) dx, **
♦, <> | € V = H0(curl,Q),
lop A
and then the problem (218) is equivalent tofindingE(o>) in V with: (220)
a(E(a>),(|>) = - fQ J(a>). $ dx,
V <> | € V.
The sesquilinear form a($,c|>) is coercive on V for all o> with Im a> * 0, thus (220) has a unique solution for all o> with Im o> * 0.
6.3 CAUSAL PROBLEMS
291
Now we integrate with respect to n, and take the real part of (220) with <> | = E; using (216) in ft, and the Cauchy-Schwarz inequality, we obtain with Cx = Ce0y: |E(c*)| 2 dxdn
(221) CJ^
with IIJII2=/ftxR I J(«)| 2 dx dn. Thus we have: (222) E(o>) € L2(C1 x R,,)3, and X curl E(o>) € L2(C1 x R / , with o>. that is: (223)
n
+ iy,
E € L^X), X = L2(Q)3, and E(t) = f / ^ E(s)ds satisfies: curl E € LJQC).
This implies: (224) thus:
H(o>) € L2(Q x R / , and X curl H(a>) € L2(Q x Fty3,
(225)
H € Lj(X), and H(t) = f / ^ H(s) ds satisfies: curlH € L^X).
Furthermore since 2s(ci>) = - X E(o>) and H(v>) * - ]jj H(w), (with |
IE € L 2 ^ , H0(curl,Q)), H € L 2 ^ , H(curl,Q)), I thus: E € Lj(H0(curl,a)), H € Lj(H(curl,Q)).
Proposition 20. The causal evolution problem (217) with delay, that is in a domain Cl occupied in part by free space and in part by a linear anisotropic medium with permittivity and permeability (E,JI) satisfying H5) and H14) chap. 1, has a unique solution (E, D, B, H) in L 2 (X Q ), with X = L2(C1)\ for a given electric current J in L2 (X), and this solution satisfies (226). y
O
3.2.4. A causal waveguide problem Like in section 3.1.4, let Q + = Qx Rz+ be a free space domain, which is a semiinfinite cylinder bounded on its lateral boundary by a perfect conductor (modelling a waveguide), with a regular bounded cross-section Cl (with boundary T) in R2. We assume that a tangential electric field is given at the end of the waveguide. Then we consider the evolution problem: find (E,H) in the waveguide, with (E,H) € L^(X3 x X 3 ), X = L2(Q+), satisfying: i ) - £ 0 ^ + curlH = 0, (227)
i * 0 ^ + curlE = 0 inQ + = Q + xR t ,
ii)nAE| =0 onl^TxR^xRt, iii)nAE| 2 =nAE° onZ0 = QxRt, Ex€l^(H0(curlT,Q)).
6 EVOLUTION PROBLEMS
292
Wefirstwrite Maxwell equations like in (30) chap. 4, using the same notations: € 9t SET 9SHT gradT H
I^
"
3" °
* °*
|ii)d 3 E T -grad T E3 + >i()atSHT«0 inQ+xR^ (where 8 t = ^ ) with: (229)
OcurljHj-co^Ea* 0 *
^curljEr+^a^sOinQ+xRt,
and with the boundary conditions: (230)
O^AEr! =0, E 3 | =0,
© B ^ = E£.
From now on we drop the subscript T for differential operators. The determination of the electromagneticfieldwill be made in several steps. Determination of the transverse electricfieldET In the same way as in chap. 4, wefirstobtain that E T satisfies: OafEj + A E T - c - ^ E T s O , (231)
inQ+xR^
ii)n T AE T | =0, (divEjH^O, iii) Ej(0) s E j o n I 0 , i^ErC^X^X^L2^);
the boundary condition on the divergence is a consequence, with (230)i), of: (232)
div
x,x 3 E = 8 3 E 3 +
div
TETs0-
Let AQ be the positive self adjoint operator in L (Q) defined by: AQU * - Au, with (233) ^ i V H u c H ^ Q ) 2 , curluand divueH 1 ^), nAu| r =0, (divu)|r=0} (see (147) chap. 2 with n = 3). Then using the spectral decomposition (X2.,*.) of AQ, with c = 1, we define the operator AT from its Laplace transform DA^pJ^-Gj^p), 9jSS(p2 + X?)1/2, withRee j>y , v^Iv^. inLj(XT), (234)
u)Dy(AT) = {v€Lj(XT),XT = L2(Q)2, Z/Rlflj^jVj^diK+oo, p = y + in}. We can prove that AT is the infinitesimal generator of a contraction semigroup (G^Xj)) in L2y(X,.), X^L^Q) 2 , of class C°. Thus like in section 3.1.4, the problem (231) has a unique solution Ej. which is given thanks to (G^x^) by:
6.3 CAUSAL PROBLEMS (235)
293
E^^G^Ef
2/ V 2x Then we can verify thanks to the Laplace transformation that E T € L~(X ).
