Clifford Algebras and Maxwell's Equations in Lipschitz Domains *

Home , Potential theory

Cli ord algebras and Maxwell's equations

in Lipschitz domains

Alan McIntosh and Marius Mitrea

April, 1998

Abstract

We present a simple, Cli ord algebra based approach to several key

results in the theory of Maxwell's equations in non-smo oth sub domains

of R . Among other things, wegive new pro ofs to the b oundary energy

estimates of Rellichtyp e for Maxwell's equations in Lipschitz domains

from [20] and [10], discuss radiation conditions and the case of variable

wavenumb er.

1 Intro duction.

It has b een long recognized that there are fundamental connections b etween

electromagnetism and Cli ord algebras. Indeed, understanding Maxwell's

equations was part of Cli ord's original motivation.

A more recent trend concerns the treatmentofsuch PDE's in domains

with irregular b oundaries. See [11] for an excellent survey of the state of

the art in this eld up to early 1990's and [13], [18] for the role of Cli ord

algebras in this context.

Following work in the context of smo oth domains ([24], [2], [4], [23]), the

three-dimensional Maxwell system

curl E ik H =0 inR n ;

curl H + ik E =0 inR n ; (M )

n E = f 2 L (@ );

1991 Mathematics Subject Classi cation. Primary 42B20, 30G35; Secondary 78A25.

Key words and phrases. Cli ord algebras, Maxwell's equations, Rellichidentities, radiation

conditions, Hardy spaces, Lipschitz domains. 1

(plus appropriate radiation conditions at in nity) in the complementofa

b ounded Lipschitz domain R has b een solved in [20], while the higher

dimensional version (involving di erential forms)

dE ik H =0 inR n ;

(M ) H + ik E = 0 in R n ;

n ^ E = f 2 L (@ );

(plus suitable radiation conditions) for arbitrary Lipschitz domains R ,

has b een solved in [10] (precise de nitions will b e given shortly). See also

[19], [16] for related developments.

One of the key ingredients of the approachin[20], [10]was establishing

estimates to the e ect that the so-called voltage-to-current map, taking n E

into n H and n ^ E into n _ H , resp ectively, is an isomorphism b etween

appropriate b oundary spaces. Ultimately, this comes down to proving norm

equivalences of the following form

(1) kn E k 2 + khn; H ik 2 kn H k 2 + khn; E ik 2

L (@ ) L (@ ) L (@ ) L (@ )

in R and, more generally,

(2) kn ^ E k + kn ^ H k kn _ E k + kn _ H k

2 2 2 2

L (@ ) L (@ ) L (@ ) L (@ )

in R . As shown in [20, Theorem 4.1] (for Im k>0) and [10, Theorem

8.5] (for arbitrary k 6= 0), the ab ove estimates lead to the invertibilityof

certain magnetostatic (and electrostatic) b oundary integral op erators which,

in turn, are used to solve(M ), m 3.

The aim of this pap er is to shed new light on the basic estimates (1), (2)

and to present an alternative, natural approachtoproving such b oundary

energy estimates, develop ed in the framework of Cli ord algebras. This is

an extension of work in [18] where such an approach has rst b een used

for certain PDE's in Lipschitz domains. Besides its intrinsic merit, the

relevance of this work should b e most apparentforthenumerical treatment

of electro-magnetic scattering problems in non-smo oth domains. For the

smo oth context and di erenttechniques see [8].

The departure p oint for us (whichinfactgoesbacktoMaxwell himself;

it has also b een reinvented by M. Riesz) is to write the Maxwell system

as a single equation which, in fact, expresses the Cli ord analyticityofa

certain Cli ord algebra-valued function. Sp eci cally,ifE and H are the 2

electric and magnetic comp onents of an electro-magnetic waveinR , then

the entire Maxwell system is equivalent to the Maxwell-Dirac equation

(3) (D + ke )(H + iE e )= 0:

4 4

Here E and H are regarded as Cli ord algebra-valued functions in R ,and

D + ke is a p erturb ed Dirac op erator.

This is b oth mathematically convenient and physically relevant. In

fact, the same idea has b een extensively used byphysicists who employ

the Lorentz metric in the four dimensional ( at Minkowskian) space-time

domains in order to write the 3D Maxwell system for E =(E ) and

i i=1;2;3

H =(H ) solely in terms of the electromagnetic eld-strength tensor.

i i=1;2;3

Due to its skew symmetry, the latter de nes the exterior di erential form

(sometimes referred to as a Faradaybi-vector eld)

! := (E dx + E dx + E dx ) dt

1 1 2 2 3 3

+H dx dx + H dx dx + H dx dx

1 2 3 2 3 1 3 1 2

(typically, electrical forces are vectorial while magnetic ones are 2-tensors).

See, e.g., the monographs [5], [31], [9], and [28].

The radiation conditions for E and H can also b e naturally rephrased

in terms of ! .Parenthetically, let us note that it is also p ossible to work

with the non-homogeneous form E + H and write Maxwell's equations in

the form (D + ik )(E + H )= 0 (cf.[23], [10]) but this setup app ears to b e

less physically relevant.

However, the most attractive asp ect of this algebraic context is that it

preserves many of the key features of the classical complex function the-

ory.For us, Hardy spaces of monogenic functions in Lipschitz domains and

Cauchyvanishing formulas will play a basic role in the sequel. Based on

these, if R is the complement of a b ounded Lipschitz domain and if

n n e + n e + n e denotes the outward unit normal to @ , we shall

1 1 2 2 3 3

show that for any solution ! = H + iE e of (3) there holds

kn! !nk k! k ;

2 2

L (@ ) L (@ )

mo dulo residual terms. The Rellichtyp e estimate (1) follows from this. In

fact, a similar argument applies in the higher dimensional case.

Before commencing the ma jor developments we shall intro duce here some

notation. Call a b ounded domain Lipschitz if its b oundary is lo cally given

by graphs of Lipschitz functions in appropriate rectangular co ordinates. We 3

let d denote the surface measure on @ and set n for the outward unit

normal to @ . For a (p ossibly algebra-valued) function u de ned in , the

nontangential maximal function u is given by u (X ):=supfju(Y )j; Y 2

; jX Y j 2dist(Y; @ )g. Also, the nontangential boundary trace on @

of a function u de ned in is taken as the p ointwise nontangential limit

almost everywhere with resp ect to the surface measure on the b oundary.

