clifford algebras and maxwell's equations in lipschitz domains

Mathematical Methods in the Applied SciencesMath. Meth. Appl. Sci., 22, 1599}1620 (1999)MOS subject classi"cation: primary 42 B20; 30 G35; secondary 78 AB25; 76D 05

Cli4ord Algebras and Maxwell:s Equations inLipschitz Domains

Alan McIntosh*1 and Marius Mitrea2

1CMA, Australian National University, Canberra ACT 0200, Australia2 Department of Mathematics, University of Missouri-Columbia, 202 Mathematical Sciences BuildingColumbia, MO 65211, U.S.A.

Communicated by W. Spro? ssig

We present a simple, Cli!ord algebra-based approach to several key results in the theory of Maxwell'sequations in non-smooth subdomains of Rm. Among other things, we give new proofs to the boundaryenergy estimates of Rellich type for Maxwell's equations in Lipschitz domains from [20, 10], discussradiation conditions and the case of variable wave number. Copyright ( 1999 John Wiley & Sons, Ltd.

KEYWORDS: Cli!ord algebras; Maxwell's equations; Rellich identities; radiation conditions; Hardy spaces;Lipschitz domains

1. Introduction

It has been long recognized that there are fundamental connections betweenelectromagnetism and Cli!ord algebras. Indeed, understanding Maxwell's equationswas part of Cli!ord's original motivation.

A more recent trend concerns the treatment of such PDEs in domains with irregularboundaries. See [11] for an excellent survey of the state of the art in this "eld up toearly 1990s and [13, 18] for the role of Cli!ord algebras in this context.

Following work in the context of smooth domains [24,2, 4, 23], the three-dimensional Maxwell system

(M3) G

curlE!ikH"0 in R3C)1 ,

curlH#ikE"0 in R3C)1 ,

n]E"f3¸2(R)),

(plus appropriate radiation conditions at in"nity) in the complement of a boundedLipschitz domain )LR has been solved in [20], while the higher-dimensional

*Correspondence to: A. McIntosh, CMA, Australian National University, Canberra ACT 0200, Australia.e-mail: [email protected]

CCC 0170}4214/99/181599}22$17.50 Received 29 April 1999Copyright ( 1999 John Wiley & Sons, Ltd.

version (involving di!erential forms)

(Mm) G

dE!ikH"0 in RmC)1 ,

dH#ikE"0 in RC)1 ,

n?E"f3¸2 (R)),

(plus suitable radiation conditions) for arbitrary Lipschitz domains )LRm, has beensolved in [10] (precise de"nitions will be given shortly). See also [19, 16] for relateddevelopments.

One of the key ingredients of the approach in [20, 10] was establishing estimates tothe e!ect that the so-called voltage-to-current map, taking n]E into n]H and n?Einto n@H, respectively, is an isomorphism between appropriate boundary spaces.Ultimately, this comes down to proving norm equivalences of the following form:

En]EEL2(R))

#ESn, HTEL2(R))

+En]HEL2(R))

#ESn, ETEL2(R))

(1)

in R3 and, more generally,

En?EEL2(R))

#En?HEL2(R))

+En@EEL2(R))

#En@HEL2(R))

(2)

in Rm. As shown in [20, Theorem 4.1] (for Im k'0) and [10, Theorem 8.5] (forarbitrary kO0), the above estimates lead to the invertibility of certain magnetostatic(and electrostatic) boundary integral operators which, in turn, are used to solve (M

m),

m*3.The aim of this paper is to shed new light on the basic estimates (1), (2) and to

present an alternative, natural approach to proving such boundary energy estimates,developed in the framework of Cli!ord algebras. This is an extension of work in [18]where such an approach has "rst been used for certain PDEs in Lipschitz domains.Besides its intrinsic merit, the relevance of this work should be most apparent for thenumerical treatment of electro-magnetic scattering problems is non-smooth domains.For the smooth context and di!erent techniques see [8].

The departure point for us (which in fact goes back to Maxwell himself; it has alsobeen reinvented by M. Riesz) is to write the Maxwell system as a single equationwhich, in fact, expresses the Cli!ord analyticity of a certain Cli!ord algebra-valuedfunction. Speci"cally, if E and H are the electric and magnetic components of anelectromagnetic wave in R3, then the entire Maxwell system is equivalent to theMaxwell}Dirac equation

(D#ke4) (H#iEe

4)"0. (3)

Here E and H are regarded as Cli!ord algebra-valued functions in R3, and D#ke4

isa perturbed Dirac operator.

This is both mathematically convenient and physically relevant. In fact, the sameidea has been extensively used by physicists who employ the Lorentz metric in thefour-dimensional (#at Minkowskian) space}time domains in order to write the 3DMaxwell system for E"(E

i)i/1,2,3

and H"(Hi)i/1,2,3

solely in terms of the electro-magnetic "eld-strength tensor. Due to its skew symmetry, the latter de"nes the

1600 A. McIntosh and M. Mitrea

Math. Meth. Appl. Sci., 22, 1599}1620 (1999)Copyright ( 1999 John Wiley & Sons, Ltd.

exterior di!erential form (sometimes referred to as a Faraday bi-vector-"eld)

u :"(E1dx

1#E

2dx

2#E

3dx

3) dt

#H1dx

2dx

3#H

2dx

3dx

1#H

3dx

1dx

2

(typically, electrical forces are vectorial while magnetic ones are 2-tensors). See e.g. themonographs [5, 31, 9, 28].

The radiation conditions for E and H can also be naturally rephrased in terms of u.Parenthetically, let us note that it is also possible to work with the non-homogeneousform E#H and write Maxwell's equations in the form (D#ik) (E#H)"0 (cf.[23, 10]) but this setup appears to be less physically relevant.

However, the most attractive aspect of this algebraic context is that it preservesmany of the key features of the classical complex function theory. For us, Hardyspaces of monogenic functions in Lipschitz domains and Cauchy vanishing formulaswill play a basic role in the sequel. Based on these, if )-R3 is the complement ofa bounded Lipschitz domain and if n,n

1e1#n

2e2#n

3e3

denotes the outward unitnormal to R), we shall show that for any solution u"H#iEe

4of (3) there holds

Enu$unEL2(R))

+EuEL2(R))

,

modulo residual terms. The Rellich type estimate (1) follows from this. In fact,a similar argument applies in the higher-dimensional case.