Determination ofEy so that
First assuming that E 3 is given at x 3 = 0, we have to find E 3
i ) 8 3 E 3 + AE 3 -c*- 2 3 2 E 3 :=0 ii)E 3 | (236)
H
inQ + xR t ,
=0,
iii) E 3 1 2 = E 3 , with given E^ in L2y(L2(Q)), iv)E 3 €L?(X),X = L2(Q+). Then E 3 is obtained thanks to the semigroup (G(x3)) of section 3.1.4 (with generator A defined by (189), (189)') by: (237)
E 3 (X 3 ) = G ( X 3 ) E 5 .
Then we have to find E 3 . It is obtained (like in chap.4) from (232) by: (238)
33E3(0) = AE 3 = - div Ej.
We have div E j € L^(H~ l(Cl)) c (D(A))\ and thus, since A is an isomorphism from D(A)ontoLj(L 2 (Q)) (239)
E° * - A* W
Ej) € L}(L2(Q», thus E 3 € LJ(X).
Determination ofHy We get H 3 from <> | = curlj. E T by time integration of (229)ii). Applying curlj. to (231)i), then by taking the scalar product of (228)i) with the normal nT, we get
inQ+xR^
dip ,
I iii)
2,i 2, This gives H 3 e L~(I/(G)) and H3 e L^(X).
6 EVOLUTION PROBLEMS
294
Determination ofHT Now we can obtain H T by time integration of (228)ii); we have to verify that H- is in L 2(X2). We can prove this result like in the stationary case, using the Hodge decomposition for H p see (68) chap .4 (we can also use an inverse Laplace transformation on the formulas of chap. 4 taken with complex 03). Causal Calderon operator of the waveguide. Now we are especially interested in having the boundary value of H- at the end of the waveguide, i.e., for x3 = 0. This is also obtained thanks to (228)ii) with x3 = 0: (242)
8tSHT(.,0,t)=ATET(.,0,t) + gradA-ldivE!J<.,0,t).
We get a formula similar to (61) chap.4 (with (29) and (38) chap. 4): (243)
HrCAt) — £ ^ [SATE^.,0,s) + curlTA"1 divTE^<.,0,s)] ds.
We can also obtain a form similar to (68), chap.4, more useful for spaces. The operator C:E£—H^.,0,.)
(orCiSEj^SH^.A.))
is the causal Calderon operator ofthe waveguide. Of course we have to specify its domain. We can do so thanks to the operators Aj and A; we do not detail this question here. The interest of this operator is (like all previous Calderon operators) that it contains all informations at hand for the domain exterior to the waveguide. Furthermore we can prove that the causal Calderon operator satisfies the positivity properties: (244)
(CSEj, E£) = - (CEj, SE£) £ 0
which we get from t 245 * - ^ x R ^ o f
E +
>*of - W e - ^ d x d t ^ ^ d i v t H A ^ e - ^ d x d t ,
and integrating by parts: (246)
y / ^ x ^ ( £ 0 E 2 + M 0 H 2 )e- 2 y t dxdt=/ 2 o x R ^nAH.Ee- 2 Y t dI 0 dt.