Acknow ledgments. This researchwas initiated while MM was visiting AM at

Macquarie University. It is a pleasure to have the opp ortunity to thank here

this institution for its hospitalityaswell as the Australian Research Council

for its supp ort. We also wish to thank Rene Grognard for his interesting

and helpful comments.

2 Cli ord algebra rudiments.

Recall that the (complex) Cli ord algebra asso ciated with R endowed with

the usual Euclidean metric is the minimal enlargementofR to a unitary

complex algebra A ,which is not generated (as an algebra) byany prop er

m m

2 2

subspace of R and such that x = jxj , for any x 2 R . This identity

readily implies that, if fe g is the standard orthonormal basis in R ,

j =1

then

= 1 and e e = e e for any j 6= k: e

j k k j

In particular, we identify the canonical basis fe g from R with the algebraic

basis of A .Thus, any element u 2A can b e uniquely representedinthe

m m

form

X X

(4) u = u e ; u 2 C;

I I I

l=0

jI j=l

where e stands for the pro duct e e :::e if I =(i ;i ;:::;i )(wemake

I i i i 1 2 l

1 2

the convention that e := 1). For this multi-index I wecall l the length of I

;

and denote it by jI j.We shall adopt the convention that indicates that

the sum is p erformed only over strictly increasing multi-indices I .

The Cli ordconjugation on A is de ned as the unique complex-linear

involution on A for whiche e = e e = 1 for anymulti-index I . In

m I I I I

P P

particular, if u = u e 2A , thenu = u e . Note that

I I m I I

I I

l(l+1)

u =(1) (5) u; 4

for any u 2 .For the complex conjugation on A weset

X X

:= (6) u e : u = u e

I I I I

I I

We de ne the scalar part of u = u e 2A as u := u , and endow

I I m 0

;

c 2 c c

A with the natural Euclidean metric hu; u i = juj := (uu ) =(u u) .

m 0 0

Note that (uv ) =(vu) = hu; vi, for any u; v 2A . Another useful

0 0 m

observation is that

(7) jauj = juaj = jujjaj

for any u 2A and a 2 R .We also de ne

u := (8) fu ug

for each u 2A .

With u as in (4), de ne u := u e and denote by the range of

l I I

jI j=l

: A !A . Elements in will b e referred to as l -vectors or di erential

l m m

forms of degree l . The exterior and interior pro duct of forms are de ned for

1 l

2 , u 2 , resp ectively,by

(9) ^ u := ( u) and _ u := ( u):

l+1 l1

By linearity, these op erations extend to arbitrary u 2A .

Two identities which are going to b e of imp ortance in the sequel are

(10) u = ^ u _ u;

and

(11) u =(1) ( ^ u + _ u) ;

1 l

valid for arbitrary 2 and u 2 .

. Then D = We shall work with the Dirac op erator D := e

j =1

@ m

e = D ,and D = 4, the negative of the Laplacian in R .For

j =1

reasons which will b ecome more apparent shortly,we shall nd it useful to

embed everything into a larger Cli ord algebra, sayR A A .If

m m+1

for k 2 Cwenowset

(12) ID := D + ke

k m+1

then ID = 4 + k .Furthermore,

(13) ID = D + ke = ID and ID = D ke = ID :

m+1 k m+1 k

k 5

If u is a A -valued function de ned in a region of R ,set

m+1

+ ke u ID u := e

m+1 k j

j =1

and

e + kue : uID :=

j m+1 k

j =1

Call u left (right, two-sided) k -monogenic if ID u (uID ,orbothID u and

k k k

uID ) = 0 in . Note that each comp onentofak -monogenic function is

annihilated by the Helmholtz op erator 4 + k . A function theory for the

p erturb ed Dirac op erator ID whichisrelevant for us here has b een worked

out in [21] and [22].

The exterior derivative op erator d acts on a -valued function u by

du := (Du). It extends by linearity to arbitrary A -valued functions.

l+1 m

Its formal transp ose, ,isgiven by u := (Du)if u is -valued. Once

again, extends by linearity to arbitrary A -valued functions. Since D =

2 2

d + , it follows that d =0, = 0 and (d + d)= 4,asiswell known.

More detailed accounts on these matters can b e found in [1], [7], [12], [18].

Further references can b e found in [14].

3 A distinguished fundamental solution.

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

Dirac op erator ID . The departure p oint is deriving an explicit expression

for a fundamental solution of the Helmholtz op erator 4 + k in R . If

k = i, this can b e taken to b e minus the kernel of the Bessel p otential

J =(4 + I ) corresp onding to = 2. This kernel is known (cf. [27],

p. 131) to have the form

1 jX j dt

(X )= exp t (14) :

m=2 m=2

(4 ) t

For general k 2 iR we simply re-scale the expression of to obtain that

1 jX j dt

(X ):= (15) exp k t

m=2 m=2

(4 ) t

is a fundamental solution for the op erator 4 + k in R . Clearly this also

continues to hold true for all k 2 CwithImk>jRe k j and, in fact, the 6

ab ove expression can b e continued analytically to the op en upp er-half plane

Im k>0 with the prop ertythat (X )= O (exp fjX j Im k g)as jX j! 1.

For instance, when Im k>0andm 2, one maytake

1 (ik )

m ik jX jt 2

(X )= dt; X 2 R n 0; e (t 1)

! (m 2)!

where ! is the area of the unit sphere in R . Itisworth p ointing out that

the ab ove de nition can b e further extended to fk 2 C n 0; Im k 0g by

setting, for X 2 R n 0,

2 >

@ 1 1

ik r

; if m is o dd; e

2r @r 2ik

r =jX j

(X )= (16)

(1)

i 1 @

H (kr) ; if m is even ;

2r @r 4

r =jX j

(1)

where H is the usual zero-order Hankel function of the rst kind. We

shall continue to denote this extension by .For this and a more detailed

discussion see the excellent treatmentin[6], pp. 59-74.