Before commencing the major developments we shall introduce here some notation.Call a bounded domain ) Lipschitz if its boundary is locally given by graphs ofLipschitz functions in appropriate rectangular co-ordinates. We let dp denote thesurface measure on R) and set n for the outward unit normal to R). For a (possiblyalgebra-valued) function u de"ned in ), the non-tangential maximal function u* is givenby u*(X) :"supMDu(>) D; >3), DX!> D)2dist(>, R))N. Also, the non-tangentialboundary trace on R) of a function u de"ned in ) is taken as the pointwisenon-tangential limit almost everywhere with respect to the surface measure on theboundary.

2. Cli4ord algebra rudiments

Recall that the (complex) Cli+ord algebra associated with Rm endowed with theusual Euclidean metric is the minimal enlargement of Rm to a unitary complex algebraA

m, which is not generated (as an algebra) by any proper subspace of Rm and such that

x2"!Dx D2, for any x3Rm. This identity readily implies that, if MejNmj/1

is the standardorthonormal basis in Rm, then

e2j"!1 and e

jek"!e

kejfor any jOk.

In particular, we identify the canonical basis MejN from Rm with the algebraic basis of

Am. Thus, any element u3A

mcan be uniquely represented in the form

u"m+l/0

+@DI D/l

uIeI, u

I3C, (4)

Cli!ord Algebras and Maxwell's Equations 1601


where eIstands for the product e

i1ei22

eilif I"(i

1, i

2,2, i

l) (we make the convention

that e0 :"1). For this multi-index I we call l the length of I and denote it by DI D. Weshall adopt the convention that +@ indicates that the sum is performed only overstrictly increasing multi-indices I.

The Cli+ord conjugation on Am

is de"ned as the unique complex-linear involutionon A

mfor which eN

IeI"e

IeNI"1 for any multi-index I. In particular, if

u"+IuIeI3A

m, then uN "+

IuIeNI. Note that

uN "(!1)l(l`1)@2u, (5)

for any u3"l. For the complex conjugation on Am

we set

u#"A+I

uIeIB

#:"+

I

uNIeI. (6)

We de"ne the scalar part of u"+IuIeI3A

mas u

0:"u0 , and endow A

mwith the

natural Euclidean metric Su, uN #T"DuD2 :"(uuN #)0"(uN #u)

0. Note that (uv)

0"(vu)

0"Su, vN T,

for any u, v3Am. Another useful observation is that

Dau D"Dua D"Du D Da D (7)

for any u3Am

and a3Rm. We also de"ne

u$

:"12Mu$uN N (8)

for each u3Am.

With u as in (4), de"ne %lu :"+ @DID/l

uIeI

and denote by "l the range of%

l:A

mPA

m. Elements in "l will be referred to as l-vectors or di!erential forms of

degree l. The exterior and interior product of forms are de"ned for a3"l, u3"l,respectively, by

a?u :"%l`1

(au) and a@u :"!%l~1

(au). (9)

By linearity, these operations extend to arbitrary u3Am.

Two identities which are going to be of importance in the sequel are

au"a?u!a@u (10)

and

ua"(!1)l (a?u#a@u), (11)

valid for arbitrary a3"1 and u3"l.We shall work with the Dirac operator D :"+m

j/1ejR/Rx

j. Then DM "+m

j/1eNjR/Rx

j"!D, and D2"!*, the negative of the Laplacian in Rm. For reasons which

will become more apparent shortly, we shall "nd it useful to embed everything intoa larger Cli!ord algebra, say Rm-A

m-A

m`1. If for k3C we now set

Dk:"D#ke

m`1(12)



then !D2k"*#k2. Furthermore,

D#k"D#kM e

m`1"D

k1and D1

k"!D!ke

m`1"!D

k. (13)

If u is a Am`1

-valued function de"ned in a region ) of Rm, set

Dku :"

m+j/1

ej

RuRx

j

#kem`1

u

and

uDk:"

m+j/1

RuRx

j

ej#kue

m`1.

Call u left (right, two-sided) k-monogenic if Dku (uD

k, or both D

ku and uD

k)"0 in ).

Note that each component of a k-monogenic function is annihilated by the Helmholtzoperator *#k2. A function theory for the perturbed Dirac operator D

kwhich is

relevant for us here has been worked out in [21, 22].The exterior derivative operator d acts on a "l-valued function u by du :"%

l`1(Du).

It extends by linearity to arbitrary Am-valued functions. Its formal transpose, d, is

given by du :"%l~1

(Du) if u is "l-valued. Once again, d extends by linearity toarbitrary A

m-valued functions. Since D"d#d, it follows that d2"0, d2"0 and

!(dd#dd)"*, as is well known. More detailed accounts on these matters can befound in [1, 7, 12, 18]. Further references can be found in [14].

3. A distinguished fundamental solution

The aim is to construct a convenient, explicit fundamental solution for the Diracoperator D

k. The departure point is deriving an explicit expression for a fundamental

solution of the Helmholtz operator *#k2 in Rm. If k"i, this can be taken to beminus the kernel of the Bessel potential Ta"(!*#I)~a@2 corresponding to a"2.This kernel is known (cf. [27], p. 131) to have the form

'(X)"!

1

(4n)m@2 P=

0

expA!t!DX D24t B

dt

tm@2. (14)

For general k3 iR`

we simply re-scale the expression of ' to obtain that

'k(X) :"!

1

(4n)m@2 P=

0

expAk2t!DX D24t B

dt

tm@2. (15)

is a fundamental solution for the operator *#k2 in Rm. Clearly this also continues tohold true for all k3C with Im k'DRek D and, in fact, the above expression can becontinued analytically to the open upper-half plane Im k'0 with the property that'

k(X)"O(expM!DX DIm kN) as DX DPR. For instance, when Im k'0 and m*2, one



may take

'k(X)"!

1

um

(!ik)m~2

(m!2)! P=

1

e*k DXDt (t2!1)(m~3)@2dt, X3RmC0,

where um

is the area of the unit sphere in Rm. It is worth pointing out that the abovede"nition can be further extended to Mk3CC0; Im k*0N by setting, for X3RmC0,

'k(X)"G A

!

