Singularities of electromagnetic fields in

Singularities of electromagnetic fields in polyhedral domains Martin COSTABEL and Monique DAUGE

Abstract. In this paper, we investigate the singular solutions of time harmonic Maxwell equations in a domain which has edges and polyhedral corners. It is now well known that in the presence of non-convex edges, the solution elds have no square integrable gradients in general and that the main singularities are the gradients of singular functions of the Laplace operator. We show how this type of result can be derived from the classical Mellin analysis, and how this analysis leads to sharper results concerning the singular parts which belong to H 1 : For the generating singular functions, we exhibit simple and explicit formulas based on (generalized) Dirichlet and Neumann singularities for the Laplace operator. These formulas are more explicit than the results announced in our note [9].

Introduction Solutions of time-harmonic three-dimensional Maxwell equations

curl E ? i! H = 0 and curl H + i! "E = J with electric or magnetic boundary conditions have singularities near corners and edges of the boundary of the domain. This well-known fact has, for example, important consequences for the construction of numerical approximation of the solution. Just as for other elliptic boundary value problems, the singularities can be analyzed by standard techniques [15, 18, 13, 10] that go back to Kondratev's technique of Mellin transformation. The special form of Maxwell equations allow to go further in the analysis of the regularity and the singularities. Some known results are:  the H 1 regularity for convex domains, Saranen [22],  the H 1=2 regularity for Lipschitz domains, Costabel [7],  a description of singular functions for corners with a smooth basis, Saranen [21],

 corner singularities for the corresponding two-dimensional problem, Moussaoui

[19], see also section 3. Further regularity results can be found in Kricek - Neittaanmaki [17], Hazard Lenoir [14], Amrouche - Bernardi - Dauge - Girault [1]. In all these cases, the main singularity is the gradient of a singular function of the Dirichlet or Neumann problem for the Laplace operator, thus is reduced to a problem that can be considered as well known [13, 10]. This relation can be extended to a more general class of piecewise smooth domains that have \screen" or \crack" parts, Birman - Solomyak and Filonov [3, 4, 11]. We will show that, in such a class of domains, not only the rst, but all singular functions for the Maxwell boundary value problems can be obtained in simple ways from Dirichlet or Neumann singular functions of the Laplace operator. More generally, we will assume that the domain is polyhedral, that is, its boundary consists of plane faces, straight edges, and corner points: such a polyhedron needs not to be Lipschitz nor simply connected. Thus can have screen parts, in which case there is only H s regularity with s < 1=2 . We nd another type of non-Lipschitz domains with the same bad regularity: These domains are, as exempli ed by a domain between two cones with the same vertex, not locally simply connected. Here the simple equation \Maxwell regularity = Dirichlet regularity ?1 " is violated: The Dirichlet problem can even have H 2 regularity in such a case. The singular functions at such a corner are generated by topological objects: the elements of the cohomology space of the base of the cone, see section 7. For a better understanding of this new phenomenon, consider the case of a domain between two circular cones with the same vertex and the same axis. In spherical coordinates we have with 0 < 0 < 1 <  in a ball B (0; 0) :

\ B (0; 0) = f(; ; ') j  2 (0; 0); 0 <  < 1; ' 2 [0; 2)g: We consider the functions (; ; ') = log tan 2 and (; ; ') = ': p Both functions are harmonic, and with r = x2 + y2 =  sin  we have  yz ; ?1T and v := grad = ? y ; x ; 0T : ; u := grad  = 1 xz r2 r2 r2 r2 Both u and v are harmonic vector elds: curl u = curl v = 0; div u = div v = 0: On @ near the vertex, u is normal and v is tangential. Thus u satis es \electric" and v \magnetic" boundary conditions. Both functions are in H s near the vertex for s < 1=2 . They correspond to electrostatic and magnetostatic singularities, respectively. 2

Maxwell's equations are not an elliptic system. But the elimination of one of the two elds E or H yields a variational formulation in a special \physical" energy space which is X := H (div) \ H (curl) , the space of square integrable vector elds with square integrable curl and divergence. The underlying boundary value problem is then a second order elliptic system, but set in a non-standard space. Instead of this space X , also the Sobolev space H 1 could be used in a similar variational formulation. Both variational formulations are based on the same bilinear form and both have unique solutions which, for non-convex domains, will not coincide in general. The second one is the projection in X of the rst one onto H 1 . But the most important fact is that only the formulation in the space X gives back a solution of the original Maxwell equations. This situation is important for numerical approximations: If one uses standard nite elements that are contained in H 1 , then the \true" solution in X cannot be approximated, and mesh re nement at the corners and edges does not help. A possible solution is to augment the nite element space by the explicitly known singular functions, Assous - Ciarlet - Sonnendru cker [2] and Bonnet - Hazard - Lohrengel [5]. In such a method, the approximation is determined by the regularity of the regular part, i.e. the solution minus the singular function. This regularity can be quite dierent for dierent choices of the singular function. In particular, if the singular function is not constructed by our explicit formulas, but from abstract principles, then this regularity can be quite low, typically H s with s < 4=3 , see sections 3 and 8.

0.a Plan

The outline of our paper is as follows: In section 1, we start from the classical Maxwell equations with general bounded permeability  and permittivity " and deduce equivalent variational formulations in spaces of X -type and we conclude this section by an important result about the H 1 regularity of the divergence (Theorem 1.3). In section 2, assuming that  and " are piecewise smooth in a polyhedral partition of the domain , we obtain formulations of the problems in a boundary value form. From then on, we assume that " and  are smooth. In subsection 2.c, we introduce the variational formulation in H 1 -type subspaces of the X -type spaces. For the analysis of singularities, we can then even freeze coecients at the corners and restrict ourselves to the simplest case " =  = 1 . Before presenting in section 4 the general principles of elliptic regularity and singularities in three-dimensional domains with edges and corners in the X -type spaces, we devote section 3 to simpler and more explicit results for two-dimensional domains (polygons). In section 5, we describe the two-dimensional generators of the edge singularities (Lemma 5.1). In section 6, we prepare for the Maxwell 3D corner singularities by recalling the properties of the Laplace operator in 3D cones for Dirichlet and Neumann conditions. All our explicit formulas for all the Maxwell corner singularities are constructed in section 7, and our results are summarized in subsection 7.e: we have got a classi cation in three main types, for example concerning the electric eld, 3

 The gradients of Dirichlet Laplace singularities,  The divergence free elds whose curls are gradients of Neumann Laplace singularity,  The elds whose divergences are Laplace Dirichlet singularities.

When the domain is not locally simply connected in the neighborhood of the corner, the rst two types are enriched by topological singularities of similar structure (the notion of Dirichlet and Neumann Laplace singularities has to be extended in a suitable way), but this concerns only the singularity exponents ?1 and 0 , respectively. The other singularity exponents are those of the Laplace Neumann problem and those of the Laplace Dirichlet problem + ?1 (Table 1). Concerning the magnetic eld, the roles of Neumann and Dirichlet conditions are interchanged. Finally if we go back to the original Maxwell equations, we nd out that only the rst two types are present (Table 2). We conclude our paper in section 8 by results about the main singularities in nonconvex domains in the X -type spaces and in H 1 -type spaces. While they are gradients of Laplace singularities in the rst case, they can only be described as sorts of Stokes singularities in the second case, see Kozlov - Mazya - Rossmann [16].

0.b Domains and Sobolev spaces

We end this introduction by a few de nitions about domains and spaces. We want to consider rather general piecewise smooth domains which need not to be Lipschitz, in general: such domains can easily appear in applications and some of them have already been studied as mentioned above. As in [10], the de nition of the classes of domains is recursive. Let B (x; r) denote the ball of center x and radius r . In R2 we de ne:  The 2D corner domains as bounded domains in R2 or S2 such that in each point x of the boundary there exists rx > 0 small enough such that to each connected component x;i , of \ B (x; rx) belongs a dieomorphism x;i transforming x;i into a neighborhood of the corner 0 of a plane sector of opening in (0; 2] , x being sent into 0 .  The pseudo-polygonal domains as the 2D corner domains with straight sides (indeed any bounded domain whose boundary is a nite union of segments). In R3 we de ne:  The 3D corner domains as bounded domains in R3 such that in each point x of the boundary there exists rx > 0 small enough such that to each connected component x;i of \ B (x; rx) belongs a dieomorphism x;i transforming x;i into a neighborhood of the corner 0 of a cone ?x;i of the form fx 2 R3; x=jxj 2 Gx;ig with Gx;i a 2D corner domain of S2 , x being sent into 0 .  The pseudo-polyhedral domains as the 3D corner domains with straight faces (indeed any bounded domain whose boundary is a nite union of polygons). 4

We say that in one of these classes is locally simply connected if for any x in its boundary, \ B (x; rx) is simply connected. The space H 1( ) is the space of complex-valued distributions u 2 D 0( ) which belong to L2( ) and such that each component of their gradients belongs to L2( ) . The space H 1=2(@ ) is the space of traces of H 1( ) where it is understood that @ is the \unfolded" boundary of , that is, in the neighborhood of each point x 2 n in the topological boundary of , @ is the disjoint union of the parts of the boundaries of

x;i which are contained in n . The space H ?1=2(@ ) is the dual space of H 1=2(@ ) . Moreover, we introduce the spaces H (curl ; ) and H (div ; ) :

H (curl ; ) = fu 2 D 0( )3 j u 2 L2( )3; curl u 2 L2( )3g

(0.1)

H (div ; ) = fu 2 D 0( )3 j u 2 L2( )3; div u 2 L2( )g: (0.2) More generally, for  a 3  3 matrix with L1 elements on , we introduce the space H (div ;  ; ) : H (div ;  ; ) = fu 2 D 0( )3 j u 2 L2( )3; div(u) 2 L2( )g:

(0.3)

Note that the space H (div ; ) de ned above is a particular case of the previous de nition with  = I3 . For any u 2 H (curl ; ) , the tangential trace u  n is well de ned in H ?1=2(@ )3 due to the Green formula:

8v 2 H

1 ( )3

Z

:



u
curl v ? curl u
v = u  n; v@ :

(0.4)

Similarly, for any u 2 H (div ;  ; ) , the normal trace (u)
n is well de ned in H ?1=2(@ ) by the Green formula:

8' 2 H

1 ( )

:

Z



div(u) ' + u
grad ' = (u)
n; '@ :

(0.5)

1 Variational formulations of Maxwell's equations In this section, we assume that is a 3D corner domain as de ned above and quite weak hypotheses about the permeability  and the permittivity " . The results of this section are also true for general Lipschitz domains, but since we are only interested in piecewise smooth domains later on, we do not try to look at the most general possible class of domains here. After the formulation of the classical time harmonic Maxwell equations, we investigate variational formulations for either the electric eld or the magnetic eld. We prove equivalences and a regularity result for the divergences of the elds (or for a Lagrange multiplier { pseudo-pressure { in the case of a saddle-point formulation). 5