It is now clear that

(17) e ke E := ID = ID =

j m+1 k k k k k k

j =1

is a (two-sided) fundamental solution for ID . When k =0 then E reduces

k k

to the usual radial fundamental solution for the Laplace op erator. If Im k>

0 then E decays exp onentially at in nity.Furthermore, if F (X )issothat

k k

ik jX j

E (X )= F (X )e then

k k

F (X )= O jX j (18) ; as jX j! 1:

This follows from the de nition of plus a straightforward computation.

In the case when m is even, the classical asymptotic expansion (in the sense

of Poincare; cf., e.g., [26, (9.13.1), p. 166])

1=2

1 1 9 1 9 25 2

(1)

iz i =4 2 3

e 1 i H + i i + ::: (z )

3 6 2 9 3

z 1!2 z 2!2 z 3!2 z

as jz j! 1, is also used.

A more delicate analysis is required when k 2 R n 0 b ecause in this case

there are two canonical decaying fundamental solutions. Sp eci cally, for

k 2 R n 0, we de ne 7

E (19) := ID = ID :

k k k k

are so that = E .Now, if the functions F In passing let us note that E

k k

ik jX j

E (X )= F (X )e then, along similar lines as b efore,

k k

1 X

F (X )= c + o jX j ie ; as jX j! 1;

m;k m+1

jX j

(20)

where c 2 C n 0, and

m;k

(X )= o jX j (21) ; as jX j! 1: rF

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

ik jX j

(X )= (22) e

4 jX j

so that, for k 2 R n 0,

X ik X ke 1

ik jX j

E (23) e : (X )=

3 2

4 jX j jX j jX j

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

4 Function theory for p erturb ed Dirac op erators.

Let b e a b ounded Lipschitz domain in R and let n stand for the outward

unit normal to . As in [18], [22], [7], weintro duce the Hardy typ e space

H ( ;k) of left k -monogenic functions in by

lef t

2 2

( ;k):= fu : !A ; u is left k monogenic in ;u 2 L (@ )g: H

m+1

lef t

Similarly,we de ne H ( ;k), the Hardy space of right k -monogenic func-

right

tions in by

2 2

H ( ;k):= fv : !A ; v is right k monogenic in ;v 2 L (@ )g:

m+1

right

We endow these spaces with the natural norms kuk 2 := ku k 2 ,

L (@ )

H ( ;k )

lef t

and kv k := kv k , resp ectively.

2 2

L (@ )

H ( ;k )

right 8

Equivalent norms are obtained by taking the L norms of the b oundary

trace; see [12] and the theory of monogenic Hardy spaces develop ed in [18].

2 2

Also, note that H ( ;k) \H ( ;k) consists precisely of two-sided k -

lef t right

monogenic functions in whichhave a square-integrable non-tangential

maximal function.

Prop osition 4.1 Let bea bounded Lipschitz domain in R , k 2 C, and

2 2

assume that u 2H ( ;k), v 2H ( ; k ). Then u and v have (non-

right lef t

tangential) boundary limits at almost every point on @ and

(24) unv d =0:

2 2 1

Also, if u 2H ( ;k) \H ( ;k) and 2 C (R ; A ),then

m+1

right lef t

Z ZZ

c c c

(25) unu d = u([D; ]u 2(Re k ) e u )

m+1

where [D; ] is the commutator between D ,actingfrom the left, and multi-

plication by to the right.

Formula (24) can b e considered as the Cli ord analogue of the classical

Cauchy's vanishing theorem in complex analysis. The second identity, (25),

is a p erturbation of this result. We state it here since we shall need it

shortly. In b oth cases, the integrands must b e interpreted in the sense of

multiplication in the Cli ord algebra.

Pro of.

The existence of the b oundary trace for u and v is proved in the usual way.

As for the vanishing formula, supp ose rst that u and v are smo oth up to

and including the b oundary of . Consequently, Gauss's formula gives

Z ZZ ZZ

(26) unvd = (uD ) v + u (Dv )= (uID ) v + u (ID v ):

k k

In this case, (24) follows based on the monogenicity assumptions. The gen-

eral case is then seen from this and a standard limiting argument (see, e.g.,

[30] for details in similar circumstances).

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

c c c

ID (u ) = [ID ;]u +(ID u )

k k k

c c

(27) = [D; ]u 2(Re k ) e u :

m+1 9

The pro of is complete. 2

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

repro ducing formula.

Prop osition 4.2 Let bea bounded Lipschitz domain in R and k 2 C.

Then every function u 2H ( ;k) has (nontangential) boundary limits at

lef t

almost every point on @ and

(28) u(X )= E (X Y ) n(Y ) u(Y ) d (Y ); X 2 :

A similar statement is valid for functions in H ( ;k).

right

Pro of.

In a smo other context, the pro of go es along well known lines; cf., e.g., [1]

for the case when k = 0. Note, however, that we utilize here the fact that

E (X Y )ID = E (X Y )ID =0forX 6= Y . Then, as b efore,

k k;Y k k;X

passing to domains with Lipschitz b oundaries is done via a routine limiting

argument. 2

In the second part of this section we shall discuss the exterior domain

version of Prop osition 4.2. The novelty is that u should satisfy a (necessary)

decay condition at in nity. In the case when Im k>0, based on (18), it is

not to o dicult to see that

Im k jX j

u(X ) e = o(jX j (29) ); as jX j! 1;

is a (necessary and) sucient condition to prove the exterior domain version

of (28). In turn, the latter implies that actually u decays exponential ly at

in nity.Parenthetically, let us also note that, e.g., u(X )= O (1) at in nity

always entails (29).