1

2nr

RRrB

(m~1)@2

A1

2ike*krBK

r/DXD

A!1

2nr

RRrB

(m~2)@2

A!i

4H(1)

0(kr)BK

r/DXD

if m is odd,

if m is even,(16)

where H(1)0

is the usual zero-order Hankel function of the "rst kind. We shall continueto denote this extension by '

k. For this and a more detailed discussion see the

excellent treatment in [6], pp. 59}74.It is now clear that

Ek:"!D

k'

k"!'

kD

k"!

m+j/1

R'k

Rxj

ej!ke

m`1'

k(17)

is a (two-sided) fundamental solution for Dk. When k"0 then E

kreduces to the usual

radial fundamental solution for the Laplace operator. If Im k'0 then Ek

decaysexponentially at in"nity. Furthermore, if F

k(X) is so that E

k(X)"F

k(X)e*kDXD then

Fk(X)"O ( DX D~(m~1)@2) as DX DPR. (18)

This follows from the de"nition of 'kplus a straightforward computation. In the case

when m is even, the classical asymptotic expansion (in the sense of PoincareH ; cf., e.g.,[26, (9.13.1), p. 166])

H(1)0

(z)&A2

nzB1@2

e*z~*n@4G1!i1

1!23z#i2

1 ) 9

2!26z2!i3

1 ) 9 ) 25

3!29z3#2H

as Dz DPR, is also used.A more delicate analysis is required with k3RC0 because in this case there are two

canonical decaying fundamental solutions. Speci"cally, for k3RC0, we de"ne

E$

k:"!D

k'

$k"!'

$kD

k. (19)

In passing let us note that E`k"E

k. Now, if the functions F$

kare so that

E$

k(X)"F$

k(X)e$*kDXD then, along similar lines as before,

F$

k(X)"c

m,k

1

DX D(m~1)@2 A$X

DX D!ie

m`1B#o ( DX D~(m~1)@2) as DX DPR,

(20)



where cm,k

3CC0, and

+F$

k(X)"o ( DX D~(m~1)@2) as DX DPR. (21)

In the physically relevant case when m"3 and Im k*0,

'k(X)"!

1

4nDX De*kDXD (22)

so that, for k3RC0,

E$

k(X)"!

1

4n AX

DX D3G

ikX

DX D2!

ke4

DX DBe$*kDXD. (23)

In particular, (20) and (21) are trivially checked.

4. Function theory for perturbed Dirac operators

Let ) be a bounded Lipschitz domain in Rm and let n stand for the outward unitnormal to ). As in [18, 22, 7], we introduce the Hardy-type space H2

-%&5(), k) of left

k-monogenic functions in ) by

H2-%&5

(), k) :"Mu : )PAm`1

; u is left k-monogenic in ), u*3¸2(R))N.

Similarly, we de"ne H23*')5

(), k), the Hardy space of right k-monogenic functions in) by

H23*')5

(), k) :"Mv : )PAm`1

; v is right k-monogenic in ), v*3¸2 (R))N.

We endow these spaces with the natural norms EuEH2-%&5 (),k)

:"Eu*EL2(R))

, andEvEH2

3*')5 (),k):"Eu*E

L2(R)), respectively.

Equivalent norms are obtained by taking the ¸2 norms of the boundary trace; see[12] and the theory of monogenic Hardy spaces developed in [18]. Also, note thatH2

-%&5(), k)WH2

3*')5(), k) consists precisely of two-sided k-monogenic functions in

) which have a square-integrable non-tangential maximal function.

Proposition 4.1. ¸et ) be a bounded ¸ipschitz domain in Rm, k3C , and assume thatu3H2

3*')5(), k), v3H2

-%&5(),!k). ¹hen u and v have (non-tangential) boundary limits

at almost every point on R) and

PR)unvdp"0. (24)

Also, if u3H23*')5

(), k)WH2-%&5

(), k) and h3C1 (Rm, Am`1

), then

PR)unu#h dp"PP)

u([D, h]u#!2(Rek)em`1

u#h) (25)

where [D, h] is the commutator between D, acting from the left, and multiplication by h tothe right.



Formula (24) can be considered as the Cli!ord analogue of the classical Cauchy'svanishing theorem in complex analysis. The second identity, (25), is a perturbation ofthis result. We state it here since we shall need it shortly. In both cases, the integrandsmust be interpreted in the sense of multiplication in the Cli!ord algebra.

Proof. The existence of the boundary trace for u and v is proved in the usual way. Asfor the vanishing formula, suppose "rst that u and v are smooth up to and includingthe boundary of ). Consequently, Gauss's formula gives

PR)unvdp"PP)

(uD)v#u (Dv)"PP)(uD

k)v#u(D

~kv). (26)

In this case, (24) follows based on the monogenicity assumptions. The general case isthen seen from this and a standard limiting argument (see, e.g., [30] for details insimilar circumstances).

The proof of (25) is once again based on (26). This time, however,

D~k

(u#h)"[D~k

, h]u##(D~k

u#)h

"[D, h]u#!2(Re k)em`1

u#h. (27)

The proof is complete. K

We shall also need the following Cli!ord algebra version of Cauchy's reproducingformula.

Proposition 4.2. ¸et ) be a bounded ¸ipschitz domain in Rm and k3C. Then everyfunction u3H2

-%&5(), k) has (nontangential) boundary limits at almost every point on R)

and

u (X)"PR)Ek(X!>)n(>)u(>) dp(>), X3). (28)

A similar statement is valid for functions in H23*')5

(), k).

Proof. In a smoother context, the proof goes along well known lines; cf., e.g., [1] forthe case when k"0. Note, however, that we utilize here the fact that E

k(X!>)

D~k,Y

"!Ek(X!>)D

k,X"0 for XO>. Then, as before, passing to domains with

Lipschitz boundaries is done via a routine limiting argument. K

In the second part of this section we shall discuss the exterior domain version ofProposition 4.2. The novelty is that u should satisfy a (necessary) decay condition atin"nity. In the case when Im k'0, based on (18), it is not too di$cult to see that

u(X)e~I.kDXD"o ( DX D~(m~1)@2) as DX DPR (29)

is a (necessary and) su$cient condition to prove the exterior domain version of (28). Inturn, the latter implies that actually u decays exponentially at in"nity. Parenthetically,let us also note that, e.g. u(X)"O (1) at in"nity always entails (29).