1.a Time harmonic Maxwell's equations Let " and  two complex 3  3 matrices with L1 elements on such that their symmetric part is positive in the sense that there exist 0 < 0  1 such that for all x 2 and for all  2 C 3 : 0jj2  Re("(x)
)  1jj2 and 0jj2  Re((x)
)  1jj2: The classical time harmonic Maxwell equations at the frequency ! in a body occupying , with permeability  and permittivity " are curl E ? i! H = 0 and curl H + i! "E = J in : (1.1a) Here E is the electric part and H the magnetic part of the electromagnetic eld. The right hand side J is the current density. The exterior boundary conditions on @ are those of the perfect conductor ( n denotes the unit outer normal on @ ): E  n = 0 and H
n = 0 on @ : (1.1b) If the body is formed by several dierent homogeneous media, " and  are piecewise constant and there are internal transmission conditions at the interfaces contained in the functional formulation (see x2). Equations (1.1a) hide equations on the divergence of the elds, as soon as ! is not 0 : taking the divergence of (1.1a) leads to (1.1c) div("E ) = i!1 div J and div(H ) = 0: In order to prepare for the Mellin approach of singularities, we prefer an elliptic formulation of the equations, which can be clearly explained in this framework of very general " and  . We obtain such a formulation by the elimination of one of the elds (either H or E ). But, before writing a system of partial dierential equations with boundary conditions (see x2), we have to describe variational formulations, the reverse order of steps leading almost surely to wrong formulations. Let us assume that J 2 H (div ; ) , i.e. J belongs to L2( )3 and its divergence div J belongs to L2( ) . The equations (1.1a) and (1.1c) yield immediately that if E and H are in L2( )3 , then they belong respectively to the following spaces E 2 H (curl ; ) \ H (div ; " ; ) and H 2 H (curl ; ) \ H (div ;  ; ): Taking account of the boundary conditions (1.1b), we obtain that E 2 XN and H 2 XT ; with XN = fu 2 H (curl ; ) \ H (div ; " ; ) j u  n = 0 on @ g and XT = fu 2 H (curl ; ) \ H (div ;  ; ) j (u)
n = 0 on @ g: These spaces are our variational spaces. 6

1.b Variational formulation for the electric eld f 2 X . As a consequence of the assumptions,  is invertible. Thus E 2 XN . Let E N

f. E , and the second versus i! E Let us integrate the rst equation of (1.1a) versus T?1 f f2X Since for E N and H 2 H (curl ; ) , there holds:

Z



we obtain

E 2 X N ; 8E 2 X N ; f

H
curl E dx = f

Z

Z



curl H
Ef dx;

?1 curl E
curl f E ? !2 "E
f E = i!

Z

f: J
E

(1.2)

Taking account of the equation (1.1c) on the divergence of E , we introduce a parameter s > 0 and the new right hand sides Z Z s f [J ; s](v) = i! J
v + i! div J div "v and g[J ] = i!1 div J : (1.3)



Then we de ne the following variational problem

u 2 XN ; 8v 2 XN ;

Z



?1 curl u
curl v + s div "u div "v ? !2 "u
v = f (v);

(1.4)

and its saddle-point version, which involves a Lagrange multiplier p (a pseudo-pressure): (u; p) 2 XN  L2( ); 8(v; q) 2 XN  L2( ); 8 Z > > > < > > > :



?1 curl u
curl v + s div "u div "v + p div "v ? !2 "u
v = f (v); Z



div "u q =

Z



(1.5)

g q:

The following statement yields the conditions of equivalence between problems (1.1), (1.4) and (1.5) if f and g are de ned in (1.3). The essential argument relies on the properties of the operator 1
Dir : H ( ) ?! H ?1( ) " (1.6)  7?! div " grad : The assumption about " implies that the sesquilinear form associated with ?
Dir " is 1 coercive on H ( ) : Z Re " grad 
grad   0jj2H ( ) : 1



Thus the operator ?
Dir " is invertible from its Dirichlet domain 

1 Dir 2 D(
Dir " ) = f 2 H ( ) j
"  2 L ( )g onto L2( ) and has a discrete spectrum.

7

Theorem 1.1 We assume ! 6= 0 . Let J 2 H (div ; ) ; for a xed s > 0 , f = f [J ; s] and g = g[J ] as de ned in (1.3). (i) If (E ; H ) solves (1.1), then u = E solves (1.4) and (u; p) = (E ; 0) solves (1.5). (ii) If u solves (1.4) and !2=s is not an eigenvalue of the Dirichlet operator ?
Dir " on ? 1

, then (E ; H ) = (u; (i!) curl u) solves (1.1). (iii) If (u; p) solves (1.5), then p = 0 and (E ; H ) = (u; (i!)?1 curl u) solves (1.1). Proof. (i) was proved while stating problems (1.4) and (1.5).

(ii) In (1.4) let us take as test functions all elds v = grad R with  2 D(
Dir " ) , which   ensures that grad  2 XN . Let us denote by a; b := a b the hermitian scalar product on L2( ) . Using the expression (1.3) of the right hand side, and the identities, 1 valid for  2 H ( )















"u ; grad  = ? div "u ;  and J ; grad  = ? div J ;  ; we easily arrive at

2  div "u ? g ; s
Dir " +!  =0 2 Dir for all  2 D(
Dir " ) . Thus if ! =s is not an eigenvalue of ?
" , we nd that div "u = g . We deduce that u solves problem (1.2). Thus: curl ?1 curl E ? !2 "E = i!J : Setting H = (i!)?1 curl u , we arrive at (1.1). (iii) We obtain similarly that

 8 2 D(
Dir p ;
Dir " ); "  = 0; ?1 whence p = 0 , since
Dir " is invertible. Thus u solves (1.2), and (u; (i!) curl u) solves (1.1). Both formulations (1.4) and (1.5) are strongly elliptic in the following sense. The norm k
kXN of XN is given by Z

kukXN = j curl uj2 + j div "uj2 + juj2: 2



The principal part of the sesquilinear form associated with problem (1.4)

a(u; v) =

Z



?1 curl u
curl v + s div "u div "v

is coercive on XN . Moreover, concerning the saddle-point formulation (1.5), we introduce Z b(p; v) = p div "v

8

and, as an easy consequence of the invertibility of
Dir " , we have the Babuska{Brezzi inf-sup condition, for a constant > 0 : 8p 2 L2( ); sup b(p; v)  kpk : v2XN

kvkXN

L2 ( )

Thus, being in presence of strongly elliptic systems, we can be sure that, with some adaptation due to the non-standard variational space, the classical Mellin approach will be applicable in the situation of smooth coecients and domains with edges and corners.

1.c Variational formulation for the magnetic eld

We describe the situation for the magnetic eld H with less details, since there are numerous symmetries and Theorem 1.1 yields already information for H . We have already seen that a suitable variational space for H is XT . The rst variational formulation is obtained by integrating the second equation of (1.1a) versus T"?1 H f , and the rst one versus i! H f , for any H f 2X : T Z



f ? ! 2 H
H f "?1 curl H
curl H

=

Z



f =: h(H f ): "?1J
curl H

(1.7)

Taking account of the equation (1.1c) div H = 0 , we obtain the following variational problem

u 2 XT ; 8v 2 XT ;

Z



"?1 curl u
curl v + s div u div v ? !2 u
v = h(v);

(1.8)

and its saddle-point version: (u; p) 2 XT  L2( )=C ; 8(v; q) 2 XT  L2( )=C ; 8 Z > > > < > > > :

"?1 curl u
curl v + s div u div v + p div v ? !2 u
v = h(v); (1.9)

Z div u q = 0:

The Laplace-like operator which plays a similar role as
Dir " is the Neumann operator Neu
 de ned from its domain 1 2 D(
Neu  ) = f 2 H ( ) j div  grad  2 L ( ) and @n  = 0 on @ g Neu Neu by
Neu   = div  grad  . The operator ?
 is invertible from D(
 )=C onto L20( ) (the subspace orthogonal to constants) and has a discrete spectrum.

9

Theorem 1.2 We assume ! 6= 0 . Let J 2 H (div ; ) ; h is de ned from J in (1.7).

For a xed s > 0 : (i) If (E ; H ) solves (1.1), then u = H solves (1.8) and (u; p) = (H ; c) solves (1.9) for any c 2 C . (ii) If u solves (1.8) and !2=s is not an eigenvalue of ?
Neu  on , then (E ; H ) = i ? 1 ( ! " (curl u ? J ); u) solves (1.1). (iii) If (u; p) solves (1.9), then p is a constant and ( !i "?1(curl u ? J ); u) solves (1.1).

1.d Symmetric roles of divergences and pressures. Regularity

Under a weak assumption of regularity on the right hand sides, we obtain the H 1 regularity for the divergence div "u in (1.4) and the pressure p in (1.5). This comes from the fact that div "u and p are solutions of independent boundary value problems. Let " = T " be the hermitian adjoint of " . Theorem 1.3 Let s > 0 , f 2 L2( )3 and g 2 L2( ) . We assume that !2=s is not an eigenvalue of the Dirichlet operator ?
Dir " on . (i) If u solves (1.4), then div "u is the solution q of the Dirichlet problem 8
> > < > > > :



?1 curl u
curl v ? grad p
"v ? !2 "u
v = f0(v); Z

? "u grad q =

For any s > 0 and v 2 XN , we de ne, cf (1.3)

Z

fs(v) = f0(v) + s g div "v:

11

Z



g q:

(1.13)



Then, if f0 2 L2( )3 and g 2 H 1( ) , we can prove, as a consequence of Theorem 1.3 and of the density of D ( )3 in H (curl; ) that (u; p) solves (1.13) if and only if (u; p) solves (1.5) with f = fs .

2 Second order elliptic version of Maxwell's equations As a second step towards the analysis of singularities, we are going to formulate problems (1.4) and (1.5) (or (1.8) and (1.9)) as elliptic boundary value problems. Here we leave the most general framework, and assume that " and  are piecewise smooth, the regions j , j = 1; : : : ; J , where they are smooth forming a polyhedral partition of the domain which is supposed itself to be pseudo-polyhedral in the sense of the de nition in subsection 0.c. We denote by Fjk the (open) faces of j , by F the set of the interior faces (the Fjk inside ) and by F0 the set of the faces contained in @ . As our aim is to concentrate on the singularities of the solutions, we can now take advantage of elliptic theory and drop the non-principal parts in the elliptic variational problems (1.4) and (1.5), which corresponds to setting ! = 0 . We x also s = 1 . In order to give sense to natural boundary conditions, we also take more regular right hand sides: f or h 2 L2( )3 and g 2 H 1( ) .

2.a Transmission problems for the electric eld Let us begin with problem (1.4), which is now

u 2 XN ; 8v 2 XN ;

Z



?1 curl u
curl v + div "u div "v =

Z

f
v;

(2.1)

with f 2 L2( )3 . This problem has a unique solution. The conversion of (2.1) to a boundary value problem is obtained by a judicious choice of test functions. We rst note that if " is discontinuous, the space XN does not contain C 1 ( )3 , but the piecewise smooth elds v , whose restrictions to j are denoted vj , satisfying:

vj 2 C 1 ( j )3; j = 1; : : : ; J [ v  n ] = 0 ; 8 F 2 F and v  n = 0; 8F 2 F0; F > F > : [("v)
n]F = 0; 8F 2 F ; 8 > >
> > < > > > :



?1 curl u
curl v + div "u div "v + p Z



div "v =

div "u q =

Z Z



f
v;

(2.4)

g q:

2.b Transmission problems for the magnetic eld

As for problem (1.8), we write its principal part as follows

u 2 XT ; 8v 2 XT ;

Z



"?1 curl u
curl v + div u

div v =

Z

h
v;

(2.5)

with h 2 L2( )3 . This problem has a unique solution. The conversion of (2.5) into a transmission problem is performed in the same way as before. The interior equations are:

curl "?j 1 curl uj ? j grad div j uj = fj in j ; j = 1; : : : ; J: 13

(2.6a)

The stable boundary conditions are: 8 < [u  n]F = 0; 8F 2 F ; : [(u)
n] = 0; 8F 2 F and u
n = 0; 8F 2 F0: F F The natural boundary conditions are: 8
>
> : u 2 L2( )2; curl u 2 L2( ) and div u 2 H 1( ):

(3.2)

We are going to investigate the regularity of the solutions u in the scale of Sobolev spaces when the eld f itself belongs to a Sobolev space H s?1( )2 , with s  1 . Since the boundary value problem (3.2) is an elliptic system, the solution u belongs to H s+1 (V \ )2 for any neighborhood V s.t. V does not meet any corner of . The singular behavior of u is attached to corners.