The b ehavior at in nity requires a more elab orate analysis when k 2 R n 0

in which situation (29) is no longer the appropriate decay condition. In fact,

the whole case when k 2 C n 0hasImk 0canbecovered by insisting that

u satis es a decay condition of radiation typ e. Sp eci cally,we shall ask that

(30) ); as jX j! 1; 1 ie X u(X )= o(jX j

m+1

X m

where wesetX := , X 2 R n 0.

jX j

When k 2 R n 0, the opp osite choice of sign in (30) will also do. Thus,

in this case, (30) may b e replaced by 10

(31) ); as jX j! 1: X u(X )= o(jX j 1 ie

m+1

The source of (31) is the expression (20). As exp ected, for real non-zero

k , the functions E satisfy (31) b ecause

^ ^ ^ ^

X X ie (32) 1 ie X : X 1 ie =0= 1 ie

m+1 m+1 m+1 m+1

Since 1 ie X are, as (32) shows, zero-divisors, one cannot deduce a decay

m+1

condition of the same order for u itself based solely on (31) and algebraic ma-

nipulations. In fact, owing to the classical Rellich's lemma (cf. App endix),

any solution u of the Helmholtz equation ( + k )u = 0 for non-zero real k

) in a (connected) neighb orho o d of in nity which satis es u(X )= o(jX j

in R must vanish identically. Hence, generally sp eaking, for (31) to hold

when k 2 R n 0, certain internal cancel lations must o ccur when multiplying

X ). Nonetheless, wedohave the following. u with (1 ie

m+1

Prop osition 4.3 Let beabounded Lipschitz domain in R and let k 2

1 2

C n 0 have Im k 0. Assume that u 2 C (R n ; A ) has u 2 L (@ )

m+1

and satis es (30) or, if k is real, either condition in (31). Also, suppose

that ID u =0 in R n . Then we have the integral representation formula

(33) (X Y ) n(Y ) u(Y ) d (Y ); X 2 R n ; u(X )= E

where the choice of the sign depends on the form of the radiation condition.

In particular, the function u decays exponential ly if Im k>0 while, for

non-zeroreal k ,

u(X )= O (34) ; as jX j! 1:

jX j

Similar results are valid for functions annihilated by the right-handedac-

tion of ID (with the radiation conditions appropriately adjusted; seebelow).

Pro of.

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

(35) juj d = O (1); as R !1:

jX j=R 11

To this end, we note that

i h

c 2

^ ^ ^

X ) X )uu (1 ie X )uj = Re (1 ie j(1 ie

m+1 m+1 m+1

h i

2 c

= Re (1 ie X ) uu

m+1

i h

X )uu = Re (2 2ie

m+1

2 c

= 2juj 2Im u Xe u :

m+1

Hence,

Z Z

2 2

j(1 ie X )uj d = 2 juj d

m+1

jX j=R jX j=R

(36) u Xe ud : 2Im

m+1

jX j=R

We use Gauss's formula (26) to further transform the last integral ab ove.

To this e ect, for 0

0 0

write

Z ZZ

c c c

u Xe ud = [(u ID )(e u)+u ID (e u)]

m+1 k m+1 k m+1

jX j=R R

+ (37) u Xe ud:

m+1

jX j=R

+ Next, observe that ID (e u)= e ID u = 0 and that ID = ID

k m+1 m+1 k k k

c c

2i(Im k )e . The latter identity implies (u ID )(e u)=2i(Im k )u u

m+1 k m+1

u u. Utiliz- so that the solid integral in (37) b ecomes 2i(Im k )

ing this backin(36)andinvoking the radiation condition gives

Z Z

c 2

Xe ud u limsup juj d = Im

m+1

jRj!1

jX j=R jX j=R

(38) juj : 2(Imk )

jX j>R

The last term ab ove is zero when k 2 R and negative when Im k>0. Thus,

in anyevent, (35) is proved. 12

With (35) at hand, it is an easy matter to tackle (33). Sp eci call y,

working on the b ounded Lipschitz domain fX 2 R n ; jX j

invoking Prop osition 4.2, matters are readily reduced to showing that

Z Z

ik jX Y j

^ ^

jE (X Y ) Yu(Y )j d = jF Yu(Y )j d (X Y ) e

k k

jY j=R jY j=R

(39) = o(1); as R !1;

uniformly for X in compact subsets of R .To this end, when Im k>0,

this follows trivially (for the \plus" choice of the sign) from (29), (35) and

Schwarz's inequality. On the other hand, when k 2 R n 0, (39) b ecomes,

thanks to (35) and (20)-(21), a simple exercise whichweomit.

Finally, the decayofu at in nity is easily established from (33) and the

decayofE . 2

Let us remark that, as insp ection of the ab ove pro of shows, (31) can b e

weakened to

j(1 ie X )uj d = o(1); as R !1:

m+1

jX j=R

In passing, let us also p oint out that any Cli ord-Cauchytyp e integral of the

form E (X Y )f (Y ) d (Y ), corresp onding to an absolutely integrable

@ k

A -valued function f on @ , radiates at in nity. This follows from (20)-

m+1

(21) and elementary estimates.

In the light of the ab ove results, it seems natural to consider the exterior

Hardy space

2 m m m

H (R n ;k) := fu :R n !A ; ID u = 0 in R n ;

m+1 k

lef t

u 2 L (@ ) and u satis es (30)g

for any k 2 C n 0 with Im k 0. As alluded to b efore, when Im k>0,

replacing (30) by (29) or, say, u = O (1) at in nity, yields the same space.

However, if k 2 R n 0, then there are two disjointtyp es of exterior Hardy

spaces. Concretely,we set

m m m

(R n ;k) := fu :R n !A ; ID u = 0 in R n ; H

m+1 k

lef t

u 2 L (@ ) and u satis es (31)g: 13

In this setting, Prop osition 4.3 b ecomes the natural exterior domain ana-

logue of Prop osition 4.2. Of course, similar considerations apply to the space

H (R n ;k), k 2 R n 0, provided (31) is replaced by

right

(40) ); as jX j! 1: X = o(jX j u(X ) 1 ie

m+1

We conclude this section by discussing the exterior domain version of

Cauchy's vanishing formula (24). Concretely,wehave the following.

Prop osition 4.4 Let bea bounded Lipschitz domain in R and k 2 R n 0.

2;+ 2;

m m

Then, for any u 2H (R n ; k ), v 2H (R n ;k) there holds

right lef t

(41) unv d =0:

A similar conclusion is valid for any function u 2H (R n ; k ),

right

2;+

v 2H (R n ;k).

lef t

Pro of.

