A boundary condition with memory in electromagnetism

A Boundary Condition with Memoryin Electromagnetism

MAUROAURO FABRIZIOABRIZIO & ANGELONGELO MORROORRO

Communicated by H. TIERSTENIERSTEN

1. Introduction

Let X be a region, occupied by an electromagnetic solid, in the three-dimensional Euclidean point space E. The position vector in X is denoted byx and n denotes the unit outward normal to @X. Physically, the boundary @Xis regarded as a conductor. For time-harmonic fields the conductor ismodelled by letting the amplitudes ~Es;

~Hs of the tangential electric field Es

and magnetic field Hs be related by

~Es�x;x� � k�x;x� ~Hs�x;x� � n�x�; x 2 @X;�1:1�

where k is an appropriate, possibly complex, scalar and x is the angularfrequency [1–4]. Owing to the dependence of k on x, as it stands, (1.1) cannotbe taken over to arbitrary time-dependent fields Es�x; t�;Hs�x; t� [5–7]. Norcan we write (1.1) by letting k be independent of x because we could notexplain the experimental results about the dependence of k on x (cf. [1, § 59];[3, § 6.7]).

It is the purpose of this paper to show that the appropriate generalizationof (1.1) to time-dependent fields at dissipative boundaries is given by ahereditary model of the form

Es�x; t� � g0�x�Hs�x; t� � n�x� �Z

1

0

g�x; s�Hst�x; s� � n�x� ds; x 2 @X;

�1:2�

where H ts�x; s� :� Hs�x; t ÿ s�; s 2 R�

; is the history of the tangential mag-netic field. The modelling equation (1.2) reduces to (1.1) if time-harmonicfields are considered.

We show that a general dissipative boundary condition results in thevalidity of the inequality

Arch. Rational Mech. Anal. 136 (1996) 359–381. Springer-Verlag 1996

Z

d

0

Es�x; t� � Hs�x; t� � n�x� dt > 0; x 2 @X;�1:3�

for every non-trivial cycle on �0;d�; namely for every non-constant time-dependent collection of state functions whose initial value (at t = 0) and finalvalue (at t = d) coincide. Incidentally, PHILLIPSHILLIPS [8] and WILCOXILCOX [9] consideredthe dissipativity condition Es�x; t� � Hs�x; t� � n�x� > 0 for each time t 2 R;

which is stronger than (1.3). We prove that the dissipativity (1.3) of theboundary results in

g0�x� �Z

1

0

g�x; s� cos xs t ds > 0 8x 2 R�

:�1:4�

We also show that, as a consequence of (1.4), there exists a boundary free-energy functional W such that

ddt

W�Ht�x��2Es�x; t� � Hs�x; t� � n�x�;

where Ht is the integrated history of the tangential magnetic field.We investigate the initial-boundary-value problem for Maxwell’s equa-

tions in materials with linear law along with the boundary inequality (1.2).We prove existence and uniqueness of the solution in X� �0; T�; T < 1;

under the assumption that g0 > 0. Next, the inequality (1.4) is shown toimply existence and uniqueness of a weak solution to the same problem inX� �0;1�: Moreover, if the sources satisfy a decay condition as t !1, thenthe solution proves to be asymptotically stable even if the domain X is oc-cupied by a non-dissipative material. Finally we show through a counter-example that if the thermodynamic requirement (1.4) is weakened (3 insteadof >), then asymptotic stability no longer holds.

2. Boundary condition

Throughout, R� and R�� are the sets of non-negative and positive realnumbers while C�� is the set of complex numbers with positive real part. LetQ be the Cartesian product X�R� of the space and time domain. Wedenote by r the derivative with respect to x 2 X and by a superposed dot thederivative with respect to t 2 R�. The electric field E, the magnetic field H ,the electric displacement D, the magnetic induction B and the (induced)electric current density J are C1 functions on Q. They satisfy the local formsof Maxwell’s equations

r� E � ÿ

_B; r � B � 0;�2:1�

r � H �

_D � J; r � D � q:�2:2�

M. FABRIZIOABRIZIO & A. MORROORRO360

Let S be a material boundary of X: Denote by ÿ and + the interior andexterior of S and let n be the unit normal to S from the ÿ side to the + side.Also denote by IfJ the jump of any function f on Q across S; it is thedifference between the limit values of f from the + and ÿ sides at corre-sponding points. Application of the global formulation of Maxwell’s equa-tions to S yields the jump conditions

IDJ � n � R;�2:3�

IEJ� n � 0;�2:4�

where R is the surface charge density at S. According to (2.4), the tangentialcomponent of E is continuous across S. By (2.3), the normal component ofD is continuous if R � 0:

As a consequence of Maxwell’s equations, for any region P 2 E; theenergy balance (Poynting’s theorem) holds in the form

ÿ

Z

@P

E � H � m da �Z

P

�E� _D � H �

_B � E � J � dv�2:5�

where m is the unit outward normal to @P: We restrict attention to isothermalprocesses and hence express the second law of thermodynamics (cf. [10, 11])by saying that, for every cycle on �0;d� and every region P � X,

Z

d

0

Z

P

�E �

_D � H �

_B � E � J �dv dt30:

Hence, by (2.5),Z

d

0

Z

@P

E � H � m da dt20:�2:6�

For any element of area A on @X; identify P with the part of S cut bythe lines through @A and parallel to n. Of course (2.6) holds. Now let Pshrink to the surface A and let E;H remain bounded in the limit process. Thecontribution of the lateral surface approaches zero and hence

Z

@P

�E � H� � n da !