3.a Corners

Let o be an element of the unfolded boundary of , corresponding to a point x 2 n . Thus o belongs to the boundary of one of the x;i and we say that o is a corner of if the corresponding sector ?x;i is non-trivial (opening 6=  ) and we denote ?x;i by ?o and x;i by Vo . Let O be the set of the corners o of . At each corner o , we associate local polar coordinates such that ?o = f(ro ; o) j ro > 0; 0 < o < !o g with !o the opening of ?o . The regularity and the structure of the solution u only depend on special \eigenspaces" of pseudo-homogeneous functions Z (?o ) de ned below.

3.b Pseudo-homogeneous functions If ? is a plane sector and  2 C , let S (?) be the space S (?) =

n

u 2 XN

loc (? )

j

Q o X  u(x) = r logq r U q () ; q=0

(3.3)

where u 2 XNloc(?) means that for any bounded open set V such that 0 62 V , u belongs to XN (? \ V ) . Thus u  n is zero on @ ? . The singularities arising at the 16

corner of ? belong to nite-dimensional subspaces Z  (?) of S (?) , with  belonging to a discrete set (?) that we are going to introduce now. We rst need the auxiliary space Y (?) de ned as the subspace of S (?) : n Y (?) = u 2 S (?) j div u = 0 on @ ?; o (3.4) curl curl u ? grad div u is polynomial : If  is a positive integer, Y (?) contains the space P (?) of homogeneous polynomial (thus non-singular) elds u of degree  satisfying the boundary conditions u  n = 0 and div u = 0 . That is why we introduce a complement Z  (?) of P (?) in Y (?) Y (?) = Z  (?)  P (?): (3.5) Of course, if  is not a positive integer, Z  (?) is simply n o u 2 S (?) j div u = 0 on @ ?; curl curl u ? grad div u = 0 : We denote by (?) the set of  2 C such that Z (?) is not reduced to f0g .

3.c Explicit form of singularities

We want to give now the set of exponents and a basis of singular functions for any sector ? with opening ! 6=  . We postpone the proofs to section 5 where the edge singularities are treated. As the two-dimensional corner singularities are a part of them, we can extract from Lemma 5.1 the relevant results. Lemma 3.1 Let ? be a plane sector of opening ! 6=  and let Dir(?) be the set of Laplace Dirichlet exponents on ? : Dir(?) = f k! ; k 2 Zg if ! 6= 2 and Dir(?) = f k2 ; k oddg if ! = 2: Then the corresponding set of (electric) Maxwell exponents is (?) = f 2 R n f1g j  ? 1 or  + 1 belongs to Dir(?)g: The corresponding spaces of singular functions are generated: (i) For  + 1 2 Dir(?) : if  62 N , by (r sin ; r cos ) , if  2 N , by (r (log r sin  +  cos ); r(log r cos  ?  sin )) . These functions are the gradients of the Dirichlet singular functions of the Laplace operator, thus have zero curls and regular divergences. (ii) For  ? 1 2 Dir(?) : if  62 N , by (r sin ; ?r cos ) , if  2 N , by (r (log r sin  +  cos ); ?r(log r cos  ?  sin )) . The divergences of these functions are the Dirichlet singular functions of the Laplace operator. 17

3.d Regularity and singularities in X

From now on, let s denote the regularity exponent. We assume s  1 . For each o 2 O introduce the following subset of admissible exponents s(X )(?o ) which correspond

to admissible singular functions contained in the \energy space", cf (2.3d) and which do not belong to H s+1( )2 . Let o = (o ) be a smooth cut-o function with its support in Vo and equal to 1 in a neighborhood of o . We set n s(X )(?o ) =  2 (?o) j ?1 < Re   s; o 9u 2 Z  (?o ); u 6= 0; ou 2 XN (?o ); div(ou) 2 H 1(?o ) : From Lemma 3.1, we can deduce that n o Dir (?) s(X )(?) =  ? 1 j  2 (0 ; s + 1) \  o (3.6) S n  + 1 j  2 (0; s ? 1) \ Dir(?)

and that for any  2 s(X )(?o ) , the space Z (?o ) is one or two-dimensional: let Uo;? be the corresponding generators. Here follows the statement of regularity and singularity for problem (3.1). Theorem 3.2 Let s  1 and let the data f belong to H s?1 ( )2 . (i) If for any o 2 O , s(X )(?o ) is empty, then the solution of problem (3.1) belongs to H s+1 ( )2 . (ii) If for any o 2 O , s(X )(?o ) does not meet the straight line Re  = s , then the solution u of problem (3.1) can be split into X u = ureg +

o;? o(ro ) Uo;?(ro; o) (3.7) +

+

with ureg 2 H s+1 ( )2 and the

o2O ; 2s(X )(?o) + coecients o;? 2 C

+

.

3.e Dierent choices of regular and singular parts

Theorem 3.2 shows the existence of a splitting into singular functions and a regular part that is as regular as desired. If the singular functions are not constructed according to our explicit formulas in Lemma 3.1, then additional singular terms can be exchanged between the \singular" and \regular" parts: we describe this phenomenon for the simple but important case of the rst singularity in XN n HN where, according to (2.7), HN is the space fv 2 H 1( )2; v  n = 0 on @ g . Let us consider the expansion (3.7) for s = 1 : only one singular function Uo=! ?1 for each reentrant corner does not belong to H 1 . If we put the other terms of the expansion (3.7) into the regular part, we obtain an expansion of the type X u = u(H ) +

o wo ; (3.8) o

o2O ; o non?convex

18

with wo 2 XN n HN and u(H ) 2 HN . The question of the regularity of this pseudo regular part u(H ) is an important problem if one wants to use the splitting (3.8) for an approximation of u by a singular function method, that is, by trial functions that are composed of the singular functions wo and regular functions (e.g. piecewise polynomials). The convergence rate of the whole method is then determined by the regularity of u(H ) . We compare ve constructions for wo and u(H ) . In each case, u(H ) will have a decomposition itself, and its regularity is determined by its rst singular function. For simplicity, we assume that there is just one reentrant corner of opening ! >  situated at the origin 0 and 0 denotes a smooth cut-o function equal to 1 in a neighborhood of 0 . In this case, u(H ) 2 H +1 with  <  , where  is the exponent of the rst singular function in u(H ) . Thus  will directly yield the convergence rate of a singular function method for the approximation of u . (i) According to Lemma 3.1, the natural choice for w is grad(0 r=! sin ! ) . In this case, the next exponent in s(X )(?) is  = 2! ? 1 if ! 6= 2 , and  = 3! ? 1 = 21 if ! = 2 . Thus  spans the whole interval (0; 1) . (ii) We can choose a divergence free form of w : curl(0 r=! cos ! ) . Indeed, this expansion is identical to (i), so  is the same.  (iii) A more abstract construction for w is described in [19] and [5]. Let  2 H 1( ) be the solution of the problem
 = SD , where SD is the rst dual singular function of the Dirichlet problem, i.e. SD 6= 0 satis es the orthogonality condition Z





SD
= 0 8 2 H 1( ) \ H 2( ):

Then the choice of w = grad  provides a splitting of u where w is orthogonal in XN to the curl-free elds in HN . Thus if curl f = 0 , this is an orthogonal decomposition within the spaces of curl-free elds. It is well known that SD has a singular part r?=! sin ! . Therefore, besides the main part c0 r=! sin ! with non-zero c0 ,  contains a singularity of exponent 2 ? ! . Thus w contains a term of exponent 1 ? ! . For ! 6= 32 ; 2 , this does not belong to s(X ) . Thus u(H ) must contain this singularity, too, and we have  = minf 2! ? 1; 1 ? ! g 2 (0; 31 ]: This is less regular than the choice (i) if ! < 32 . (iv) A similar construction in [19] and [2] is w = curl  with  2 H 1 ( ) , @n = 0 , a solution of
 = SN , SN the rst dual singularity of the Neumann problem. This gives a splitting with w orthogonal in XN to the divergence-free elds in HN . Thus if div f = 0 , this is an orthogonal decomposition within the spaces of divergence-free 19

elds. This singular function w is linearly independent of the one in (iii). But modulo H 1( ) , these two functions are proportional, and their  are also the same. (v) Another natural construction is the orthogonal decomposition of u with respect to the inner product in XN , with the part u(H ) in HN and the residual 0w in HN? . Thus u(H ) is the solution of the variational problem in HN and 0w is the dierence between these two solutions of the same Maxwell boundary value problem. In this case, u(H ) contains a singularity of exponent 1 ? ! , see (3.9) just below. Thus  is the same as in (iii) and (iv). Note that even if curl f = 0 or div f = 0 , this decomposition is in general, not the same as the one in (iii) or (iv) , respectively.

3.f Singularities in H 1 . Comparison

If instead of (3.1), we consider the variational problem in HN , the corresponding boundary value problem is the same as (3.2) except the regularity requirements: we have now u 2 H 1( )2 and no special regularity on div u , which is only L2( ) . The corresponding admissible sets of exponents are n o s(H )(?o) =  2 (?o ) j ? 1 < Re   s; 9u 2 Z (?o ); u 6= 0; ou 2 HN (?o ) : Then instead of (3.6), we can deduce from Lemma 3.1 n

o

s(H )(?) =  ? 1 j  2 (1 ; s + 1) \ Dir(?) o S n  + 1 j  2 (?1; s ? 1) \ Dir(?) :

(3.9)

Thus we see that both sets s(X )(?) and s(H )(?) are dierent if and only if Dir(?) has elements in the interval [?1; +1] , i.e. if ! >  (non-convex corner). Then n n o o s(X ) = ! ? 1 S (s(X ) \ s(H )) and s(H ) = 1 ? ! S (s(X ) \ s(H )):

4 Regularity and singularities on polyhedra (general theory) As explained in the previous section, we continue our investigations of \model situations", now in dimension 3 . So we assume that  R3 is pseudo-polyhedral with " and  equal to the identity matrix. Thus the \electric" problem under consideration is:

u 2 XN ; 8v 2 XN ;

Z



Z

curl u
curl v + div u div v = f
v;

(4.1)

and the \magnetic" problem is the same with the variational space XT instead XN . The corresponding boundary value problem (2.3) is 8 > curlcurl u ? grad div u = f in ; > < u  n = 0 and div u = 0 on @ ; (4.2) > > : u 2 L2( )3; curl u 2 L2( )3 and div u 2 H 1( ); 20

and the magnetic problem is the same with the boundary conditions u
n = 0 and curl u  n = 0 on @ instead. Of course, the interior operator curlcurl ? grad div is the vector Laplace operator. But we prefer to keep the formulation via rst order operators, because this formulation corresponds better to the weak formulation (4.1), which is also re ected in the boundary conditions. We assume from now on that the right-hand side f belongs to H s?1 ( )3 , with s  1 . As in dimension 2 , the solutions u are regular in any neighborhood which does not meet any corner or edge of . So the singular behavior of u is attached to corners and edges. In this section, we extract from Kondrat'ev [15] and Dauge [10] general information concerning the regularity and the main singularities of u : these are determined by sets of admissible exponents sN and sT and their corresponding admissible singularity spaces ZN and ZT , for each corner c and each edge e . The admissible exponents can be seen as forming a spectrum and the admissible singularities as eigenvectors. The particular study of the above sets and spaces is postponed to sections 6 and 7.