Making use of the Prop osition 4.1 in the context of the b ounded Lipschitz

domain fX 2 R ; jX j R; X2 = g gives

Z Z

u(X ) Xv(X ) d u(X ) n(X ) v (X ) d =

jX j=R @

h i

^ ^

= (42) u(X ) X ie v (X )+ u(X ) X ie v (X ) d:

m+1 m+1

jX j=R

Next, from the radiation conditions (31), (40) and the decay conditions

(m1)=2 (m1)=2

u(X )= O (jX j ), v (X )= O (jX j ) at in nitywe obtain

h i

(m1)

u(X ) X ie v (X ) = o(jX j ); as jX j! 1

m+1

and

h i

(m1)

u(X ) v (X )= o(jX j ); as jX j! 1: X ie

m+1

These, in turn, prove that the last integral in (42) is o(1) as R !1. The

desired conclusion now follows easily. 2

Remark. The de nition of the spaces H (R n ;k) can b e naturally

lef t

(R n ;k) extends for extended for Im k 0, whereas that of H

right

Im k 0. In particular, Prop osition 4.4 continues to hold true for all

Im k 0. 14

5 Rellichtyp e identities for Cli ord k -monogenic

functions.

In this section we present several Rellich-typ e identities which, in turn, will

b e used later to prove b oundary L -energy estimates. The starting p ointis

the following theorem.

Theorem 5.1 Let beabounded Lipschitz domain in R , k 2 Cand

2 2 1

u 2H ( ;k) \H ( ;k). Then, for any C -vector eld with real

lef t right

components, also identi edwitha A -valued function, we have the identi-

ties:

Z Z

c 2

u(nu) d juj hn; i d = Re

@ @

c c

Re (43) u ([D; ]u ) (Re k )ue u :

m+1

Furthermore, the surface integral in the right side of (43) can bereplacedby

Z Z Z

c c c

d : n ud =Re u d =Re Re u n (u) (u) (un)

@ @ @

0 0 0

Pro of.

2 c

Wewrite hn; i = (n + n )andjuj =(uu ) so that

1 1

2 c c

juj hn; i = (u u n ) + (u u n )

0 0

2 2

1 1

c c

= (u u n ) + (n u u)

0 0

2 2

1 1

c c

(u u n ) + (u u n ) =

0 0

2 2

= Re (u u n ) :

At this p oint, using the p erturb ed Cauchyvanishing formula (25) wemay

continue with

Z Z Z

2 c c c

juj hn; i d = Re u u n d =Re u (nu +u n ) d

@ @ @

0 0

c c

u ([D; ]u ) 2(Rek )ue u : Re

m+1

0 15

Now, since

Z Z

c c c

Re u (nu +u n ) d = 2Re u (nu) d

@ @

0 0

u(nu) d ; = 2Re

the identity (43) follows.

The pro of of the claim in the second part of the theorem is based on

purely algebraic manipulations (as discussed in x 2) and is left as an exercise

to the reader. 2

The previous theorem has several imp ortant consequences. To b e able to

state them, we shall write A B mo d C if there exists a p ositive constant

M , indep endent of the relevant parameters, so that A M (B + C )and

B M (A + C ).

Corollary 5.2 Assume that is a bounded Lipschitz domain in R , k 2 C,

2 2 1

and u 2H ( ;k) \H ( ;k). Then, for any real C -vector eld which

lef t right

is transversal to @ , we have

kuk 2 k(nu) k 2 k(un) k 2

L (@ ) L (@ ) L (@ )

k(u ) k 2 k(u) k 2 mo dulo kuk 2 :

L (@ ) L (@ ) L ( )

l+1

In particular, if u is -valued for some 0 l m, then

kn ^ uk 2 kn _ uk 2 k ^ uk 2

L (@ ) L (@ ) L (@ )

k _ uk 2 kuk 2 mo dulo kuk 2 :

L (@ ) L (@ ) L ( )

An analogous statement is valid in the case when is replaced by the

complement of a bounded Lipschitz domain. Morespeci cal ly, in this case,

we take to becompactly supported and the corresponding equivalences are

valid modulo kuk 2 .

L ( \supp )

Pro of.

The rst set of equivalences is a simple consequence of the previous theorem

and the Cauchy-Schwarz inequality since, by assumption, hn; i c> 0 a.e.

on @ . The fact that [D; ] is a zero-order op erator is also used here.

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

l(l+1)

n ^ u; if is o dd;

(nu) = (44)

l(l+1)

is even; n _ u; if

2 16

l(l+1)

n ^ u; if is even ;

(nu) = (45)

l(l+1)

n _ u; if is o dd:

l(l+1)

is o dd then (nu) = n ^ u. To see this, let us show, for instance, that if

To this e ect, recall that

1 1

(nu + (nu +un ): (nu) = nu)=

2 2

(l+1)(l+2)

l+1 1

Now, since u 2 , n 2 ,wehavethatu =(1) u andn = n.

(l+1)

Also, nu = n ^ u n _ u and un =(1) [n ^ u + n _ u], by (10) and (11).

(l +1)(l +2)+(l + 1) + 1 has the same parity From this and the fact that

l (l + 1) + 1 (mo d 2) the conclusion easily follows. as

The case when is replaced by the complement of a b ounded Lipschitz

domain is virtually identical. Note that, in this case, only the restriction of

u to supp plays a role and, hence, the b ehavior of u at in nityisnotan

issue. 2

The interested reader may also consult the Rellichtyp e estimates from

[15].

6 A priori estimates and integral representation

formulas for solutions of Maxwell's equations.

Here we indicate how our results from x 3 can b e utilized in connection with

Maxwell's equations in Lipschitz domains. Our main interest is to derivea

priori b oundary estimates as well as integral representation formulae.

First, we deriveanintegral representation formula.

Lemma 6.1 Let bea bounded Lipschitz domain in R and consider k 2

C. Then for any u 2H ( ;k) we have

lef t

u(X ) = E (X Y )(n ^ u)(Y ) d (Y )

E (X Y )(n _ u)(Y ) d (Y ); X 2 :

(R n ;k) provided k 2 A similar statement is valid for functions in H

lef t

(R n ;k) if k 2 R n 0. C n 0 has Im k 0 and functions in H

lef t

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

handed version of ID .

k 17

Pro of.