3.2.5. Uniqueness at most In all these causal problems we have uniqueness (at most) results for the solution of the wave equation like the Maxwell equation (this is to compare to the stationary case, chap. 2.13; these results are given for a bounded domain Q, to simplify): Theorem 3. Uniqueness at most. Given a (regular bounded) domain Cl in Rn, with boundary T, the wave problem: find u so that
6.3 CAUSAL PROBLEMS
295
i) Du=f, f given in D'+(X), in Q x 1^, (247)
ii)u! 2 = u°, i J l ^ w i t h l ^ r x R , , withu°€D%(H-i/2(r)), u1 €D\(H'm(T)) liii)u€l)\(X),X = l/(Q),
has at most one solution. Theorem 4. Uniqueness at most. The Maxwell problem: find (E,H) so that i ) £ 0 ^ - c u r l H = J, (248)
j i 0 ^ + cur!E = M i n Q x R ,
with M and J given in D%(X), X = L2(Q)3 ii) n A E | 2 = M°, n A H | 2 = J° withM°andJ°€DV(H" 1/2 (div,r)) iii) (E, H) € Z)V(X), X = L2(Q)3xL2(Q)3,
has at most one solution. The essential difference with the stationary case is that these uniqueness results are not local in time: we can substitute a part Z 0 =sr o xR, with T0 a "regular" part of T, for I . However we cannot replace R by finite time intervals, except for bounded Q, in which case these time intervals have to be large enough. This is a consequence of the finite velocity of propagation. Finding the optimal time interval for which we have uniqueness is of great interest in control problems (see for instance Lions [3]). These uniqueness results imply that the causal Calderon operators and the causal Calderon projectors have the same importance as in the stationary case: these operators on the boundary T of a domain Q contains all the informations on the wave inside the domain Cl that we consider. Of course these operators are not very easy to handle especially for numerical implementations. So they are often replaced by approximate boundary conditions. However they are very useful in obtaining a priori estimates. CONCLUSION We have solved many evolution problems, but we would have to deal with many other linear interesting problems, which are not a simple application of the above theory, for example with a moving obstacle, and a moving antenna... Furthermore there are very often parameters giving different scales, so that an asymptotic analysis is necessary, with singular perturbations.
APPENDIX
DIFFERENTIAL GEOMETRY FOR ELECTROMAGNETISM
Using differential geometry in electromagnetism is quite natural, atfirstin the modelling of Maxwell equations by differential forms, according to the right transformation laws when changing a coordinate system into another one with a different orientation (this explains the notion of "polar vector" often used in electromagnetism). The notion of cohomology allows a deep understanding of many points, notably in the study of the differential operators grad, curl, div with their kernels and their images, in an open domain Q; and then in the trace spaces on the boundary T of Q for electromagneticfieldsoffiniteenergy. We have to use the manifold structure of T, and the differential operators gradp, curlp diVp on T, with the technical difficulty of having enough regularity in order to apply usual tools generally developed for very regular manifolds. Differential geometry is also useful in numerical methods in electromagnetism with Whitney forms (see Bossavit [1]). We suppose only elementary prerequisites onfinitedimensional manifolds. We review some simple notions in order to set notations, following Morrey [1], Arnold [1], Abraham-Marsden-Ratiu [1]. We also need some elementary notions on variational framework (the notion of a V-coercive bilinear form) and we often use the Peetre lemma (see chapter 2). We emphasize that this appendix is not a course on differential geometry; we refer to Malliavin [1] or to Kobayashi-Nomizu [1] for instance for that. 296
A.l INTRODUCTION
297
1. INTRODUCTION. MATHEMATICAL FRAMEWORK
1.1. M anifold with boundary The notion of manifold is supposed to be known, with the notions of charts, of mutually C^a-compatible charts, of atlases and of C*'a-compatible atlases (we refer to Dieudonne [1], Bourbaki [1]). We recall the standard definition of an n-dimensional manifold M with boundary T of class C**", k € N, 0 £ a <* 1 (C00, analytic). Each point of M is contained in some open set Fof M, which is the homeomorphic image either of the unit ball in Rn, or of the part of it for which ^ ^ 0, the points where ^ = 0 correspond to the boundary T of M; any two coordinate systems are related by a transformation of class C^a (C°°, analytic). Let Q be an open subset of Rn; Q. may be considered as an n-dimensional manifold with boundary T; following Grisvard [1], we then define: Definition 1. We say that Q. is cm n-dimensional continuous (resp Lipschitz, continuously differentiable, of class C**", k €N, 0 £ a < 1) submanifold with boundary in Rn, if for every x€T, there are a neighborhood V of x in Rn and an injective map yfrom Vinto Rn so that: i) Q n K= {y € Q, <>| (y) < 0}, where <>| (y) denotes the n-th component ofitfy); ii) <>| together (j)"1 defined on <|>(V) is continuous (resp. Lipschitz, continuously differentiable, ofclass C^a). o The boundary r of M = Cl is then locally defined by (> | (y) = 0. The interior M = Q, is a C°° (even analytic) manifold without boundary! The boundary T of an ndimensional continuously differentiable (resp.C , C '*) manifold M, has an induced structure of (n - l)-dimensional continuously differentiable (resp. ) manifold, according to the implicit function theorem. In the case of Lipschitz regularity, this is not true (see Grisvard [1]). Example 1. Polyhedrons, which are in common use for numerical applications, are only Lipschitz manifold, so we often have to make direct proofs of the main properties, like for the Stokes theorem! Let M be a regular (differentiable) manifold, we will use the following notations in this appendix.