4.a Corners and edges

Let C be the set of the corners c of , that we de ne similarly to the corners of a polygon, with the requirement that for any c 2 C the corresponding cone ?x;i , that we denote ?c , is a non-trivial cone (i.e. it is neither a half space nor a dihedron). The corresponding neighborhood x;i is denoted by Vc . In local spherical coordinates, the cone ?c is: ?c = f(c; #c) j c > 0; #c 2 Gc  S2g with Gc a spherical polygonal domain. Let E be the set of the (open) edges e of : for each point x 2 e , there exists a neighborhood x;i in which coincides with the wedge ?e  R , where ?e is a plane sector given in local polar coordinates by ?e = f(re ; e) j re > 0; 0 < e < !e g with !e the opening of ?e . We obtain local cylindrical coordinates by the adjunction of a coordinate ze along the edge e . We are now going to introduce the \eigenspaces" of pseudo-homogeneous functions  ZN (?) (with  2 C and ? being alternatively ?c or ?e ) corresponding to the \electric" boundary conditions, the procedure for the \magnetic" conditions being strictly similar.

4.b Pseudo-homogeneous functions

If c is a corner, let SN (?c) be the space n

1 (? ); u(x) =  SN (?c) = u 2 XNloc(?c ) j div u 2 Hloc c c

21

Q X q=0

o

logq c U q (#c) ;

(4.3)

1 (? ) mean that for any bounded open set O such where u 2 XNloc(?c ) and div u 2 Hloc c that c 62 O , u belongs to XN (?c \ O ) and div u 2 H 1(?c \ O ) . This contains two informations:  u  n is zero on @ ?c ,  u belongs to H (curl) and div u belongs to H 1 near the edges of ?c . This de nition is slightly more complicated than in [10] (where the variational spaces are classical subspaces of H m( ) ), because we have to take into account the variational spaces and the basic regularity. The de nition of ST (?c) is similar. We now de ne YN(?c ) as the subspace of SN (?c ) : n

YN(?c) = u 2 SN (?c ) j div u = 0 on @ ?c; o (4.4) curlcurl u ? grad div u is polynomial : If  is a positive integer, YN(?c ) contains the space PN (?c ) of homogeneous polynomial elds u of degree  satisfying the boundary conditions u  n = 0 and div u = 0 . That is why we introduce a complement ZN (?c ) of PN (?c ) in YN(?c) YN(?c ) = ZN (?c)  PN (?c): (4.5) Of course, if  is not a positive integer, ZN (?c) is simply o n u 2 SN (?c ) j div u = 0 on @ ?c; curlcurl u ? grad div u = 0 : We denote by N (?c) the set of  2 C such that ZN (?c ) is not reduced to f0g . This set can be understood as a spectrum. It is discrete and has the following property: in each strip Re  2 [a; b] , the number of elements of N (?c) is nite. Moreover the eigenspaces ZN (?c) are nite-dimensional. If e is an edge, the introduction of the corresponding spaces on ?e requires the elimination of the tangential derivatives in the operator curlcurl u ? grad div u and also in the natural boundary conditions. Let ye be cartesian coordinates in the normal plane to the edge e , and we denote by a subscript y the operators from which the tangential derivatives are eliminated. We can introduce SN (?e ) as the space

SN (?e ) =

n

u 2 XN

loc (? )

e

j divy u 2 H

1  loc (?e )

;

Q o X  u(x) = re logq re U q (e) ; q=0

(4.6)

1 (? ) mean that for any bounded open set O where u 2 XNloc(?e ) and divy u 2 Hloc e such that 0 62 O , u and curly u belong to L2(?e \ O )3 and divy u 2 H 1(?e \ O ) . The de nition of ST (?e ) is similar. We now de ne YN(?e ) as the subspace of SN (?e ) : n

YN(?e ) = u 2 SN (?e ) j divy u = 0 on @ ?e ; o curly curly u ? grady divy u is polynomial ; 22

(4.7)

and choose a complement ZN (?e ) of PN (?e ) in YN(?e ) :

YN(?e ) = ZN (?e )  PN (?e ): (4.8) We denote by N (?e ) the set of  2 C such that ZN (?e ) is not reduced to f0g . Like for corners c , this set is discrete, ZN (?e ) is nite-dimensional and there holds: in each strip Re  2 [a; b] , the number of elements of N (?e ) is nite.

4.c Regularity and singularities Let  be  ?1 . We introduce the following subsets of N (?c ) and N (?e ) which

correspond to admissible singular functions contained in the \energy space", cf (2.3d). Let c = (c) be a smooth cut-o function with support in Vc and equal to 1 in a neighborhood of c . We set n

N (?c ) =  2 N (?c ) j ? 23 < Re    ? 12 ; o 9u 2 ZN (?c); u 6= 0; cu 2 XN (?c ); div(cu) 2 H 1(?c ) : Similarly, with e a smooth cut-o function equal to 1 in a neighborhood of 0 : n

N (?e ) =  2 N (?e ) j ?1 < Re   ; o 9u 2 ZN (?e ); u 6= 0; eu 2 XN (?e ); divy (eu) 2 H 1(?e ) : Note that the lower bound for Re  is but a consequence of the condition u 2 L2(?)3 . Here follows the regularity statement Theorem 4.1 Let s  1 and  2 (?1; s] . If  for any c 2 C , N (?c ) is empty,  for any e 2 E , N (?e ) is empty, then for any data f 2 H s?1 ( )3 the solution of problem (4.1) belongs to H +1 ( )3 . We are going to describe now the \ rst level" singularities, when the sets N are not all empty. For any c 2 C and  2 N (?c ) let (Uc;p)p be a basis of ZN (?c) , and similarly for any e 2 E and  2 N (?e) let (Ue;p)p be a basis of ZN (?e ) . The corner singularities are simply a linear combination of the Uc;p , whereas the edge singularities are more dicult to describe: We use a smooth function de on the closed edge e , which is equivalent to the distance to the ends of e : if for example e = fx j re = 0; ze 2 (?1; +1)g , we can take de (ze) = 1 ? ze2 . We have now to introduce the special weighted Sobolev spaces Vm(e) which are the correct spaces for the singularity coecients , de ned for m 2 N and  2 R by Vm  (e) =

n

2 L2(e) j (de)+k @zk 2 L2(e); k = 0; 1; : : : ; m e

23

o

and by interpolation for non-integers m . We also need a smoothing operator K [
] which acts like a lifting of functions on e into : in order to de ne it, we introduce the stretched variable Z z 1 dz; z~e = de(z) 0 where z = 0 corresponds to an interior point of e . The change of variable ze 7! z~e is one to one e ! R and for any function de ned on e , we set ~ (~ze) = (ze) . Then K [ ](e ; e ; ze ) is the convolution operator with respect to z~e : Z 1 ' t  ~(t ? z~ ) dt with  = re ; K [ ](e; e ; ze) = e e e de R e R where ' is a smooth function in S (R) such that R ' = 1 . We are now ready to give a statement of splitting in regular and singular parts: the assumptions (i) and (ii) below are the usual necessary ones, see [10, Thm 17.13], but the assumption (iii) only aims at simplifying the expression of the edge singularities. Theorem 4.2 Let s  1 and  2 [0; s] . If (i) For any c 2 C , N (?c) does not meet the straight line Re  =  ? 12 , (ii) For any e 2 E , N (?e ) does not meet the straight line Re  =  , (iii) For any e 2 E , N (?e ) is contained in a strip Re  2 (a; a + 1) , and for any  2 N (?e ) , ZN (?e ) has no logarithmic element, then for any data f 2 H s?1 ( )3 the solution u of problem (4.1) can be split in the following way e

u = ureg; +

X

X

X

c2C

2N (?c )

p

c;p c(c ) Uc;p(c; #c) +

X

X

X

e2E 2N (?e ) p

K [ e;p] e (e ) Ue;p(e ; e ) (4.9)

with ureg; 2 H +1 ( )3 , c;p 2 C and the edge coecients e;p 2 Vs??Re (e) .

Remark 4.3 1) In the usual statements of this sort,  is taken equal to s . Here

we allow intermediate splittings. Note that the in uence of the regularity of the data subsists in the regularity of the edge coecients. 2) If only the hypotheses (i) and (ii) hold, we still have an expansion of u in (optimal) regular and singular parts, but the edge contribution is still more complicated, due to possible Keldysh chains and supplementary \shadow" terms. In section 8, we will give more explicit splittings, based on the knowledge of the sets of admissible exponents N (?c ) and N (?e ) , and of the corresponding spaces of singular functions ZN (?c ) and ZN (?e ) . 24

5 Maxwell edge singularities Under the assumptions of the last section, our aim is to describe explicitly the spaces

Z  (4.5) and (4.8) attached to each corner and edge of . In this section, we treat the

edges. As a byproduct, we will also obtain the singularities at the corners of a plane polygonal domain.

5.a 3D Maxwell singularities in a plane sector

The boundary value problem is then two-dimensional and involves a plane sector ? of opening ! 2 (0; 2] , ! 6=  ; the polar coordinates are denoted (r; ) , the cartesian coordinates in the plane of ? are denoted y , and z is a perpendicular coordinate. Let  2 C . We search for non-polynomial solutions u 2 SN (?) of the system

curly curly u ? grady divy u = f in ?; f polynomial; with the natural boundary condition divy u = 0 on @ , and where the eld u has still 3 components, and SN (?) is the space (4.6) of pseudo-homogeneous elds of degree  . Let us note that, when  is not a positive integer, the above problem reduces to nd the non-zero solutions u 2 SN (?) of curly curly u ? grady divy u = 0 with the same boundary conditions. Since the boundary of ? is smooth outside its vertex 0 , the above boundary value problem is elliptic, and the data are smooth, we can introduce the space S (?) : n

S (?) = u = r

Q X q=0

logq r U q () j

Uq

o

2 C 1([0; !]) ;

(5.1)

and solve equivalently 8 >
: u 2 S (?)3 :

(5.2)

Let now (v; w) be the decomposition of the eld u in the system of cartesian coordinates (y; z) . Then system (5.2) is split into 2 independent problems: 8 >
: v 2 S (?)2 ; and

8 >
:  w 2 S (?): 25

(5.3)

(5.4)

Indeed problem (5.3) is also the problem attached to two-dimensional Maxwell equations in a polygonal domain. The situation for the Dirichlet problem (5.4) is well known: let

 = rei = y1 + iy2:

 if ! 6= 2 , the set Dir(?) of singular exponents is f k! ; k 2 Zg and, for  (?) is generated by  = Im   if  62 N  2 Dir(?) , the singularity space ZDir Dir and by Dir = Im   log  + !(y2= sin !) if  2 N ;  if ! = 2 , the set Dir(?) of singular exponents is f k2 ; k oddg and, for any  (?) is generated by Im   .  2 Dir(?) , ZDir

Let us consider now the two-dimensional \Maxwell-type" problem (5.3). We introduce two auxiliary scalar variables = curl v and q = div v:

(5.5)

Taking the divergence of the rst line of (5.3) yields equation (5.6a) below and the next equations (5.6b) and (5.6c) are straightforward

?
q = div f in ? and q = 0 on @ ? with q 2 S ?1(?): (5.6a) curl = grad q + f in ? with 2 S ?1(?): (5.6b) curl v = ; div v = q in ? and v  n = 0 on @ ? with v 2 S (?)2 : (5.6c) We easily see that the system of equations (5.6) is equivalent to (5.3).