This follows from the Cauchy-Cli ord repro ducing formulas (28), (33) and

the fact that nu = n ^ u n _ u. 2

In order to pro ceed, wemake the following key observation. Let E , H be

two smo oth di erential forms of degrees l and l + 1, resp ectively,0 l m,

de ned in some op en subset of R .To these, we asso ciate an A -valued

m+1

function u by setting

l+1

(46) u := H ie E = H + i(1) Ee :

m+1 m+1

For instance, if m = 3 and l = 1, the ab ove reduces to u = H + iE e .

Prop osition 6.1 With the above notation, u is (left or right) k -monogenic

if and only if E; H satisfy the Maxwel l system

dE ik H =0;

E =0;

(M )

H + ik E =0;

dH =0:

Moreover, u satis es the radiation condition (30) if and only if E , H

satisfy the Silver-Mul ler typeradiation conditions

> 2

X ^ E H = o(jX j ); as jX j! 1;

(47)

X _ H E = o(jX j ); as jX j! 1:

Pro of.

A straightforward calculation shows that

l+1

ID u = i(1) (dE + E ik H )e +(dH + H + ik E ):

k m+1

The claim in the rst part of the prop osition follows from this based on

simple degree considerations.

^ ^ ^

Similarly,(1 ie X )u =(H XE) ie (XH E ) so that the

m+1 m+1

claim in the second part of the prop osition is seen from (30) and (10). 2

Remark. The radiation conditions (31) for u translate into the Silver-

Muller typ e radiation conditions for E , H

> 2

X ^ E H = o(jX j ); as jX j! 1;

(48)

X _ H E = o(jX j ); as jX j! 1: 18

2 2

Remark. If k is nonzero then, b ecause d =0 and = 0, the equations

E =0,dH = 0 b ecome sup er uous. Nonetheless, as for k =0 Maxwell's

equations decouple (i.e. E and H b ecome unrelated), it is precisely this case

for which these two equations are relevant. Also, when m =3,l =1 (and

k 6= 0), the formulae (M ) reduce to the more familiar system of equations

(

rE ik H =0;

(M )

rH + ik E =0;

while (47) b ecomes the classical Silver-Muller radiation condition for vector

elds; see, e.g., [20] and the references therein.

We are now ready to present the estimates and the integral representa-

tion formulas alluded to earlier. They have b een rst proved in [10]witha

di erent approach(arealvariable argument).

Theorem 6.2 Let beabounded Lipschitz domain in R and assume that

1 l 1 l+1

k 2 C n 0.Also, suppose that the forms E 2 C ( ; ), H 2 C ( ; )

solve the Maxwel l system (M ) and have E ;H 2 L (@ ). Then, for any

real C -vector eld which is transversal to @ we have

kE k + kH k kn ^ E k + kn ^ H k

2 2 2 2

L (@ ) L (@ ) L (@ ) L (@ )

kn _ E k + kn _ H k

2 2

L (@ ) L (@ )

k ^ E k + k ^ H k

2 2

L (@ ) L (@ )

k _ E k + k _ H k

2 2

L (@ ) L (@ )

mo dulo kE k 2 + kH k 2 :

L ( ) L ( )

Furthermore, for X 2 ,

E (X ) = d (X Y )(n _ E )(Y ) d (Y )

+ (X Y )(n ^ E )(Y ) d (Y )

(49) (X Y )(n _ H )(Y ) d (Y ); ik

and

H (X ) = ik (X Y )(n ^ E )(Y ) d (Y )

@ 19

+ (X Y )(n ^ H )(Y ) d (Y )

(50) d (X Y )(n _ H )(Y ) d (Y ) :

Final ly, a similar set of conclusions is valid with replacedbythecom-

plement of a bounded Lipschitz domain. In this latter case, as far as (49)-

(50) areconcerned, we assume that Im k 0 and, in addition, that E , H

satisfy the (generalized) Silver-Mul ler radiation condition (47).

Pro of.

l+1

If weset u := H + i(1) Ee , then everything follows by straight-

m+1

forward calculations from Prop osition 6.1, Corollary 5.2, and Lemma 6.1.

Let us now pause and explain how these estimates relate to, for instance,

the study of b oundary value problems for the Maxwell system in R , the

Laplace op erator and the Helmholtz op erator. First, if is a Lipschitz

domain in R and (E; H ) solve the Maxwell system (M ) in , then the

ab ove theorem yields

kn E k + khn; H ik kn H k + khn; E ik

2 2 2 2

L (@ ) L (@ ) L (@ ) L (@ )

mo dulo kE k + kH k :

2 2

L ( ) L ( )

This has b een rst proved byrealvariable metho ds in [20] and the imp or-

tance of this estimate is discussed there in detail.

Furthermore, taking this time k := 0, E := 0 and H := du, where u is a

function harmonic in the Lipschitz domain R ,we obtain

kr uk 2 mo dulo kruk 2 :

tan

L (@ ) L ( )

2 @n

L (@ )

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

b oundary value problems for the Laplacian.

Finally, if the complex-valued function U satis es (4 + k )U =0in

m m

R , then for any transversal vector 2 R to @ one has

@U @U

kU k 2 + krU k 2

L (@ ) L (@ )

2 2 @n @

L (@ ) L (@ )

kr U k + kU k

2 2

tan

L (@ ) L (@ )

mo dulo krU k 2 ;

L ( ) 20

where r stands for the tangential gradienton@ . This is immedi-

tan

ately seen by applying the second part of Corollary 5.2 to the two-sided

k -monogenic function u := ID U in . In fact, one can easily check that

u satis es (30) if and only if U satis es the higher dimensional analogue of

the classical Sommerfeld radiation condition, i.e.

D E

rU (X ); X ik U (X )= o(jX j ); as jX j! 1:

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

of the Helmholtz equation.

7 More general wavenumb ers.

Consider rst the propagation of an electromagnetic wave in an isotropic,

inhomogeneous medium. In this case, the Maxwell system reads

(

dE ik H =0;

(51)

H + ik E =0;

where k ;k 2 C (R ; C)are variable parameters. As b efore, E , H are

1 2

smo oth di erential forms of degree l and l + 1, resp ectively.

