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Clifford Algebras and Maxwell's Equations in Lipschitz Domains *

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Clifford Algebras and Maxwell's Equations in Lipschitz Domains * 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 m 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 8 3 > curl E ik H =0 inR n ; < 3 curl H + ik E =0 inR n ; (M ) 3 > : 2 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) 8 m > dE ik H =0 inR n ; < (M ) H + ik E = 0 in R n ; m > : 2 n ^ E = f 2 L (@ ); m (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 (@ ) 3 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 (@ ) m 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. m 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 3 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 3 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. 4 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 3 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 4 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. m Recall that the (complex) Cli ord algebra asso ciated with R endowed with m 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 m 2 2 subspace of R and such that x = jxj , for any x 2 R . This identity m m readily implies that, if fe g is the standard orthonormal basis in R , j j =1 then 2 = 1 and e e = e e for any j 6= k: e j k k j j m In particular, we identify the canonical basis fe g from R with the algebraic j basis of A .Thus, any element u 2A can b e uniquely representedinthe m m form m X X 0 (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 l the convention that e := 1). For this multi-index I wecall l the length of I ; P 0 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 m 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) 2 u =(1) (5) u; 4 l for any u 2 .For the complex conjugation on A weset m ! c X X c := (6) u e : u = u e I I I I I I P We de ne the scalar part of u = u e 2A as u := u , and endow I I m 0 ; I 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 m for any u 2A and a 2 R .We also de ne m 1 u := (8) fu ug 2 for each u 2A . m P 0 l With u as in (4), de ne u := u e and denote by the range of l I I jI j=l l : A !A .
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