5.b Non-integral exponents

In order to solve system (5.6), we begin with the simpler situation when  is not a positive integer. Then the above system of equations reduces to

?
q = 0 in ? and q = 0 on @ ? with q 2 S ?1(?): (5.7a) curl = grad q in ? with 2 S ?1(?): (5.7b) curl v = ; div v = q in ? and v  n = 0 on @ ? with v 2 S (?)2: (5.7c) We can split the solutions of system (5.7) into three natural types: Type 1 q = 0 , = 0 and v general non-zero solution of (5.7c). Type 2 q = 0 , general non-zero solution of (5.7b) and v particular solution of (5.7c). Type 3 q general non-zero solution of (5.7a), particular solution of (5.7b) and v particular solution of (5.7c). 26

Let us study successively these three types.

Type 1.

Since curl v = 0 on the simply connected domain ? , v is the gradient of a function  . Thus we have: ( vr := cos  v1 + sin  v2 = @r  v := ? sin  v1 + cos  v2 = 1r @ ; whence (we denote by v~ the function v~(r; ) = v(y) ) ~ (r; ) ? ~ (1; 0) =

Z r 1

v~r

(r0; 0) dr0

+r

Z  0

v~ (r; 0) d0;

(5.8)

which proves that since v belongs to S (?)2 ,  is the sum of a function in S +1(?) and a constant. Therefore,  can be found in S +1(?) . Then (5.6c) is equivalent to
 = 0 in ? and  = 0 on @ ? with  2 S +1(?):

(5.9)

Hence,  + 1 belongs to Dir(?) and  belongs to the space generated by Im  +1 .

Type 2.

We easily see that

Type 3.

is zero and a particular solution of (5.7c) is v = 0 .

From equation (5.7a), we obtain that  ? 1 belongs to Dir(?) and that q belongs to ?1 (?) : thus q is proportional to Im  ?1 . Then it is easy to see that = ? Re  ?1 ZDir is a particular solution of (5.7b), and that v = 21 (Im  ; ? Re  ) is a particular solution of (5.7c).

5.c Integral exponents

When  is a positive integer, we are searching for non-polynomial solutions of system (5.6). Similarly to the case when  is not an integer, we split the solutions of the system (5.6) into the three types: Type 1 q and are polynomial and v is a non-polynomial solution of (5.6c). Type 2 q is polynomial, is a non-polynomial solution of (5.6b) and v a particular solution of (5.6c). Type 3 q is a non-polynomial solution of (5.6a), a particular solution of (5.6b) and v a particular solution of (5.6c). Now the arguments are based on the evaluation of dimensions of polynomial spaces. Let Q be the space of homogeneous polynomials of degree  , P (?) the subspace of q 2 Q with zero traces on @ ? . We divide our study into three subcases: (i) ! 6= 2 and  ? 1 does not belong to Dir (?) : In equation (5.6a) the r.h.s. div f is any polynomial in Q?3 , thus the dimension of the range is ( ? 2)+ . The dimension of P ?1(?) is ( ? 2)+ too. Moreover equation (5.6a) de nes an operator from P ?1(?) 27

into Q?3 which is one to one due to the assumption that  ? 1 does not belong to Dir(?) . Therefore this operator is onto.  The r.h.s. of equation (5.6b) is any eld in Q?2  Q?2 which is divergence free. Thus the dimension of its range is 2( ? 1) ? ( ? 2)+ which is equal to  if   2 and 0 if  = 1 . The dimension of Q?1 is equal to  and equation (5.6b) de nes an operator from Q?1 into fg 2 Q?2  Q?2; div g = 0g , which is one to one for any   2 , thus onto.  The r.h.s. ( ; q) of (5.6c) is any element of Q?1  P ?1(?) . Thus the dimension of its range is  + ( ? 2)+ = 2( ? 1) if   2 and 1 if  = 1 . The space of polynomial solutions of (5.7c) is

fv 2 Q  Q j v  n = 0 and div v = 0 on @ ?g: (5.10) Its dimension is 2( + 1) ? 4 = 2( ? 1) if   2 ; if  = 1 , its dimension is either 2 if

cos ! = 0 or 1 if not. If  + 1 does not belong to Dir(?) , we check that in any case the operator of equation (5.6c) is one to one, thus it is onto: the system (5.6) has only polynomial solutions. If  + 1 2 Dir(?) , its kernel is one-dimensional, and for   2 we add to the above polynomial space (5.10) a singular function equal to the sum of grad(Im  +1 log  ) and of a polynomial: we have found now a solution of type 1. For  = 1 nally,  + 1 2 Dir(?) only if cos ! = 0 and the operator of equation (5.6c) is onto and there is no singularity. (ii) ! = 2 : The arguments are similar. Here  ? 1 and  + 1 never belong to Dir(?) . The dimensions of the polynomial spaces involving boundary conditions are slightly dierent: dim P ?1(?) =  ? 1 and the operator of equation (5.6a) has a onedimensional kernel generated by Im  ?1 . Thus it is still onto from P ?1(?) ! Q?3 .  The situation for equation (5.6b) is unchanged.  The dimension of the space Q?1  P ?1(?) is 2 ? 1 and the dimension of the space (5.10) is 2( + 2) ? 2 = 2 . The kernel of the operator of equation (5.6c) is generated by grad Im  ?1 . Thus we have only polynomial solutions. (iii) ! 6= 2 and  ? 1 belongs to Dir(?) : Then the operator of equation (5.6a) has a one-dimensional kernel generated by Im  ?1 and it is onto from the space generated by the sum of P ?1(?) and of Dir?1 which is the sum of Im  ?1 log  and of a polynomial.  Corresponding to this new solution q , we nd a new solution of equation (5.6b) = Re  ?1 log  .  Accordingly, we nd a new solution v of equation (5.6c) in the form v = 21 (Im   log ; ? Re   log  ) , which is a non-polynomial solution of type 3.

5.d Edge singularities

We summarize the results that we have just proved: 28

Lemma 5.1 The set of the exponents N (?) attached to a plane sector of opening ! is N (?) = f 2 R n f1g j  ? 1;  or  + 1 belongs to Dir(?)g where Dir (?) is the set of the Dirichlet exponents on ? :

Dir(?) = f k! ; k 2 Zg if ! 6= 2 and Dir(?) = f k2 ; k oddg if ! = 2: The corresponding spaces ZN (?) of singular functions u = (v; w) are generated by: (i) If  + 1 2 Dir(?) , then v is a Maxwell singularity of type 1: If  62 N , v = grad Im  +1 , in other words

(v; w) = (r sin ; r cos ; 0): If  2 N : v = grad Im  +1 log  , in other words

(v; w) = (r(log r sin  +  cos ); r(log r cos  ?  sin ); 0): (ii) If  2 Dir(?) , w is a Dirichlet singularity:

(v; w) = (0; r sin ) if  62 N and (0; r(log r sin  +  cos )) if  2 N . (iii) If  ? 1 2 Dir (?) , v is a Maxwell singularity of type 3: If  26 N : (v; w) = (r sin ; ?r cos ; 0): If  2 N : (v; w) = (r(log r sin  +  cos ); ?r(log r cos  ?  sin ); 0):

According to the general theory stated in x4.c, we have to consider the subset of N (?) satisfying (with  a cut-o function which is equal to 1 in a neighborhood of the corner of ? ):

?1 <  and 9u 2 ZN (?); u 6= 0 u 2 XN (?); divy (u) 2 H 1(?):

(5.11)

We examine the eect of condition (5.11) on each of the singularities (i), (ii) and (iii) in Lemma 5.1. (i) In this case  = k! ? 1 : as k 6= 0 , the lowest possible value of k is 1 . For the corresponding singularities u = (v; 0) with v Maxwell singularity of type 1, the curl and the divergence are 0 (or polynomial if  2 N ). Thus for any k  1 such that k! belongs to Dir(?) ,  = k! ? 1 satis es (5.11). (ii) If  = k! , the conditions about the curl and the divergence of u = (0; w) mean exactly that w belongs to H 1(?) . Thus  is > 0 , and for any k  1 such that k belongs to Dir (?) ,  = k satis es (5.11). ! ! 29

(iii) If  = k! + 1 , u = (v; 0) with v a Maxwell singularity of type 3: the divergence of v is itself a non-regular pseudo-homogeneous function of degree  ? 2 . Since divy (u) 2 H 1(?) , we must have  ? 2 > 0 . Then the other conditions in (5.11) are implied, and for any k  1 such that k! > 1 and k! 2 Dir(?) ,  = k! + 1 satis es (5.11). Thus, for any value of the opening ! , the most singular among the admissible elements of the ZN (?) (admissible singular functions) is u = (grad Im  =! ; 0) corresponding to  = ! ? 1: If ! >  , this is the only admissible singular function such that u does not belong to H 1(?) .

6 Basic Laplace corner singularities In order to continue the investigation of the Maxwell singularity spaces ZN (?) and ZT (?) when ? is a three-dimensional polyhedral cone, in a preparatory step we recall well-known facts about the basic Laplace singularities for Dirichlet and Neumann boundary conditions. Here we mean that we do not investigate polynomials (either as right hands sides, nor as solutions of our problems in the pseudo-homogeneous function spaces). The polar coordinates in the polyhedral cone are denoted (; #) and ? = f(; #) j  > 0; # 2 G  S2g with G a spherical polygonal domain. In contrast with the two-dimensional situation when G is an interval (thus a smooth domain), we have to introduce spaces of pseudo-homogeneous functions where the regularity of the angular functions is xed: for  2 R n

S(?) = u = r

Q X

q=0

logq r U q () j

Uq

o

2 H  (G) :

(6.1)

Of course, the space of pseudo-homogeneous functions adapted to the variational regularity of the Laplace operator is S1(?) . The set Dir(?) of the singular exponents of the Dirichlet problem is the set of the  2 C such that there exist non-zero solutions to 8 > in ?; < ?
u = 0 (6.2) u=0 on @ ?; > :  u 2 S1 (?);  (?) is the solution space. Similarly the set Neu (?) of the Neumann and the space ZDir singular exponents is the set of the  2 C such that there exist non-zero solutions to 8 > in ?; < ?
u = 0 @ u = 0 on @ ?; (6.3) > n :  u 2 S1 (?): 30

Let
Dir G be the positive Laplace-Beltrami operator with Dirichlet conditions on Dir Dir G . The operator
Dir G is self-adjoint with a compact inverse. Let 1 < 2 


Dir Neu be its spectrum and 'j be corresponding eigenfunctions. Similarly
G denotes Neu Neu the Neumann Laplace-Beltrami operator on G , Neu 0 = 0 < 1  2 


is its spectrum and 'Neu are corresponding eigenfunctions. From the expression of
in j polar coordinates  
= 1 (@)2 + @ ?
G we obtain that the sets of exponents Dir(?) and Neu(?) are given by the roots of the equations ( + 1) = j : 2

q

n

1 Dir(?) = ? 21 + j + 4; j  1 ? Dir

o

n

q

1 Neu(?) = ? 21 + j + 4; j  0 ? Neu

o

(6.4)

and the corresponding singularity spaces by: n

q

n

q

o

 (?) = span 'Dir (#) j  = ? 1 + Dir + 1 ; ZDir j j 2 ? 4

(6.5a)

 (?) = span 'Neu (#) j  = ? 1 + Neu + 1 : ZNeu j j 2 ? 4

(6.5b)

o

7 Maxwell corner singularities We continue the investigation of the Maxwell singularity spaces ZN and ZT . After those attached to plane sectors corresponding to the edges of our polyhedral domain (or to the corners of a polygonal domain), we study in this section the spaces attached to ? where ? is a three-dimensional polyhedral cone. We recall that the polar coordinates are denoted by (; #) and ? = f(; #) j  > 0; # 2 G  S2g . According to de nitions (4.3)-(4.5) we have to solve Maxwell systems in the spaces of pseudo-homogeneous functions n

1 (? ); u(x) =  SN (?) = u 2 XNloc(?) j div u 2 Hloc

Q X q=0

o

logq  U q (#) ;

and n

1 (? ); u(x) =  ST (?) = u 2 XTloc(?) j div u 2 Hloc

Q X q=0

o

logq  U q (#) ;

i.e. to nd the non-polynomial solutions of the system

curlcurl u ? grad div u = f in ?; f polynomial; with either u 2 SN (?) and div u = 0 on @ (electric), or u 2 ST (?) and curl un = 0 on @ (magnetic).