The behavior at in"nity requires a more elaborate analysis when k3RC0 in whichsituation (29) is no longer the appropriate decay condition. In fact, the whole casewhen k3CC0 has Im k*0 can be covered by insisting that u satis"es a decaycondition of radiation type. Speci"cally, we shall ask that

(1!iem`1

XK )u (X)"o( DX D~(m~1)@2) asDX DPR, (30)

where we set XK :"X/DX D, X3RmC0.When k3RC0, the opposite choice of sign in (30) will also do. Thus, in this case, (30)

may be replaced by

(1Giem`1

XK )u (X)"o ( DX D~(m~1)@2) as DX DPR. (31)

The source of (31) is expression (20). As expected, for real non-zero k, the functionsE$

ksatisfy (31) because

(1Giem`1

XK ) ($XK !iem`1

)"0"(1$iem`1

XK )(1Giem`1

XK ). (32)

Since 1Giem`1

XK are, as (32) shows, zero-divisors, one cannot deduce a decay condi-tion of the same order of u itself based solely on (31) and algebraic manipulations. Infact, owing to the classical Rellich's lemma (cf. the appendix), any solution u of theHelmholtz equation (*#k2)u"0 for non-zero real k which satis"es u (X)"o( DX D~(m~1)@2) in a (connected) neighbourhood of in"nity in Rm must vanish identi-cally. Hence, generally speaking, for (31) to hold when k3RC0, certain internalcancellations must occur when multiplying u with (1Gie

m`1XK ). Nonetheless, we do

have the following.

Proposition 4.3. ¸et ) be a bounded ¸ipschitz domain in Rm and let k3CC0 haveIm k*0. Assume that u3C1(RmC)M , A

m`1) has u*3¸2(R)) and satis,es (30) or, if k is

real, either condition in (31). Also, suppose that Dku"0 in RmC)M . ¹hen we have the

integral representation formula

u(X)"PR)E$

k(X!>)n(>)u(>) dp(>), X3RmC)1 , (33)

where the choice of the sign depends on the form of the radiation condition.In particular, the function u decays exponentially if Im k'0 while, for non-zero real

k,

u(X)"OA1

DX D(m~1)@2B as DX DPR. (34)

Similar results are valid for functions annihilated by the right-handed action of Dk

(with the radiation conditions appropriately adjusted; see below).

Proof. First, we shall prove a seemingly weaker decay condition for u, namely

PDXD/R

DuD2dp"O(1) as RPR. (35)



To this end, we note that

D (1Giem`1

XK )u D2"Re[(1Giem`1

XK )uuN #(1Giem`1

XK )]0

"Re[(1Giem`1

XK )2uuN #]0

"Re[(2G2iem`1

XK )uuN #]0

"2 Du D2G2 Im(uN #XK em`1

u)0.

Hence,

PDXD/R

D(1Giem`1

XK )u D2 dp"2PDXD/R

Du D2dp

G2 ImAPDXD/R

uN #XK em`1

udpB0

. (36)

We use Gauss's formula (26) to further transform the last integral above. To this e!ect,for 0(R

0(R(R with R

0su$ciently large and "xed, we write

PDXD/R

uN #XK em`1

udp"PPR0:DXD:R

[(uN #Dk) (e

m`1u)#uN #D

~k(e

m`1u)]

#PDXD/R0

uN #XK em`1

u dp. (37)

Next, observe that D~k

(em`1

u)"!em`1

Dku"0 and that D

k"!D#

k#

2i(Im k)em`1

. The latter identity implies (uN #Dk)(e

m`1u)"!2i(Im k)uN #u so that the

solid integral in (37) becomes !2i(Im k)::R0:DXD:R

uN #u. Utilizing this back in (36) andinvoking the radiation condition gives

lim supDRD?=PDXD/R

Du D2 dp"$ImAPDXD/R0

uN #XK em`1

u dpB0

G2(Im k)PPDXD;R0

Du D2. (38)

The last term above is zero when k3R and negative when Im k'0. Thus, in anyevent, (35) is proved.

With (35) at hand, it is an easy matter to tackle (33). Speci"cally, working on thebounded Lipschitz domain MX3RmC)1 ; DX D(RN and invoking Proposition 4.2, mat-ters are readily reduced to showing that

PDYD/R

DE$

k(X!>)>K u (>) Ddp"PDYD/R

F$

k(X!>)e$*kDX~YD>K u (>) D dp

"o(1) as RPR, (39)



uniformly for X in compact subsets of Rm. To this end, when Im k'0, this followstrivially (for the &&plus'' choice of the sign) from (29), (35) and Schwarz's inequality. Onthe other hand, when k3RC0, (39) becomes, thanks to (35) and (20)}(21), a simpleexercise which we omit.

Finally, the decay of u at in"nity is easily established from (33) and the decay of E$

k.

K

Let us remark that, as inspection of the above proof shows, (31) can be weakened to

PDXD/R

D (1Giem`1

XK )u D2dp"o (1) as RPR.

In passing, let us also point out that any Cli!ord}Cauchy-type integral of the form:R)

E$

k(X!>) f (>) dp(>), corresponding to an absolutely integrable A

m`1-valued

function f on R), radiates at in"nity. This follows from (20) to (21) and elementaryestimates.

In the light of the above results, it seems natural to consider the exterior Hardyspace

H2-%&5

(RmC)1 , k) :"Mu :RmC)1 PAm`1

; Dku"0 in RmC)1 ,

u*3¸2(R)) and u satis"es (30)N

for any k3CC0 with Im k*0. As alluded to before, when Im k'0, replacing (30) by(29), or, say, u"O(1) at in"nity, yields the same space. However, if k3RC0, then thereare two disjoint types of exterior Hardy spaces. Concretely, we set

H2,$-%&5

(RmC)1 , k) :"Mu : RmC)1 PAm`1

; Dku"0 in RmC)1 ,

u*3¸2(R)) and u satis"es (31)N.

In this setting, Proposition 4.3 becomes the natural exterior domain analogue ofProposition 4.2. Of course, similar considerations apply to the space H2,$

3*')5(RmC)1 , k),

k3RC0, provided (31) is replaced by

u(X)(1$iem`1

XK )"o ( DX D~(m~1)@2) as DX DPR. (40)

We conclude this section by discussing the exterior domain version of Cauchy'svanishing formula (24). Concretely, we have the following.