A - DIFFERENTIAL GEOMETRY
298
TXM the tangent space at a point x of M (the space of tangent vectors at x), TM = U TxYM the tangent bundle of M; x€ M TXM the cotangent space at x € M (i.e., the dual space of TXM, or the space of covectors atx), T M = U TAYM the cotangent bundle of M (or the phase space); x€M r s * VSXM the tensor product &TXM # <8>TXM (the space of r-contravariant s-covariant tensors), T5M the tensor bundle oftype (r9s) over M,TfM = U TLM; x€M r , r ATXM and ATXM the spaces of v-covectors and of t-vectors, r * r * AT M = U ATAVM the r-exterior bundle over M (with A the exterior or wedge x€M product). A vectorfield(resp. a (r,s) tensorfield)is a section of TM (res p. of T^M), that is, a mapping v from M into TM (resp. T^M), such that « o v a l , with n the canonical projection from TM (resp. T^M) onto M; a differential form is a section of T*M, i.e., a mapping o>: M —> T*M such that it*ow = l, with n* the canonical projection from T M onto M. r r * A (differential) r-form » is a section of AT M, and an r-vector field is a r section of ATM, the regularity of these mappings being so far not specified. ft
i
ft
In the domain of any coordinate system (x 1 ,...^) we denote by Trff-fQZn (or simply by 3!,...,3 n ) the respectively associated vector fields, by dx 1 ,..., dx" ft
ft
i
the associated 1-forms, so that(-2r) ,...,(^i) is a basis of TXM, (dx1)v,...,(dxn) 0J£ X
OA
is the dual basis of TXM, with (dx*) . 3j-*y,
X
X
X
Vij.
An r-foim o> may be represented as follows: (1)
0)=
2
Wj ; dx1 A...Achr,
where (coj j) are the components of o> in that coordinate system and A denotes the exterior product. Let I be a sequence I = {ip.. ,,ir} with 1 < ij <.. .< i,. < n. We often abbreviate notations by: (iy
a) = 2 (Ojdxj. I
A A INTRODUCTION
299
We assume that M is an n-dimensional orientable manifold (i.e., there is a continuous n-form co on M such that
,~x
V1
,x
/
/ ~ x% <*vK » • • • » *
w,
)
^"■ ^ »^r?i'
dco^^dx'Adx1, I,a
1
or (3)'
9&>~
da> = 2 (ckO.*^, (*>),» 2 ( - i f " 1 — p , J
J
v=l
JV=J\UVK
g^v
dcox(v0,...,vr)= S^lrtD^x-ViXvo,...,^,...^,),
Vje^M,
where v{ denotes that Vj is missing, and Do>x is the derivative of co in charts. Note also d2 = 0, i.e.,d2a> = 0, V
gx(v,v)>0,
VveT x M,v*0.
In a local coordinate system x 1 ,..., x!\ g (which is often written as ds2) is given with fe«) = (& (9i > 9j)) a positive definite matrix, by: (5)f
g^gy^1®^-
300
A - DIFFERENTIAL GEOMETRY
A manifold M with a Riemannian stucture is called a Riemannian manifold. IfM is of class C^", g(x) is of class C*"1,01. For all x in M, e is an inner product onTxM; wenote: (6) gx (h,k) = (h,k) = 2 g ( x ^ , for all tangent vectors h, k of TXM, given in a local coordinate system x 1 ,..., x11 by h = Ih^i, k = Ikiai. Recall that (aj) is the dual basis of (dx i ),i=l,...,n. The length of the tangent vector h € TXM is defined by: |h|=(h,h) 1/2 . g x A tangent vector is said to be unitary if |h| = 1 (we say that h is a unit vector). For all x € M, the metric g assigns to each tangent vector h € TXM the covector a)h = Gx(h) € TxMby: (7)
(8)
coh(k)3SGx(hXk)«gx(h,k),
Vh,k € TXM,
and Gx is an isomorphism from TXM onto T*XM. The cotangent space T* M is therefore also equipped with a positive definite bilinear form (thus an inner product, corresponding to a tensorfieldg* on M): -1
(9)
-1
*
(cop
V»i, o)2 €TXM.