31

7.a Splitting of the problem

Here we concentrate rst on the case when  is not a positive integer. Thus, in the electric case, the problem reduces to nding non-zero solutions to 8 > < curlcurl u ? grad div u = 0 in ?; u  n = 0; div u = 0 on @ ?; (7.1) > :  u 2 SN (?): Like in the case of plane sectors, we introduce the auxiliary unknowns = curl u and q = div u: In contrast to the two-dimensional case, it is impossible to require the same type of regularity for u , and q on G , because of the edge singularities which appear as 1 (? ) contained in singularities at the corners of G . From the condition div u 2 Hloc the de nition of SN (?) , we are lead to use the space S1(?) adapted to q = div u , see (6.1). Concerning the eld , we remark that the equality = curl u implies that div = 0 and since u  n = 0 on @ ? , then
n = 0 on @ . Moreover, from the equality curl = grad div u , and from the H 1 regularity of div u , we obtain that curl belongs to L2loc(?) . Thus the natural space for is ST (?) and it is now clear that problem (7.1) is equivalent to nd non-zero solutions to the system ?
q = 0 in ? and q = 0 on @ ? with q 2 S1?1(?): (7.2a) (7.2b) curl = grad q and div = 0 in ? with 2 ST?1(?): (7.2c) curl u = and div u = q in ? with u 2 SN (?): Now we see that the \electric" and \magnetic" boundary conditions appear simultaneously inside (7.2). Thus we have better to treat both conditions together. The \magnetic" problem corresponding to (7.1) is to nd non-zero solutions to 8 > < curlcurl u ? grad div u = 0 in ?; u
n = 0; curl u  n = 0 on @ ?; (7.3) > :  u 2 ST (?); and it is equivalent to nd non-zero solutions to the system of three problems ?
q = 0 in ? and @nq = 0 on @ ? with q 2 S1?1(?): (7.4a) curl = grad q and div = 0 in ? with 2 SN?1(?): (7.4b) curl u = and div u = q in ? with u 2 ST (?): (7.4c) Like for the plane sectors, the solutions of systems (7.2) and (7.4) belong to one of three types: 32

Type 1 q = 0 , = 0 and u general non-zero solution of (7.2c), resp. (7.4c). Type 2 q = 0 , general non-zero solution of (7.2b), resp. (7.4b) and u particular solution of (7.2c), resp. (7.4c). Type 3 q general non-zero solution of (7.2a), resp. (7.4a), particular solution of (7.2b), resp. (7.4b) and u particular solution of (7.2c), resp. (7.4c).

7.b Explicit solutions of rst order problems

In the case of plane sectors, we have exhibited the singular functions (Lemma 5.1). Here, on three-dimensional cones, the Laplace singularities have no more explicit description than (6.5), but, relying on them, we are able to provide completely explicitly the three types of Maxwell singularities. This section is devoted to the description of solution formulas for the rst order problems (7.2) and (7.4). All these formulas are based on the scalar product or the vector product with the vector x , with x denoting the vector of cartesian coordinates (x1; x2; x3) . We begin with three series of formulas. First we give product laws: a and b denoting vector elds and being a scalar function on R3 , we have grad(a
b) = (a
grad) b + (b
grad) a + a  curl b + b  curl a; (7.5a) curl(a  b) = (b
grad) a ? (a
grad) b + a div b ? b div a; (7.5b) div(a  b) = b
curl a ? a
curl b; (7.5c) curl( a) = curl a + grad  a; (7.5d) div( a) = div a + grad
a: (7.5e) Now, using the above formulas for the eld x which satis es div x = 3; curl x = 0; x
grad = @ and grad x = I; we obtain for any eld a and scalar q grad(a
x) = (@ + 1)a + x  curl a; curl(a  x) = (@ + 2)a ? x div a; div(a  x) = x
curl a; curl(qx) = grad q  x; div(qx) = (@ + 3)q: Finally, with = 2 and a = grad q , (7.5d) and (7.5e) yield curl(2 grad q) = ?2 grad q  x; div(2 grad q) = 2@q + 2
q: 33

(7.6a) (7.6b) (7.6c) (7.6d) (7.6e) (7.6f) (7.6g)

These formulas allow to solve rst order problems in the subspaces of homogeneous elements of our pseudo-homogeneous spaces S1 , SN and ST : 

n

o

S1(?) = u = U (#) j U 2 H 1(G) ; and

(7.7)



n

o



n

o

1 (? ); u(x) =  U (#) ; SN (?) = u 2 XNloc(?) j div u 2 Hloc 1 (? ); u(x) =  U (#) : ST (?) = u 2 XTloc(?) j div u 2 Hloc

(7.8) (7.9)

As an easy consequence of formula (7.6a), we can solve the equation grad  = u with Dirichlet or Neumann boundary conditions: Lemma 7.1 Let u belong to+1SN (?) or ST (?) . (i) Then u
x belongs to S1 (?) . (ii) We assume that   6= ?1 and that curl u = 0 . If moreover u is homogeneous, i.e.  u 2 SN (?) , resp. ST (?) , then  de ned as  = u+
x1 2 S1+1(?); 

(7.10)

solves the equation grad  = u , with zero Dirichlet, resp. Neumann boundary conditions on @ ? . (i) A rst consequence of formula (7.6a) is that grad(u
x) belongs to thus u
x has the correct regularity outside the corner of ? . 0  (ii) As an obvious consequence of the fact that if u belongs to S (?) , then @ u = u , we obtain that grad  = u . Moreover, if u  n = 0 on @ ? , then u
x = 0 on @ ? as a simple consequence of the fact that x is a tangential eld. As for the Neumann boundary condition in the case when u
n = 0 , it is only a consequence of the formula @n = n
u . Similarly formulas (7.6b) and (7.6c) yield a solution of the equation curl u = : Lemma 7.2 Let belong to ST?1(?) , resp. SN?1(?) . (i) Then  x belongs to SN (?) , resp. ST (?) . (ii) We assume that  6= ?1 and that div = 0 . If moreover is homogeneous, i.e. 2 ST?1(?) , resp. SN?1(?) , then u de ned as Proof.

S (?)3 ,

  u =  + 1x 2 SN (?) resp. ST (?); 1 x
curl . solves the equation curl u = . Moreover div u = +1

34

(7.11)

u is a direct consequence of formulas (7.6b) and (7.6c). The boundary condition n  (  x) = 0 is satis ed if n
= 0 on @ ? due to the equality n  (  x) = (n
x) ? x(n
) . And the boundary condition n
(  x) = 0 is satis ed if n  = 0 on @ ? due to the equality n
(  x) = x
(n  ) . Part (i)

Proof. The regularity of

is proved and part (ii) is now obvious. The third step is the solution of the equations curl u = 0 , div u = q , which is done with the help of formulas (7.6d)-(7.6g): Lemma 7.3 Let q belong to S1?1(?) , such that
q 2 S0?3(?) and satisfying Dirichlet, resp. Neumann boundary conditions on @ ? . (i) Then 2qx + 2 grad q belongs to SN (?) , resp. ST (?) . (ii) We assume that  6= ? 21 and that
q = 0 . If moreover q is homogeneous, i.e.  q 2 S1?1(?) , then u de ned as

q 2 S  (?) resp. S  (?); u = 2qx +4+grad N T 2 2





(7.12)

solves the equations curl u = 0 and div u = q .

7.c The three types of Maxwell singularities generated by the Laplacian

In the case of plane sectors, we have seen that only two types of Maxwell singularities do exist and that they are generated by the Laplace operator: type 1, corresponding to the exponents k! ? 1 and the singular functions of the form grad  with  Dirichlet singularity for the Laplace operator, and type 3, corresponding to the exponents k! +1 . Now for three-dimensional cones, relying on the solution formulas (7.10)-(7.12) we are going to exhibit the three types in the case when they are generated by the Laplacian (Dirichlet or Neumann). In the next subsection, we will describe the remaining singularities which are generated by the topology of ? . In the following lemmas, we show the link between the sets of Maxwell singularity exponents N (?) and T (?) and those of the Laplacian, see Dir(?) and Neu(?) in (6.4). We also prove that the singularities of type 1, 2 and 3 can be expressed with  (?) and Z  (?) , the help of the Laplace singular functions, cf (6.5) for the spaces ZDir Neu except in particular geometrical situations when  = ?1 . Lemma 7.4 We assume that  6= ?1 . Then (i) is equivalent to (ii): (i) u 2 SN (?) is a solution of (7.2) of type 1, +1(?) . (ii)  + 1 belongs to Dir(?) and u = grad  where  belongs to ZDir Similarly, (iii) is equivalent to (iv): (iii) u 2 ST (?) is a solution of (7.4) of type 1, +1 (?) . (iv)  + 1 belongs to Neu(?) and u = grad  where  belongs to ZNeu

35

Proof.

1. In a rst step, we investigate the non-zero homogeneous solutions of (7.2)

of type 1, i.e. solutions of



curl u = 0 and div u = 0 in ?

with u 2 SN (?): 1 u
x . Thus Using Lemma 7.1, we immediately obtain that u = grad  with  = +1

 2 S1+1 and  = 0 on @ ?: Moreover the condition div u = 0 yields that
 = 0 . In other words,  is a Dirichlet +1(?) . The converse singularity for
, thus  + 1 belongs to Dir(?) and  2 ZDir +1(?) , u de ned as grad  is a singularity statement is straightforward: for any  2 ZDir of type 1. Concerning the magnetic boundary condition, the same arguments lead to grad  = u , where  is still de ned by (7.10) and satis es  2 S1+1 and @n = 0 on @ ?; and
 = 0 . Thus  + 1 belongs to Neu(?) . 2. In a second step, we prove that there is no logarithmic term in any solution of type 1. It suces to study a solution of type 1 with one logarithmic term, i.e. of the form 

u = u0 + u1 log ; with u0; u1 2 SN (?):  Since curl u is the sum of curl u1 log  and of a eld with each component in S ?1(?) , we deduce that curl u1 = 0 . Thus we obtain that u1 is itself a solution of type 1 of the same problem. Then instead of (7.10), we set

x;  = u+
x1 ? (u+
1) 2 1

(7.13)

and we deduce from the previous remark that grad  = u , and  satis es the Dirichlet (or Neumann) conditions on @ ? and
 = 0 . Therefore  belongs to Z +1(?) , but since we do not consider polynomial right hand sides here, there is no logarithmic term in  , hence u1 = 0 . If u is a singularity of type 2, then is a singularity of type 1 with a permutation of the roles of electric and magnetic boundary conditions. Moreover, when is known, Lemma 7.2 provides a formula for u (note that here div = 0 , thus formula (7.11) yields a divergence free u ). Thus we obtain: Lemma 7.5 We assume that  62 f?1; 0g . Then (i) is equivalent to (ii): (i) u 2 SN (?) is a solution of (7.2) of type 2,  (?) . (ii)  belongs to Neu(?) and curl u = grad where belongs to ZNeu Similarly (iii) is equivalent to (iv): (iii) u 2 ST (?) is a solution of (7.4) of type 2, 36

 (?) . (iv)  belongs to Dir(?) and curl u = grad where belongs to ZDir 1 grad  x . In each case, representatives of type 2 are given by u = +1

Finally we have directly from equations (7.2a) and (7.4a) the necessary conditions for the existence of a non-zero q and we combine lemmas 7.2 and 7.3 to obtain formulas for and u : Lemma 7.6 We assume that  62 f? 21 ; 0g . Then (i) is equivalent to (ii): (i) u 2 SN (?) is a solution of (7.2) of type 3, ?1(?) . (ii)  ? 1 belongs to Dir(?) and div u = q where q belongs to ZDir Similarly (iii) is equivalent to (iv): (iii) u 2 ST (?) is a solution of (7.4) of type 3, ?1 (?) . (iv)  belongs to Neu(?) and div u = q where q belongs to ZNeu In each case, representatives of type3 are given by = 1 grad q  x and by u = (21+1) (2 ? 1)qx ? 2 grad q .