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 b e

valid only mo dulo certain residual terms. Our aim is to prove the following.

Theorem 7.1 Let R beanarbitrary, bounded Lipschitz domain and

let k ;k 2 C (R ) becomplex-valued, non-vanishing functions. Then the

1 2

2 2

map assigning (n ^ E; n ^ H ) 2 L (@ ) L (@ ) to each pair (E; H ) which

solves (51) in and also has E ;H 2 L (@ ), has a closedrange and a

nite dimensional kernel. In particular, it is semi-Fredholm between appro-

priate spaces.

Pro of.

The idea is to show that the mapping in the statement of the theorem is

b ounded from b elow mo dulo compact op erators. To this end, assume for a

moment that the estimate

kE k + kH k C kn ^ E k + C kn ^ H k

2 2 2 2

L (@ ) L (@ ) L (@ ) L (@ )

(52) +C kE k 2 + C kH k 2

L ( ) L ( ) 21

holds uniformly in (E; H ) as in the statement of the theorem. Then, since

E and H (separately) satisfy a strongly elliptic, second order, variable co-

ecient PDE with a formally self-adjoint principal part, it follows (cf. the

results in x 3of[17]) that

(53) kE k + kH k C kE k 2 + C kH k 2 :

1=2;2 1=2;2

L (@ ) L (@ )

W ( ) W ( )

Now, the desired conclusion follows from the compactness of the emb edding

1=2;2 2

W ( ) ,! L ( ). Therefore, it remains to prove the a priori estimate

(52).

To this e ect, x (E; H ) as b efore and once again consider the homoge-

neous form u := H ie E . Since, this time,

m+1

dk dk

2 1

(54) ^ H and E = _ E in ; dH =

k k

1 2

it follows that

uID (55) C juj pointwise in ; j =1; 2: + ID u

k k

j j

In particular, a natural adaptation of the Cauchy repro ducing formula from

x 4 yields in this case

(56) ku k C kuk + C kuk :

2 2 2

L (@ ) L (@ ) L ( )

Next, we turn attention to the Rellichtyp e identities of x 5. The rst ob-

servation is that, as is well known, it is p ossible to cho ose a compactly

supp orted, C -variable vector eld with real comp onents in R and so

that hn; i c> 0 almost everywhere on @ . Using this, invoking (55) and

paralleling the pro ofs of Theorem 5.1 and Corollary 5.2, we arriveat

2 2

(57) : C kuk kn ^ uk + C kuk kuk

2 2

L (@ ) L (@ )

L ( ) L (@ )

2 1 2

Using the usual trick to the e ect that ab "a +(4") b for any "> 0,

we nally get

(58) kuk C kn ^ uk + C kuk :

2 2 2

L (@ ) L (@ ) L ( )

Now, the estimate (52) follows bycombining (56) and (58). This completes

the pro of of the theorem. 2

Our next result deals with the case when (51) is equipp ed with a pair of

b oundary conditions involving an arbitrary (Lipschitz) transversal eld in

place of the exterior unit normal. 22

Theorem 7.2 Let and k , k beasbefore and assume that is a Lipschitz

1 2

vector eld in R which is transversal to @ . Then the operator assigning

2 2

( ^ E; ^ H ) 2 L (@ ) L (@ ) to each pair (E; H ) which solves (51)

in and also satis es E ;H 2 L (@ ), has a closedrange and a nite

dimensional kernel. In particular, this operator is also semi-Fredholm.

Pro of.

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

(59) kuk C k ^ uk + C kuk

2 2 2

L (@ ) L (@ ) L ( )

which, in turn, follows by paralleling the corresp onding argument in Corol-

lary 5.2. 2

Finally, let us p oint out that analogous results are valid in the case of

anisotropic, inhomogeneous media, in whichcasek , k in (51) are matrix-

1 2

valued functions. For example, when m = 3, the problem b ecomes

(

curl E i! H = 0 in ;

curl H + i! E = 0 in ;

where ! 2 C plays the role of the frequency, while =( ), =( )

ij ij

are complex invertible matrices corresp onding to the p ermeability and the

dielectricity of the medium, resp ectively.Weleave the details of this matter

to the motivated reader.

8 App endix.

Here, for the convenience of the reader, we recall a short pro of of the fol-

lowing.

Lemma 8.1 (Rellich's lemma [25 ]) Assume that a complex-valued func-

tion u,de nedinthecomplement of some bal l in R , satis es ( + k )u =0

for some non-zeroreal k as wel l as

(60) juj d = o(1); as R !1:

jX j=R

Then u must vanish identical ly. 23

Pro of.

Let u(X )= a (jX j)Y (X ) b e the expansion of u in spherical har-

n; n;

monics, where

(61) a (r ):= u(r! )Y (! ) d! :

n; n;

Since the Laplacian in R is related to the Laplacian on the unit sphere

S R by

m 1 1 @ @

+ + ; =

2 2

@r r @r r

it follows from ( + k )u =0, Y = n(n + m 2)Y ,(61)and

n; n;

integration by parts that each a satis es the second order, linear ODE

@ a @a n(n + m 2) m 1

n; n;

+(k (62) + )a =0:

2 2

@r r @r r

As is well known (cf. [3, (20)-(23), p. 96]), for each xed n, the solutions of

1m=2 2

(62) are linear combinations of r J (kr), where := [(m=2 1) +

1=2

n(n + m 2)] and J is the Bessel function. In particular, the co ecients

a have the asymptotic b ehavior

ik r

(63) 1+ O ( ) ; as r !1: a (r )= c e r

n; k ;m;n;

On the other hand, Parseval's identity gives

m1 2 2

(64) r ja (r )j = ju(X )j d

n; r

jX j=r

m1 2

so that, by (60), r ja (r )j = o(1) as r !1. Utilizing this backin(63)

nally yields a =0 foreach n; .Thus, u = 0 as desired. 2

References

[1] F. Brackx, R. Delanghe, and F. Sommen, Cli ordAnalysis, Pitman Advanced

Publ. Program, 1982.