Proposition 4.4. ¸et ) be a bounded ¸ipschitz domain in Rm and k3RC0. ¹hen, for anyu3H2,`

3*')5(RmC)1 , !k), v3H2,~

-%&5(RmC)1 , k) there holds

PR)unvdp"0. (41)

A similar conclusion is valid for any functions u3H2,~3*')5

(RmC)1 , !k),v3H2,`

-%&5(RmC)1 , k).



Proof. Making use of the Proposition 4.1 in the context of the bounded Lipschitzdomain MX3Rm; DX D)R, XN)1 N gives

PR)u (X)n (X)v (X) dp"PDXD/R

u(X)XK v (X) dp

"

1

2 PDXD/R

[u(X) (XK $iem`1

)v(X)#u (X)(XK Giem`1

)v (X)] dp. (42)

Next, from the radiation conditions (31), (40) and the decay conditions u (X)"O ( DX D~(m~1)@2), v (X)"O ( DX D~(m~1)@2) at in"nity we obtain

u(X)[(XK Giem`1

)v(X)]"o ( DX D~(m~1)) as DX DPR

and

[u(X) (XK $iem`1

)]v(X)"o ( DX D~(m~1)) as DX DPR.

These, in turn, prove that the last integral in (42) is o (1) as RPR. The desiredconclusion now follows easily. K

Remark. The de"nition of the spaces H2,$-%&5

(RmC)1 , k) can be naturally extended for$Im k*0, whereas that of H2,$

3*')5(RmC)1 , k) extends for $Im k)0. In particular,

Proposition 4.4 continues to hold true for all Im k*0.

5. Rellich-type identities for Cli4ord k-monogenic functions

In this section we present several Rellich-type identities which, in turn, will be usedlater to prove boundary ¸2-energy estimates. The starting point is the followingtheorem.

Theorem 5.1. ¸et ) be a bounded ¸ipschitz domain in Rm, k3C andu3H2

-%&5(), k)WH2

3*')5(), k). ¹hen, for any C1-vector ,eld h with real components, also

identi,ed with a Am-valued function, we have the identities:

1

2 PR)Du D2Sn, hTdp"$ReAPR)

hu(nu)#$

dpB0

GReAPP)1

2u([D, h]u#)!(Re k)ue

m`1u#hB

0

. (43)

Furthermore, the surface integral in the right-hand side of (43) can be replaced by

ReAPR)(un)#

$uhdpB

0

"ReAPR)(hu)#

$nN uN dpB

0

"ReAPR)uN nN (uh)#

$dpB

0

.



Proof. We write Sn, hT"12(nN h#hnN ) and Du D2"(uuN #)

0so that

Du D2Sn, hT"12(uuN #nN h)

0#1

2(uuN #hnN )

0

"12(uuN #nN h)

0#1

2(nN hu#uN )

0

"12(uuN #nN h)

0#1

2(u#uN nN h)

0

"Re(uuN #nN h)0.

At this point, using the perturbed Cauchy vanishing formula (25) we may continuewith

PR)Du D2Sn, hTdp"ReAPR)

uuN #nN h dpB0

"ReAPR)u ($nu#PuN #nN )hdpB

0

GReAPP)u ([D, h]u#)!2(Re k)ue

m`1u#hB

0

.

Now, since

ReAPR)u($nu##uN #nN )hdpB

0

"2ReAPR)u(nu)#

$h dpB

0

"$2ReAPR)hu(nu)#

$dpB

0

.

identity (43) follows.The proof of the claim in the second part of the theorem is based on purely algebraic

manipulations (as discussed in Section 2) and is left as an exercise to the reader. K

The previous theorem has several important consequences. To be able to statethem, we shall write A+B modC if there exists a positive constant M, independent ofthe relevant parameters, so that A)M(B#C) and B)M (A#C).

Corollary 5.2. Assume that ) is a bounded ¸ipschitz domain in Rm, k3C, andu3H2

-%&5(), k)WH2

3*')5(), k). ¹hen, for any real C1-vector ,eld h which is transversal to

R), we have

EuEL2(R))

+E(nu)$

EL2(R))

+E(un)$

EL2(R))

+E(uh)$

EL2(R))

+E(hu)$

EL2(R))

modulo EuEL2())

.

In particular, if u is "l`1-valued for some 0)l)m, then

En?uEL2(R))

+En@uEL2(R))

+Eh?uEL2(R))

+Eh@uEL2(R))

+EuEL2(R))

modulo EuEL2())

.

An analogous statement is valid in the case when ) is replaced by the complement ofa bounded ¸ipschitz domain. More speci,cally, in this case, we take h to be compactlysupported and the corresponding equivalences are valid modulo EuE

L2()W 4611h).



Proof. The "rst set of equivalences is a simple consequence of the previous theoremand the Cauchy}Schwarz inequality since, by assumption, Sn, hT*c'0 a.e. on R).The fact that [D, h] is a zero-order operator is also used here.

The second set of equivalences follows from the "rst one and the identities

(nu)`"G

n?u

!n@u

ifl (l#1)

2

ifl (l#1)

2

is odd,

is even,(44)

(nu)~"G

n?u

!n@u

ifl (l#1)

2

ifl (l#1)

2

is even,

is odd.(45)

To see this, let us show, for instance, that if l (l#1)/2 is odd then (nu)`"n?u. To this

e!ect, recall that

(nu)`"1

2(nu#nu)"1

2(nu#uN nN ).

Now, since u3"l`1, n3"1, we have that uN "(!1)(l`1)(l`2)@2u and nN "!n. Also,nu"n?u!n@u and un"(!1)(l`1)[n?u#n@u], by (10) and (11). From thisand the fact that 1

2(l#1)(l#2)#(l#1)#1 has the same parity as 1

2l(l#1)#

1(mod2) the conclusion easily follows.The case when ) is replaced by the complement of a bounded Lipschitz domain is

virtually identical. Note that, in this case, only the restriction of u to supph plays a roleand, hence, the behaviour of u at in"nity is not an issue. K

The interested reader may also consult the Rellich-type estimates from [15].