Remark 7.7 For the sake of comparison, let us consider the case when ? is a dihedron of opening ! . Then Dir(?) = Neu(?) = f k! + `; k; ` 2 Z; k = 6 0g , cf [10, Ch.18.C].

In this case, for nding explicit expressions for the singular functions, one can choose between the formulas given in Lemma 5.1 and those of Lemma 7.6. They do, however, not give the same results because of the non-trivial in uence of homogeneous polynomials in the tangential variable z along the edge.

7.d The Maxwell singularities generated by the topology

It essentially remains to investigate the solutions of type 1 for  = ?1 , i.e. the elements u in SN?1(?) , resp. ST?1(?) with zero curl and divergence. The existence of such solutions depends on the topology of the spherical domain G which generates the cone ? . We are going to prove that we have singularity spaces in  = ?1 , ZN?1 (?) and ZT?1 (?) , if and only if G is not simply connected, and that their dimensions are equal to the dimension of the homology space of G . Lemma 7.8 Let us assume that G is simply connected. If u belongs to SN?1(?) , resp. ? 1 ST (?) and satis es curl u = 0 and div u = 0 , then u = 0 .

curl u = 0 that u is the gradient of some function  . Then we use a formula of integration of u along paths like (5.8): we x x0 2 ? and write in polar coordinates (; #) = (jxj; x=jxj) and (0; #0) = (jx0j; x0=jx0j) :

Proof. Since ? is simply connected, we derive from the condition

(; #) ? (0; #0) =

Z 

0

x0
u(0; # ) d0 +  Z 0 

(#0 ;#)

0

37

u(; #0)
d#0;

(7.14)

where the second integral is a path integral along a curve (#0; #) from #0 to # in G . From (7.14), we nd that  belongs to S10(?) . The condition div u = 0 yields that
 = 0 and the boundary conditions on u give either the Dirichlet conditions on  , or the Neumann condition. In the rst case, we nd that  = 0 since the eigenvalues Dir j are all > 0 , and in the second case, we nd that  is a constant, thus in any case u = 0. If the spherical domain G is not simply connected, its boundary @G is not connected. Let @j G , j = 1; : : :; J + 1 be its connected components ( J  1 ). We assume that G itself is connected (if not, the cones corresponding to each of its connected components can be considered separately). Then there exist J regular and non-intersecting cuts j , j = 1; : : : ; J such that G0 := G n [Jj=1 j is simply connected. The singularities of degree ?1 that we are investigating are closely linked with the kernels (7.15) and (7.16) of the tangential curl and divergence, curl > and div > . These operators are tangential to the sphere S2 and can be de ned with the help of the usual curl and div on three-dimensional elds: we rst introduce L2> (G) as the subspace of L2(G)3 spanned by the elds v tangential to the sphere, i.e. satisfying v
x = 0 . If v belongs to L2> (G) , we can introduce vb as any homogeneous extension of v to the cone ? : we x any  and vb (; #) is de ned as  v(#) . Then  div >v is the restriction on  = 1 of div vb ,  curl > v is the restriction on  = 1 of x
curl vb . Then the two kernels are de ned in a classical way: n o KN (G) = v 2 L2> (G) j curl >v = 0; div >v = 0 in G , v  n = 0 on @G (7.15) and n o KT (G) = v 2 L2> (G) j curl > v = 0; div > v = 0 in G , v
n = 0 on @G : (7.16)

Their description involves the tangential gradient grad > , and also (alternatively) the tangential vectorial curl curl > , which are de ned for any scalar function  in L2(G) with the help of any homogeneous extension b of  to ? as follows:  grad >  is the restriction on  = 1 of grad b ? (grad b
x) x ,  curl >  is the restriction on  = 1 of curl(b x) . There holds the following description of the spaces KN (G) and KT (G) (see [6] for a classical presentation and [1] for the case of less regular domains). In the de nitions (7.17) and (7.18) below, the cj denote arbitrary constant functions, nj is a unitary normal to j in S2 and [
]j the jump across j along nj : Lemma 7.9 (i) The space KN (G) is generated by the tangential gradients grad >  where  spans the space PDir(G) de ned as n PDir(G) =  2 H 1(G) j
G = 0 in G; o (7.17)  = cj on @j G, 1  j  J + 1 : 38

The dimension of PDir(G) is J + 1 and the dimension of KN (G) is J . g  to G of the tangential (ii) The space KT (G) is generated by the L2 extensions grad > gradients grad >  on G0 where  spans the space PNeu(G) de ned as n

PNeu (G) =  2 H 1(G0 ) j
G = 0 in G; @n  = 0 on @G; o []j = cj ; [@nj ]j = 0; 1  j  J :

(7.18)

The dimension of PNeu(G) is J and the dimension of KT (G) is J , too.

Relying on the de nitions of the \tangential" operators curl > , grad > , curl > , div > and on the relations (7.6b)-(7.6d), it is easy to show that for any  2 L2(G) and any v 2 L2> (G)

curl >  = (grad >)  x; div >v = ? curl >(v  x); curl > v = div >(v  x):

Then, we can prove the following g  Corollary 7.10 The space KN (G) is generated by the extended tangential curls curl > where  spans the space PNeu (G) and the space KT (G) is generated by curl >  where  spans the space PDir(G) . Moreover, we have the relations

KT = x  KN and KN = x  KT : Thus, the kernels KN (G) and KT (G) are gradients (or curls) of harmonic functions belonging to the spaces PDir(G) and PNeu(G) . We extend these spaces to homogeneous functions of degree 0 on the cone ? and thus de ne the spaces PDir(?) and PNeu(?) . We note that any  2 PDir(?) has its traces constant on each connected component of @ ? and similarly that the jumps of any  2 PNeu(?) across the cuts j of ? corresponding to j are constant too. For any  2 PDir(?) , the gradient grad  is an homogeneous function of degree ?1 whose radial component is 0 : we have grad (; #) = 1 grad >(#): The eld u = grad  belongs to SN?1(?) and satis es curl u = 0 like all gradients, and div u =
 = ?2
G = 0 by construction. Similarly for  2 PNeu(?) , the extended g  belongs to S ?1 (?) and satis es curl u = 0 and div u = 0 . gradient u = grad T Lemma 7.11 Let us assume that G is not simply connected. Then ZN?1 (?) is the space ? 1 of the elds of the form u = grad  , where  2 PDir (?) . Correspondingly, ZT (?) is g  , where  2 P the space of the elds of the form u = grad Neu (?) . 39

We have just proved that any eld of the form u = grad  , where  is a non-zero element of PDir(?) is a non-trivial element of ZN?1 (?) . Conversely let u belong to ZN?1 (?) . Then, for a non-zero eld uP , P  0 , we have Proof.





u = ?1 u0 +


+ logP  uP : Since curl u = 0 , u is a gradient grad  in the simply connected domain ?0 generated by the spherical domain G0 . Using formula (7.14) with paths (#0; #) contained in G0 , we obtain that  belongs to S10(?0 ) and can be expanded into  = 0 +


+ logQ Q ; with P  Q  P + 1: Since u  n is zero on @ ? , the traces of  on @ ? are constant on each of its connected components @j ? . Thus the traces of 0 are constant on each @j G and the traces of q for q  1 are zero. Since curl u and div u are zero in ? , the jumps of u across the cuts j generated by j are zero. Thus []j is constant and [@nj ]j is zero. With the conditions on the traces on @j ? , this yields that []j is zero. Moreover div u = 0 in ? gives
GQ = 0 . Therefore Q belongs to PDir(G) , and if Q  1 we have moreover that the traces of Q on @G are all zero, thus Q = 0 . Whence Q = 0 and 0 belongs to PDir(G) , so u has the desired form. The proof for ZT?1 (?) is similar.

7.e Corner singularities: a synthesis

Let us consider the electric boundary condition. According to the general theory stated in x4.c, we have to consider the subset of N (?) satisfying (with  a cut-o function which is equal to 1 in a neighborhood of the corner of ? ): ? 32 <  and 9u 2 ZN (?); u 6= 0; u 2 XN (?); div(u) 2 H 1(?): (7.19) We examine the eect of condition (7.19) on each of the singularities of types 1, 2 and 3, cf Lemmas 7.4 - 7.6 and 7.11. Type 1 In this case  + 1 belongs to Dir(?) (subtype 1
) or  = ?1 if G is Dir not simply connected (subtype 1Top ): according to (6.4), as Dir 1 > 0 ,  (?) \ [?1; 0] is empty. Thus  avoids [?2; ?1] for type 1
. Whence  > ?1 . The +1 (?) for type 1 or corresponding singularities are grad  with  either in ZDir
in PDir(?) for type 1Top . Type 2 In this case = curl u is a singularity of type 1 with magnetic boundary conditions and condition (7.19) on u requires that  belongs to L2(?) , thus  ? 1 > ? 23 , i.e.  > ? 12 . Then we have the subtype 2
corresponding to  (?) and  2 Neu (?) (thus  > 0 ) and the subtype = grad with 2 ZNeu g with 2 P 2Top corresponding to = grad Neu (?) and  = 0 (if G is not simply connected). 40

Type 3 In this case q = div u is a Dirichlet singularity of the Laplace operator and  ? 1 belongs to Dir(?) . Thus condition (7.19) on u requires that q belongs to H 1(?) , thus  > 21 , whence  > 1 . Thus the most singular among the admissible elements of the ZN (?) is either q

+1 (?);  = ? 3 + Dir + 1 : u = grad  with  2 ZDir 1 2 4

if ? is simply connected, or the singularity of type 1Top if not. We summarize all the results in the following table, where we omit the reference to the cone ? in the notation of spaces:

Type 1
2
3




u

> Generator

+1  + 1 2 Dir ?1  2 ZDir

 2 Neu

0

 ? 1 2 Dir 1

 2 ZNeu

q

grad 

0

0

grad  x

grad

0

+1

2 ?1 (2 ? 1)q x ?  grad q grad q  x q q 2 ZDir (2 + 1) 

1Top

?1

 2 PDir

grad 

0

0

2Top

0

2 PNeu

g x grad

g grad

0

 2 PDir

 grad 

Alternative formulation 2Top

0

Table 1

x  grad  0 

This alternative formulation is obtained with the help of Corollary 7.10. The adaptation of this table to magnetic boundary conditions is left to the reader. 41

Going back to the primitive Maxwell equations (1.1), we see that for a regular current density J , the divergences of the electric and magnetic elds E and H are regular too, thus only the singularities of types 1 and 2 can occur and they exchange each other between the electric and magnetic elds (here  denotes the degree of homogeneity of the generator and is either the degree of E or H ):

Type

Generator



E

H


(electric)

  2 ZDir

 2 Dir

grad 

grad   x


(magnetic)

x  2 ZNeu  2 Neu ? grad i!( + 1)

Top (electric)

 2 PDir

0

grad 

Top (magnetic) 2 PNeu

0

? gradi!  x

i!( + 1)

grad grad   x i!

g

g grad

Table 2

This table gives the principal parts of the singularities (as can be seen from (1.2) or (1.7) the operators are not homogeneous and according to the general theory [15, 10] the singularities themselves have an asymptotic expansion).