[2] A. P.Calderon, The multipole expansion of radiation elds, J. Rat. Mech.

Anal., 3 (1954), 523{537.

[3] B. C. Carlson, Special Functions of Applied Mathematics, Academic Press,

New York, San Francisco, London, 1997. 24

[4] D. Colton and R. Kress, Integral equation methods in scattering theory,Wiley,

New York, 1983.

[5] H. Flanders, Di erential forms with applications to the physical sciences,A.P.

Press, 1963.

[6] H. G. Garnir, Les problemes aux limites de la physique mathematique, Basel,

Birkhauser Verlag, 1958.

[7] J. Gilb ert, and M. A. Murray, Cli ordAlgebras and Dirac Operators in Har-

monic Analysis, Cambridge Studies in Advanced Mathematics, 1991.

[8] K. Gurleb eck and W. Sproig, Quaternionic Analysis and El liptic Boundary

Value Problems, Birkhauser Verlag, Basel, 1990.

[9] B. Jancewicz, Multivectors and Cli ordAlgebras in Electrodynamics,World

Scienti c, 1988, Singap ore/ New Jersey/ London.

[10] B. Jawerth and M. Mitrea, Higher dimensional scattering theory on C and

Lipschitz domains, American Journal of Mathematics, Vol. 117, No. 4 (1995),

929{963.

[11] C. Kenig, Harmonic Analysis Techniques for Second Order El liptic Bound-

ary Value Problems, CBMS Series in Mathematics, No. 83, Amer. Math. So c.,

1994.

[12] C. Li, A. McIntosh and S. Semmes, Convolution singular integrals on Lipschitz

surfaces, J. Amer. Math. So c., 5 (1992), 455{481.

[13] A. McIntosh, Cli ord algebras, Fourier theory, singular integrals, and har-

monic functions on Lipschitz domains, in the Pro ceedings of the Conference

on Cli ord Algebras in Analysis, J. Ryan ed., Studies in Advanced Mathe-

matics, C.R.C. Press Inc., 1995, pp. 33{88.

[14] A. McIntosh, Review of \Cli ord algebra and spinor-valued functions, a func-

tion theory for the Dirac operator", by R. Delanghe, F. Sommen and V.

Soucek, Bull. of A.M.S., Vol. 32, No. 3, (1995), 344{348.

[15] A. McIntosh, D. Mitrea and M. Mitrea, Rel lich type identities for one-sided

monogenic functions in Lipschitz domains and applications, in \Analytical and

Numerical Metho ds in Quaternionic and Cli ord Algebras", W. Sprossig and

K. Gurleb eck eds., the Pro ceedings of the Sei en Conference, Germany, 1996,

pp. 135{143.

[16] D. Mitrea, M. Mitrea and J. Pipher, Vector potential theory on non-smooth do-

mains in R and applications to electromagnetic scattering, Journal of Fourier

Analysis and Applications, Vol. 3, No. 2 (1997), 131-192.

[17] D. Mitrea, M. Mitrea and M. Taylor, Layer potentials, the Hodge laplacian

and global boundary problems in non-smooth Riemannian manifolds, preprint,

(1997). 25

[18] M. Mitrea, Cli ord Wavelets, Singular Integrals, and Hardy Spaces, Lecture

Notes in Mathematics, No. 1575, Springer-Verlag, Berlin Heidelb erg, 1994.

[19] M. Mitrea, Electromagnetic scattering on nonsmooth domains, Mathematical

ResearchLetters,Vol. 1, No. 6 (1994), 639{646.

[20] M. Mitrea, The method of layer potentials in electro-magnetic scattering theory

on non-smooth domains,Duke Math. Journal, Vol. 77, No. 1 (1995), 111-133.

[21] M. Mitrea, Hypercomplex variable techniques in Harmonic Analysis, in Pro-

ceedings of the Conference on Cli ord Algebras in Analysis, J. Ryan ed.,

Studies in Advanced Mathematics, C.R.C. Press Inc., 1995, pp. 103{128.

[22] M. Mitrea, Boundary value problems and Hardy spaces for the Helmholtz equa-

tion in Lipschitz domains, Jour. Math. Anal. Appl., Vol. 202 (1996), 819{842.

[23] M. Mitrea, R. Torres, and G. Welland, Regularity and approximation results

for the Maxwel l problem on C and Lipschitz domains, in Pro ceedings of the

Conference on Cli ord Algebras in Analysis, J. Ryan ed., Studies in Advanced

Mathematics, C.R.C. Press Inc., 1995, pp. 297{308.

[24] C. Muller, Foundations of the mathematical theory of electromagnetic waves,

Springer{Verlag, New York, 1969.

[25] F. Rellich, Uber das asymptotische Verhalten der Losungen von u + u =0

in unend lichen Gebieten, Jb er. Deutsch. Math. Verein., 53 (1943), 57{65.

[26] B. Spain and M. G. Smith, Functions of Mathematical Physics,Van Nostrand

Reinhold Co., London, New York, 1970.

[27] E. M. Stein, Singular integrals and di erentiability properties of functions,

Princeton Univ. Press, Princeton, N.J., 1970.

[28] M. Taylor, Partial Di erential Equations, Springer-Verlag, 1996.

[29] R. Torres and G. Welland, The Helmholtz equation and transmission problems

with Lipschitz interfaces, Indiana Univ. Math. J. 42 (1993), 1457{1485.

[30] G. Verchota, Layer potentials and boundary value problems for Laplace's equa-

tion in Lipschitz domains,J.Funct. Anal., 59 (1984), 572{611.

[31] C. Von Westenholz, Di erential Forms in Mathematical Physics,North-

Holland, Amsterdam, 1978.

|||||||| || || {

Alan McIntosh: 26

Macquarie University, Sydney

NSW 2109, Australia

e-mail: alan@mp ce.mq.edu.au

Marius Mitrea:

Institute of Mathematics

of the Romanian Academy,

P.O. Box 1-764

RO-70700 Bucharest, Romania

and

Department of Mathematics

University of Missouri-Columbia

202 Mathematical Sciences Building

Columbia, MO 65211

e-mail: [email protected] 27