Thus using notations (8): (10)
K ,
V h, k€TxM
*x
(i.e., G x carries the metric of the tangent space onto the cotangent space), or in a local coordinate system: (11)
g* = I glj di®8j, (glj) inverse matrix of (g..).
r This can be extended more generally to exterior algebras: on the space ATXM of r tangent r-vectors at x, and on the space AT*XM of r-covectors at x, we define, for all hj, kj in TXM, and all coj, 3 j in TXM (hi A ... Ahr, kx A ... AkJ ^deta^hj) ), *x *x («! A ... A(or, S j A ... A 3 r ) adetftcoj, Sj) ); (12) ©X
J
©X
r r♦ Gx defines an isomorphism (again denoted by Gx) from ATXM onto ATXM by:
A.I
(13)
INTRODUCTION
301
G x (h,A...Ah r )1i f G x (h 1 )A...AG x (h r ) = 0 ) h i A . . . A a ) v
We will often denote by (,) x the inner products (6), or (9), or (12). For r = n, we can define an odd n-form v = v* which is positive (for the chosen orientation of the orientable manifold M): (14)
^ K o j A "-^n-l^A.I.Aa^r
±
"l A '"'
Aa)
nWetg*Ka)j)r1/2,
for a family C0ly...9(on of n independent 1-forms, where lo^ A ... Aa>n| is afunction ofx defined according to (12): Iwj A ... Ao)nlx=(a>1 A ... Aa>n, oj A ...Aw n ) x . This definition is independent of the chosen family. Thus (14) defines a Lebesgue measure on M. In a coordinate system x 1 ,.. .,x n , v is given by: (14)' v = v g = ± dx1 A ... Adx^det(g*(dx\dx*))r 1/2 = ± dx1 A ... Adx n (detg..) 1/2 . We often denote by dM the measure defined by Vs (and also by g the determinant of the matrix (g..)). But the notation dM may be misleading, since in general it is not the exterior derivative of any form. We then define a scalar product on the space of (smooth) r-forms (with compact support if M is only locally compact) by: (15)
(o),n) = (o),n)g d = f / M ^(G x n x ) dM,
with G x defined by (8) and (13). We also have, from the symmetry of g: -l
(16)
-l
a>(Gn) = n(Ga>),
(o>,n) = (n,
In a coordinate system, with r-forms a>, n, o) = I(o. dx1, I
l
n = 2 n v C^K» K
K
we have: (17) o>x (Gxtix) = IgIK(x)(uI(x)nK(x), with: ghh
,K gJK = det
...
g 'A|
dM = (det(g..))1/2dx1...dx11.
^ g ^ The scalar product (15) is independent of the system of local charts. Thus we can define a Hilbert space L r(M) as the closure of the space of smooth differential r-forms with compact support in M. It is also possible to define it as the space of r-forms with square integrable coefficients in every coordinate system. The elements of the space
302
(18)
A- DIFFERENTIAL GEOMETRY
LJ(M)»(
"generalizedr-formsM, (G>,G>) <+oo}
are called square integrable r-forms. We often do not specify whether o> is an even or odd r-form. But if necessary, we shall use the notations L2r e(M) and L2„ AM). Note that from the definition (14), we have (19) (v*,v«) x =l, Vx€M, and thus (19)' (v*,^) - J ^ - I M I (finite or not). For all n-forms © on M, we have (20)
a) = (co x ,v*) x v*,
/
M
o> = (a>, v*) g .
Then we define by duality to the exterior product the inner (or interior) product «-» of a p-covector u by an r-vector k, r s p, as the (p - r)-covector iku ■ k ^ u such that: (21)
k*«i u s G ^ k ' ^ u .
We then define the Hodge transform (or * transform) of an r-form n by: (24)
(•n)x=nx^v* = ( G - \ ) ^ v x \
Therefore the Hodge transform of an even (resp. odd) r-form is an odd (resp. even) (n - r)-form, with: (25)
(nAa,v*)f«(af.n)g,
V a € Lj_r(M), n € L*(M).
Note that the Hodge transformation has the following properties: (26)
.v*«l,
.l^v*.
One of the main properties of the Hodge transformation is: (27)
w A *n = (
303
A.l INTRODUCTION giving for all <■>, n in L2r(M), even or odd: (28)
/ M M A •n»(»,n).
This is expressed in a coordinate system, with (29) by: (30)
» 2 **i d*1' n = 2 n K d x K ,