8 Conclusions Taking advantage of the information about admissible exponents and singularities collected in sections 5 and 7, we are now able to give more speci c versions of the regularity and singularity theorems 4.1 and 4.2. For any edge e 2 E and   ?1 (below, k denotes any integer  1 , with the requirement that k is odd if !e = 2 ), the set of admissible exponents is: N (?e ) =

n

k !e

? 1   j k  2 if !e = 2 42

oSn

k !e



oSn

k !e

o

+1 :

(8.1)

Neu For any corner c 2 C and   ?1 , Dir j (?c ) and j (?c ) denote the Dirichlet and Neumann Laplace singular exponents on ?c , cf (6.4) q

q

1 1 1 1 Neu Dir Neu Dir j (?c ) = ? 2 + j + 4 and j (?c ) = ? 2 + j + 4 ;

and the set of admissible exponents is oSn o n    ? 21 j  = Neu N (?c ) =  + j (?c ) j (?c ) ? 1   ?n 12 j  = Dir o S ?1; 0   ? 21 j if Gc is not simply connected : Let now be and

E

(

 !e

= min e2E 3

if !e 6= 2 if !e = 2

)

(8.2) (8.3)

Dir = min Dir(?c) and Neu = min Neu(?c): c2C 1 c2C 1

(8.4)

8.a Regularity Let s 2 [1; 25 ) and  2 (0; s + 1] (  plays the role of 1 +  ). For any data f 2 H s?1 ( )3 let u be the solution of problem (4.1). As a corollary of Theorem 4.1, u 2 H  ( )3 for all  ,  < minfE ; Dir + 21 ; Neu + 32 g . More precisely, according to

the geometrical properties of we have: (i) If has screen parts, or is not locally simply connected, then u 2 H  ( )3 for all  < 21 . (ii) If has no screen parts and is locally simply connected, but is not convex, then u 2 H  ( )3 for all  , 21 <  < minfE ; Dir + 21 g < 1 . (iii) If is convex, the monotonicity of Dirichlet eigenvalues allows then to prove that E  Dir , thus u 2 H  ( )3 for all  , 1 <  < minfE ; Neu + 23 g . (iv) If is a parallelepiped, u 2 H  ( )3 for all  < 3 .

8.b Singularities

In this whole subsection s 2 [1; 25 ) and  2 [0; s] , the data f belongs to H s?1( )3 and u is the solution of problem (4.1). We are going to give three forms (more and more particular) of the expansion (4.9). A. If  < E and  ? 12 62 N (?c) for any c 2 C , the splitting (4.9) can be written as

u = ureg; +

X

X

c2C 2N (?c)

c c(c) Uc(c; #c) +

X



X

K [ e;k ] e (e ) grade k=! sin k e ! e  e

e

e2E k2Ke

43



(8.5)

with ureg; 2 H +1 ( )3 and the edge coecients e;k 2 Vs?+1 ?k=! (e) .  Dir Neu Here, for any c 2 C , with Dir j =  j 'j (#) and a similar notation for j :  Dir if  = Dir j ? 1, Uc = grad j , if  = Neu Uc = grad Neu j , j  x, Dir  2 Dir if  = j + 1, Uc = (2 ? 1)Dir j x ?  grad j , if  = ?1 or 0, Uc is the corresponding topological singularity (case when Gc is not simply connected). For any e 2 E , n o  , k 6= 2 if ! = 2 Ke = k 2 f1; 2g j k <  + 1 ; and k = 6 1 if ! = e e ! 2 and grade is the gradient associated with the variables y~e = ye=de and ze . B. We assume now that is locally simply connected and that  < E as before, and 1 Dir Neu moreover that  < minf + 2 ;  + 32 g . Then the above splitting (8.5) only involves gradients in the singular part and can be written as X X

c;j c(c) grad Dir u = ureg; + c;j (c ; #c) c2C j 2J  X X (8.6) K [ ]  ( ) grad Dir ( ;  ) + e

Dir

e

c

e2E k2Ke

e;k

e e

e e;k e e

1 Dir with Jc = fj  1; Dir c;j <  + 2 g and the obvious de nition for e;k . C. If is locally simply connected and if we take  = 0 , ureg;0 is the H 1 regular part of u (which was also called u(H ) in the section 3 devoted to polygons) and denoting 1 C0 = fc 2 C ; Dir c;1 < 2 g and E0 = fe 2 E ; !e >  g , the splitting (8.6) takes the simpli ed form X u = ureg;0 + c c(c ) grad Dir c;1 (c ; #c ) c2C X (8.7) + K [ ]  ( ) grad Dir( ;  ): 0

e2E0

e

e e

e e;1 e e

Remark 8.1 In the splittings (8.6) and (8.7), the singular generators can also be expressed as curls since for any harmonic and homogeneous function  of degree  , cf (7.6b): ( + 1) grad  = curl(grad   x) and grade(e sin e) = curle (e cos e ): with curle the two-dimensional vectorial curl in the y~e plane, completed by a zero tangential component along the edge. to



We recall that ye are cartesian coordinates in the normal plane to e , ze is a coordinate tangent and de is equivalent to the distance to the ends of e .

e

44

Another interesting question in the framework of the splittings (8.6) and (8.7), is to know whether it is possible to write the singular parts as a gradients in a global way. Lemma 8.2 Let  2 [0; E ) and c 2 C . Let  = Dir c;j be a singularity of the Laplace Dirichlet problem on ?c . Then c grad  ? grad(c) 2 H +1( ) (8.8)  and := c belongs to H 1 ( ) and is such that
is in H  ( ) .

Remark 8.3 The limitation of the regularity in (8.8) and for
comes only from the brutal cut-o of the edge asymptotics of  away from the corner. A re ned cut-o procedure [10, x16.C], would yield a similar statement without any limitation on  . Lemma 8.4 Let  2 [0; E ) and e 2 E . Let  = Dir e;k be a singularity of the Laplace ?k=! (e) . Then, k Dirichlet problem on ?e , with ! ? 1 <  . Let for s   , 2 Vs?+1  e

with the coordinates y~e = ye=de   K [ ] e (~ye ) grade (~ye ) ? grad K [de ] e (~ye ) (~ye ) 2 H +1 ( ) e



(8.9)

and := K [de ] e belongs to H 1 ( ) and is such that
is in H  ( ) .

Remark 8.5 Beyond what could be done by the introduction of correct \shadow" terms, it is impossible to avoid the limitation of the regularity by the weight ? in the space containing the edge coecient . This implies that, if we apply such a re ned statement to the edge terms in (8.5), we have a sharp limitation by the smallest corner exponents which does not correspond to gradients ( Neu + 21 and Dir + 23 ). As a consequence of the expansion (8.6) and of the two previous lemmas, we obtain Theorem 8.6 Let be locally simply connected,  < minfE ; Neu + 21 ; Dir + 23 g and s   . Then for any data f 2 H s?1 ( )3 the solution u of problem (4.1) can be split in the following way u = ureg; + grad (8.10) where ureg; 2 H +1 ( )3 and can be written as X X

c;j c(c ) Dir = c;j (c ; #c ) c2C j 2J  X X (8.11) + K [d ]  ( ) Dir ( ;  ): c



Here 2 H 1 ( ) satis es

e2E k2Ke
2 H  ( ) .

e e;k

e e

e;k e e

When applied with  = 0 , the above statement can be compared with the splitting of any element of XN in the sum of an element of HN and of a term grad  with  2 H 1( ) such that
 2 L2( ) , cf [3, 4, 11]. 45

8.c Singularities in H 1

We conclude by some remarks on the regularity of the variational problem (2.8) posed in HN  H 1( )3 . Very similar remarks hold for the saddle-point variant (2.4), posed in HN  L2( ) . In section 3.f, this question was discussed for polygons. We saw that the set of exponents s(H )(?) was, in general, dierent from s(X )(?) . Still these exponents were related to those of the Dirichlet problem for the Laplace operator in a simple way. Consider now the case of a three-dimensional (pseudo-)polyhedral corner c . If the cone ?c is convex, then the XN -singular functions belong to H 1 , and for the HN singular functions, the divergence has H 1 regularity. Thus the two problems have the same singularities near the corner c . We suppose therefore that the cone ?c is not convex. There exist general results on the exponents of the singular functions for this problem in the case of a Lipschitz cone (Kozlov - Mazya - Rossmann [16]). For instance, there is a strip ?1  Re   0 that does not contain such exponents. Note that this does not imply H 3=2( ) regularity, as it would for a cone with a regular base, because we have strong edge singularities here. The lowest edge exponent is  < 1=3 , see section 3.e (iii), and this corresponds to H 1+ regularity for u and H  regularity for div u for all  <  . There is, in general, no simple relation between the singular functions of this problem in HN and those of the Dirichlet or Neumann problems for the Laplace operator. In particular, our explicit constructions in section 7 do not work here, and the classi cation into types 1, 2, 3 does not make sense. Let us explain two reasons for this. First, the solutions of the rst-order systems (7.2b) and (7.2c) do not belong to H 1 near a non-convex edge. Thus, independently of the corner exponent  , the XN -singular functions of all 3 types will not belong to H 1 . Second, the Laplace-Dirichlet problem (7.2a) for q is now posed with only L2 regularity required. This problem (with S(?) de ned in (6.1))

?
q = 0 in ? and q = 0 on @ ?

with q 2 S0?1(?)

(8.12)

does now not select a discrete set of exponents  . Proposition 8.7 Let ? be a non-convex pseudo-polyhedral cone. Then the Dirichlet problem (8.12) has non-trivial solutions for any  2 C . Proof. This is a Laplace-Beltrami eigenvalue problem on G = ? \ S2 . We look for

eigenfunctions in L2(G) . By duality (and the \very weak" de nition of the Dirichlet problem (8.12), see (2.9)), we see that such eigenfunctions span the orthogonal comple1 ment of the image of H (G) \ H 2 (G) under the adjoint operator. Now we know that in the presence of non-convex corners of G , one never has H 2 regularity for this Laplace1 Beltrami Dirichlet eigenvalue problem. Thus, in addition to the H (G) eigenfunctions 46

that may exist if  ? 1 2 Dir(?) , we nd as many L2(G) eigenfunctions and therefore solutions to (8.12) as there are non-convex edges meeting at c . As a consequence of Proposition 8.7, we lose not only the discreteness of the spectrum Dir(?) , but also the equivalence between our elliptic second order variational formulations with the original Maxwell problem:  In the presence of non-convex corners, the solutions of problems (1.4) and (1.5), posed in HN instead of XN , will depend on the parameter s .  Even under the hypotheses of Theorem 1.1, the solution of (1.4) will not satisfy div u = 0 .  The solution of (1.5) will not satisfy p = 0 .

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