6. A priori estimates and integral representation formulas for solutions of Maxwell:sequations

Here we indicate how our results from Section 3 can be utilized in connection withMaxwell's equations in Lipschitz domains. Our main interest is to derive a prioriboundary estimates as well as integral representation formulae.

First, we derive an integral representation formula.

Lemma 6.1. ¸et ) be a bounded ¸ipschitz domain in Rm and consider k3C. ¹hen forany u3H2

-%&5(), k) we have

u(X)"PR)Ek(X!>) (n?u) (>) dp (>)

!PR)Ek(X!>)(n@u) (>) dp(>), X3).



A similar statement is valid for functions in H2-%&5

(RmC)1 , k) provided k3CC0 hasIm k*0 and functions in H2,$

-%&5(RmC)1 , k) if k3RC0.

Moreover, an analogous result holds for functions annihilated by the right handedversion of D

k.

Proof. This follows from the Cauchy}Cli!ord reproducing formulas (28), (33) and thefact that nu"n?u!n@u. K

In order to proceed, we make the following key observation. Let E, H be twosmooth di!erential forms of degrees l and l#1, respectively, 0)l)m, de"ned insome open subset of Rm. To these, we associate an A

m`1-valued function u by setting

u :"H!iem`1

E"H#i(!1)l`1Eem`1

. (46)

For instance, if m"3 and l"1, the above reduces to u"H#iEe4.

Proposition 6.1. =ith the above notation, u is (left or right) k-monogenic if and only if E,H satisfy the Maxwell system

(Mm) G

dE!ikH"0,

dE"0,

dH#ikE"0,

dH"0.

Moreover, u satis,es the radiation condition (30) if and only if E, H satisfy theSilver}MuK ller-type radiation conditions

XK ?E!H"o ( DX D~(m~1)@2) as DX DPR,

XK @H!E"o ( DX D~(m~1)@2) as DX DPR.(47)

Proof. A straightforward calculation shows that

Dku"i(!1)l`1(dE#dE!ikH)e

m`1#(dH#dH#ikE).

The claim in the "rst part of the proposition follows from this based on simple degreeconsiderations.

Similarly (1!iem`1

XK )u"(H!XK E)!iem`1

(XK H!E) so that the claim in thesecond part of the proposition is seen from (30) and (10). K

Remark. The radiation conditions (31) for u translate into the Silver}MuK ller-typeradiation conditions for E, H

XK ?EGH"o ( DX D~(m~1)@2) as DX DPR,

XK @HGE"o ( DX D~(m~1)@2) as DX DPR.(48)

Remark. If k is non-zero then, because d2"0 and d2"0, the equations dE"0,dH"0 become super#uous. Nonetheless, as for k"0 Maxwell's equations decouple



(i.e. E and H become unrelated), it is precisely this case for which these two equationsare relevant. Also, when m"3, l"1 (and kO0), the formulae (M

m) reduce to the

more familiar system of equations

(M3) G

+]E!ikH"0,

+]H#ikE"0,

while (47) becomes the classical Silver}MuK ller radiation condition for vector "elds;see, e.g., [20] and the references therein.

We are now ready to present the estimates and the integral representation formulasalluded to earlier. They have been "rst proved in [10] with a di!erent approach (a realvariable argument).

Theorem 6.2. ¸et ) be a bounded ¸ipschitz domain in Rm and assume that k3CC0.Also, suppose that the forms E3C=(), "l), H3C= (), "l`1) solve the Maxwell system(M

m) and have E*, H*3¸2 (R)). ¹hen, for any real C1-vector ,eld h which is transversal

to R) we have

EEEL2(R))

#EHEL2(R))

+En?EEL2(R))

#En?HEL2(R))

+En@EEL2(R))

#En@HEL2(R))

+Eh?EEL2(R))

#Eh?HEL2(R))

+Eh@EEL2(R))

#Eh@HEL2(R))

modulo EEEL2())

#EHEL2())

.

Furthermore, for X3),

E (X)"!dAPR)'

k(X!>)(n@E) (>) dp(>)B

#dAPR)'

k(X!>)(n?E) (>) dp (>)B

!ikPR)'

k(X!>) (n@H)(>) dp(>) (49)

and

H(X)"!ikPR)'

k(X!>)(n?E) (>) dp(>)

#dAPR)'

k(X!>) (n?H) (>) dp(>)B

!dAPR)'

k(X!>) (n@H) (>) dp(>)B. (50)



Finally, a similar set of conclusions is valid with ) replaced by the complement ofa bounded ¸ipschitz domain. In this latter case, as far as (49) and (50) are concerned, weassume that Im k*0 and, in addition, that E, H satisfy the (generalized) Silver}MuK llerradiation condition (47).

Proof. If we set u :"H#i (!1)l`1Eem`1

, then everything follows by straightforwardcalculations from Proposition 6.1, Corollary 5.2, and Lemma 6.1. K

Let us now pause and explain how these estimates relate to, for instance, the studyof boundary value problems for the Maxwell system in R3, the Laplace operator andthe Helmholtz operator. First, if ) is a Lipschitz domain in R3 and (E, H) solve theMaxwell system (M

3) in ), then the above theorem yields

En]EEL2(R))

#ESn, HTEL2(R))

+En]HEL2(R))

#ESn, ETEL2(R))

modulo EEEL2())

#EHEL2())

.

This has been "rst proved by real variable methods in [20] and the importance of thisestimate is discussed there in detail.

Furthermore, taking this time k :"0, E :"0 and H :"du, where u is a functionharmonic in the Lipschitz domain )-Rm, we obtain

E+5!/

uEL2(R))

+KKRuRn KK

L2(R))

modulo E+uEL2())

.

See, e.g., [30, 11] for the relevance of this estimate in connection with boundary valueproblem for the Laplacian.

Finally, if the complex-valued function ; satis"es (*#k2);"0 in )LRm, thenfor any transversal vector h3Rm to R) one has

E;EL2(R))

#E+;EL2(R))

+KKR;Rn KK

L2(R))

+KKR;Rh KK

L2(R))

+E+5!/;E

L2(R))#E;E

L2(R))

modulo E+;EL2())

,

where +5!/

stands for the tangential gradient on R). This is immediately seen byapplying the second part of Corollary 5.2 to the two-sided k-monogenicfunction u :"D

k; in ). In fact, one can easily check that u satis"es (30) if and only if

; satis"es the higher-dimensional analogue of the classical Sommerfeld radiationcondition, i.e.

