As tends to zero the family of operators diverges somewhat like the matrices
In another paper we will study the behavior of this family of operators as the constant function tends to zero and infinity in the closed upper half plane. For now we observe that if uniquely determines the solution to (4.1), then the condition
holds if and only if
The former condition is an immediate consequence of the latter. On the other hand if the boundary condition holds, then and are solutions to (4.1) with the same tangential boundary data. By uniqueness they must agree throughout .
4.2 -Neumann Fields and Beltrami Fields
Let us first examine the case in which is an unbounded region. If is the complement of a bounded region , and is constant outside of a compact set, , then in and we can use (1.6) to define the subset of outgoing solutions to (4.1). We denote this set by . From the definition of it is easy to see that maps to itself.
In applications to fluid mechanics and plasma physics [2, 8, 9, 13] a particular subspace of solutions to (4.5) plays an important role: those solutions with vanishing normal components. If we let denote the unit outward normal vector field along , then in the exterior form representation this condition is expressed by
As their normal components vanish, these fields exert no outward force on the boundary, and are therefore called force-free fields. Once again it follows easily from the definition that preserves this condition. In this case, equation (1.3) implies that the tangent components of these fields are closed forms on the boundary . We say that a -Neumann field is topologically trivial if defines the trivial class in . One could also say that a -Neumann field is topologically trivial if defines the trivial class in , but for the moment we just consider the -component. In Section 7.4 we consider these conditions in a broader context. In the case is an unbounded region, and is a constant, then it is proved in  that an outgoing -Neumann field for which either component is topologically trivial is automatically zero.
If is a bounded domain, or if we restrict attention to outgoing solutions in , then it is known that the space of solutions to (4.1) that satisfy the boundary condition (4.13) is finite dimensional. In the case that is piecewise constant these solutions are called -Neumann fields , which we denote by , , respectively. Since preserves these subspaces, it again defines a complex structure and so these vector spaces split into the eigenspaces of
We can represent these vectors spaces as , from which it is apparent that . For real , the equations are real, so we can take bases of the forms , with real.
Now suppose that is the complement of a bounded region with boundary , and is constant. The space of -Neumann fields is well understood to be closely related to the topology of . If the total genus of the components of is , then the remarks above imply that
This will allow us to prove the following result:
If is the complement of a finite union of smoothly bounded regions, then
for all .
As shown in , an outgoing solution to THME([)k], with is uniquely determined by the data