S+;(X), XK T!ik;(X)"o ( DX D~(m~1)@2) as DX DPR.

We omit the straightforward details. See also [22, 29] for a fuller treatment of theHelmholtz equation.



7. More general wave numbers

Consider "rst the propagation of an electromagnetic wave in an isotropic, in-homogeneous medium. In this case, the Maxwell system reads

GdE!ik

1H"0,

dH#ik2E"0,

(51)

where k1, k

23C2 (Rm, C) are variable parameters. As before E, H are smooth di!eren-

tial forms of degree l and l#1, respectively.The general idea is that much of our previous analysis carries over to this context

even though the various identities and estimates are going to be valid only modulocertain residual terms. Our aim is to prove the following.

Theorem 7.1. ¸et )LRm be an arbitrary, bounded ¸ipschitz domain and let k1,

k23C1 (Rm) be complex-valued, non-vanishing functions. ¹hen the map assigning

(n?E, n?H)3¸2 (R))=¸2(R)) to each pair (E, H) which solves (51) in ) and alsohas E*, H*3¸2(R)), has a closed range and a ,nite-dimensional kernel. In particular, itis semi-Fredholm between appropriate spaces.

Proof. The idea is to show that the mapping in the statement of the theorem isbounded from below modulo compact operators. To this end, assume for a momentthat the estimate

EE*EL2(R))

#EH*EL2(R))

)CEn?EEL2(R))

#CEn?HEL2(R))

#CEEEL2())

#CEHEL2())

(52)

holds uniformly in (E, H) as in the statement of the theorem. Then, since E andH (separately) satisfy a strongly elliptic, second order, variable coe$cient PDE witha formally self-adjoint principal part, it follows (cf. the result in Section 3 of [17]) that

EEEW1@2,2 ())

#EHEW1@2,2 ())

)CEE*EL2(R))

#CEH*EL2(R))

. (53)

Now, the desired conclusion follows from the compactness of the embedding=1@2,2()))¸2()). Therefore, it remains to prove the a priori estimate (52).

To this e!ect, "x (E, H) as before and once again consider the homogeneous formu :"H!ie

m`1E. Since, this time,

dH"!

dk1

k1

?H and dE"!

dk2

k2

@E in ), (54)

it follows that

DDkju D#DuD

kjD)C Du D pointwise in ), j"1, 2. (55)

In particular, a natural adaptation of the Cauchy reproducing formula from Section4 yields in this case

Eu*EL2(R))

)CEuEL2(R))

#CEuEL2())

. (56)



Next, we turn attention to the Rellich-type identities of Section 5. The "rst observa-tion is that, as is well known, it is possible to choose a compactly supported,C1-variable vector "eld h with real components in Rm and so that Sn, hT*c'0almost everywhere on R). Using this, invoking (55) and paralleling the proofs ofTheorem 5.1 and Corollary 5.2, we arrive at

EuE2L2(R))

)CEuEL2(R))

En?uEL2(R))

#CEuE2L2())

. (57)

Using the usual trick to the e!ect that ab)ea2#(4e)~1b2 for any e'0, we "nally get

EuEL2(R))

)CEn?uEL2(R))

#CEuEL2())

. (58)

Now, the estimate (52) follows by combining (56) and (58). This completes the proof ofthe theorem. K

Our next result deals with the case when (51) is equipped with a pair of boundaryconditions involving an arbitrary (Lipschitz) transversal "eld in place of the exteriorunit normal.

Theorem 7.2. ¸et ) and k1, k

2be as before and assume that h is a ¸ipschitz vector ,eld

in Rm which is transversal to R). ¹hen the operator assigning (h?E, h?H)3¸2(R))=¸2 (R)) to each pair (E, H) which solves (51) in ) and also satis"es E*,H*3L2(R)), has a closed range and a ,nite dimensional kernel. In particular, thisoperator is also semi-Fredholm.

Proof. Essentially the same pattern of reasoning as in the previous theorem applies.One notable exception is that the role of (58) is now played by

EuEL2(R))

)CEh?uEL2(R))

#CEuEL2())

(59)

which, in turn, follows by paralleling the corresponding argument in Corollary 5.2.K

Finally, let us point out that analogous results are valid in the case of anisotropic,inhomogeneous media, in which case k

1, k

2in (51) are matrix-valued functions. For

example, when m"3, the problem becomes

GcurlE!iukH"0 in ),

curlH#iueE"0 in ),

where u3C plays the role of the frequency, while k"(kij), e"(e

ij) are complex

invertible matrices corresponding to the permeability and the dielectricity of themedium, respectively. We leave the details of this matter to the motivated reader.

Appendix

Here, for the convenience of the reader, we recall a short proof of the following.



Lemma 8.1 (Rellich:s lemma [25]). Assume that a complex-valued function u, de,ned inthe complement of some ball in Rm, satis,es (*#k2)u"0 for some non-zero real k aswell as

PDXD/R

Du D2dp"o (1) as RPR. (60)

¹hen u must vanish identically.

Proof. Let u (X)"++ an,l( DX D)>

n,l(XK ) be the expansion of u in spherical harmonics,where

an,l (r) :"P

Sm~1

u (ru)>Mn,l (u) du. (61)

Since the Laplacian in Rm is related to the Laplacian on the unit sphere Sm~1-Rm by

*Rm"

R2Rr2#

m!1

r

RRr#

1

r2*Sm~1,

it follows from (*#k2)u"0, *Sm~1>n,l"!n (n#m!2)>

n,l , (61) and integration byparts that each a

n,l satis"es the second order, linear ODE

R2an,lRr2 #

m!1

r

Ran,lRr #Ak2!

n (n#m!2)

r2 Ban,l"0. (62)

As is well known (cf. [3, (20)}(23), p. 96]), for each "xed n, the solutions of (62) arelinear combinations of r1~m@2J

$k(kr), where k :"[(m/2!1)2#n(n#m!2)]1@2 andJs

is the Bessel function. In particular, the coe$cients an,l have the asymptotic

behaviour

an,l(r)"c

k,m,n,le$*krr(1~m)@2G1#OA

1

rBH as rPR. (63)

On the other hand, Parseval's identity gives

rm~1++ Dan,l(r) D2"PDXD/R

Du(X) D2dpr

(64)

so that, by (60), rm~1 Dan,l(r) D2"o (1) as rPR. Utilizing this back in (63) "nally yields

an,l"0 for each n, l. Thus u"0 as desired. K

Acknowledgments

This research was initiated while MM was visiting AM at Macquarie University. It is a pleasure to havethe opportunity to thank here this institution for its hospitality as well as the Australian Research Councilfor its support. The work of MM has also been supported in part by the NSF grant DMSd9870018.Finally, both authors wish to thank ReneH Grognard for his interesting and helpful comments.



References

1. Brackx, F., Delanghe, R. and Sommen, F., Cli+ord Analysis, Pitman Advanced Publ. Program, London,1982.

2. CalderoH n, A. P., &The multipole expansion of radiation "elds', J. Rat. Mech. Anal., 3, 523}537(1954).

3. Carlson, B. C., Special Functions of Applied Mathematics, Academic Press, New York, San Francisco,London, 1997.

4. Colton, D. and Kress, R., Integral Equation Methods in Scattering ¹heory, Wiley, New York, 1983.5. Flanders, H., Di+erential Forms with Applications to the Physical Sciences, Academic Press, New York,

1963.6. Garnir, H. G., ¸es Problemes aux ¸imites de la Physique MatheH matique, Basel, BirkhaK user, 1958.7. Gilbert, J. and Murray, M. A., Cli+ord Algebras and Dirac Operators in Harmonic Analysis, Cambridge

Studies in Advanced Mathematics, 1991.8. GuK rlebeck, K. and SproK {ig, W., Quaternionic Analysis and Elliptic Boundary <alue Problems,

BirkhaK user Verlag, Basel, 1990.9. Jancewicz, B., Multivectors and Cli+ord Algebras in Electrodynamics, World Scienti"c, 1988,

Singapore/New Jersey/London.10. Jawerth, B. and Mitrea, M., &Higher dimensional scattering theory on C1 and Lipschitz domains', Amer.

J. Math., 117(4), 929}963 (1995).11. Kenig, C., Harmonic Analysis ¹echniques for Second Order Elliptic Boundary <alue Problems, CBMS

Series in Mathematics, No. 83, Amer. Math. Soc., Providence, RI, 1994.12. Li, C., McIntosh, A. and Semmes, S., &Convolution singular integrals on Lipschitz surfaces', J. Amer.

Math. Soc., 5, 455}481 (1992).13. McIntosh, A., &Cli!ord algebras, Fourier theory, singular integrals, and harmonic functions on Lip-

schitz domains', in : Proc. Conf. on Cli+ord Algebras in Analysis (J. Ryan ed.), Studies in AdvancedMathematics, C.R.C. Press Inc., Boca Raton, FL, 1995, pp. 33}88.

14. McIntosh, A., &Review of &&Cli!ord algebra and spinor-valued functions, a function theory for theDirac operator'', by R. Delanghe, F. Sommen and V. Soucek', Bull. Amer. Math. Soc., 32(3), 344}348(1995).

15. McIntosh, A., Mitrea, D. and Mitrea, M., &Rellich type identities for one-sided monogenic functions inLipschitz domains and applications', in: Analytical and Numerical Methods in Quaternionic andCli!ord Algebras, W. SproK ssig and K. GuK rlebeck eds., the Proceedings of the Sei!en Conference,Germany, 1996, pp. 135}143.

16. Mitrea, D., Mitrea, M. and Pipher, J., &Vector potential theory on non-smooth domains in R3 andapplications to electromagentic scattering', J. Fourier Anal. Appl., 3(2), 131}192 (1997).

17. Mitrea, D., Mitrea, M. and Taylor, M. &Layer potentials, the Hodge laplacian and global boundaryproblems in non-smooth Riemannian manifolds', preprint, (1997).

18. Mitrea, M., Cli+ord=avelets, Singular Integrals, and Hardy Spaces, Lecture Notes in Mathematics, No.1575, Springer, Berlin, 1994.

19. Mitrea, M., &Electromagnetic scattering on nonsmooth domains', Math. Res. ¸ett., 1(6), 639}646(1994).

20. Mitrea, M., &The method of layer potentials in electro-magnetic scattering theory on non-smoothdomains', Duke Math. J., 77(1), 111}133 (1995).

21. Mitrea, M., &Hypercomplex variable techniques in Harmonic Analysis', in: Proc. Conf. on Cli+ordAlgebras in Analysis (J. Ryan ed.), Studies in Advanced Mathematics, CRC Press Inc., Boca Raton, FL,1995, pp. 103}128.

22. Mitrea, M., &Boundary value problems and Hardy spaces for the Helmholtz equation in Lipschitzdomains', J. Math. Anal. Appl., 202, 819}842 (1996).

23. Mitrea, M., Torres, R. and Welland, G., &Regularity and approximation results for the Maxwellproblem on C1 and Lipschitz domains', in: Proc. Conf. Cli+ord Algebras in Analysis (J. Ryan ed.),Studies in Advanced Mathematics, CRC Press Inc., Boca Raton, FL, 1995, pp. 297}308.

24. MuK ller, C., Foundations of the Mathematical ¹heory of Electromagnetic =aves, Springer, New York,1969.

25. Rellich, F., &Uber das asymptotische Verhalten der Losungen von *u#ju"0 in unendlichenGebieten', J. Deutsch. Math. <erein., 53, 57}65 (1943).

26. Spain, B. and Smith, M. G., Functions of Mathematical Physics, Van Nostrand Reinhold Co., London,New York, 1970.

27. Stein, E. M., Singular Integrals and Di+erentiability Properties of Functions, Princeton Univ. Press,Princeton, N. J., 1970.



28. Taylor, M., Partial Di+erential Equations, Springer, Berlin, 1996.29. Torres, R. and Welland, G., &The Helmholtz equation and transmission problems with Lipschitz

interfaces', Indiana ;niv. Math. J., 42, 1457}1485 (1993).30. Verchota, G., &Layer potentials and boundary value problems for Laplace's equation in Lipschitz

domains', J. Funct. Anal., 59, 572}611 (1984).31. Von Westenholz, C. Di+erential Forms in Mathematical Physics, North-Holland, Amsterdam, 1978.



clifford algebras and maxwell's equations in lipschitz domains

Documents