Z

A

IE � HJ � n da:

The same conclusion follows if P does not shrink to A but E � H � m � 0 atthe lateral surface and the normal to the inner and outer parts of P is thesame at corresponding points.

The limit of (2.6) and the observation that �E � H� � n � �Es � Hs� � nyield

Z

d

0

Z

A

IEs � HsJ � n da dt20:

A Boundary Condition with Memory in Electromagnetism 361

By the arbitrariness of A and the continuity of the integrand on @X itfollows that

Z

d

0

IE � HJ � n dt20�2:7�

at any point of @X. In fact, the occurrence of n makes E, H contributethrough the tangential parts Es;Hs only. Hence we write (2.7) as

Z

d

0

IEs � HsJ � n dt20:�2:8�

Across layers of vanishing thickness, Es is continuous, that is IEsJ = 0,and (2.8) can be written as

Z

d

0

Es � IHsJ � n dt20:�2:9�

If the external (+ side) field Es vanishes, then (2.8) becomes

Z

d

0

Es � Hs � n dt30;�2:10�

Es;Hs being the limit values at @X from inside X.

3. Complex-valued fields

In many contexts, especially for time-harmonic dependence, it is cus-tomary to regard the fields as complex-valued and to understand that it is thereal part that has a physical significance. Accordingly we express the secondlaw as

Z

d

0

�Re E � Re _D � Re H � Re _B � Re E � Re J �dt > 0:�3:1�

Consider time-harmonic fields in the form q�x; t� � ~q�x� exp�ÿixt�;D�x; t� � ~D�x� exp�ÿixt� and so on. Hence the Maxwell equations read

r � D � q; r� H � ixD � J;

r � B � 0; r� E ÿ ixB � 0:�3:2�

Upon integration over a time period, the second law (3.1) yields

E �

_D�

� E�

�

_D � H �

_B�

� H�

�

_B � E � J�

� E�

� J > 0

where � means complex conjugate. Upon substitution and use of the Max-well equations (3.2) we obtain


r � �E � H�

� E�

� H� � ÿ

12�ix�E � D�

ÿ E�

� D � H � B�

ÿ H�

� B�

� E � J� � E�

� J� < 0:�3:3�

By arguing as in the previous section we find

IEs � H�

s � E�

s � HsJ � n < 0�3:4�

at any point of @X. Meanwhile, if the external fields vanish, we have

�Es � H�

s � E�

s � Hs� � n > 0:�3:5�

The inequality (3.4), or possibly (3.5), is the boundary condition for time-harmonic fields. Incidentally, (3.3) indicates that

S �

14�E � H�

� E�

� H�

is the time-average of the energy flux vector over a time period.By way of analogy, we remark that (3.5) is satisfied also by wave solu-

tions. Consider a conductor modelled by the constitutive equations

D � �E; B � lH ; J � rE;�3:6�

where �; l; r are positive parameters. Assume that q � 0. Application of thecurl operator to (3.2) and use of the assumptions (3.6) yield

DH � x2l�� ir=x�H � 0;�3:7�

and the same equation for E. Consider now a plane wave such that thedependence on x is through

exp�ikn � x�; Re k > 0:

Substitution in (3.7) gives

k2� x2l�� ir=x�

and hence

k � x��

lp

��

�� ir=xp

; Re��

�� ir=xp

> 0:

Also, by (3.2),

ikn � H � ÿix�� ir=x�E

whence

Es � kHs � n�3:8�

where k ��

l=�� ir=x�p

; Re k > 0: Hence, because

�n � Es� � H�

s � �n � E�

s� � Hs � 2�Rek�H2s ;

the inequality (3.5) holds. This means that a conductor satisfies the samecondition as a dissipative boundary does. The quantity k is usually called thesurface impedance. We mention that a condition similar to (3.8) has beenused in connection with the determination of �; l; r in terms of k [12, 13].

As a simpler case we let l; �; r be real and independent of x. Still theparameter k occurring in (3.8) depends on x and


limx!1

k�x� ��

l=�p

;�3:9�

in view of [14, 15] we regard (3.9) as Graffi’s condition. If, instead, we neglect� in comparison with r=x, then (cf. [1 § 59]) we have Schelkunoff’s relation[2], namely

k � �1 � i�

��

lx2r

r

:�3:10�

The limit of (3.10) as x !1 is meaningless.In either case, k approaches zero as x ! 0. This limit behavior is con-

sistent with the static conditionk�x; 0� � 0:

4. Material boundaries with memory

The boundary condition (3.8) can be written as

~Es�x;x� � k�x;x� ~Hs�x;x� � n�x�; x 2 @X;�1:1�

where ~Es;~Hs are the vector amplitudes of (tangential) time-harmonic fields.

A similar form of the boundary condition has been taken in the form [5–7]

Es�x; t� � f�x�Hs�x; t� � n�x�; x 2 @X;�4:1�

for arbitrary fields Es�x; t�;Hs�x; t� depending on space and time. The rela-tion (4.1), which is applicable to any time-dependent field, cannot be viewedas the appropriate generalization or counterpart of (1.1). Rather, we can viewk�x� ~Hs�x� as the product of two Fourier transforms, which then is thetransform of a convolution. This suggests that, for materials with memory,we consider a constitutive relation of the form

Es�x; t� � g0�x�Hs�x; t� � n�x� �Z

1

0

g�x; u�H ts�x; u� � n�x� du;

x 2 @X:

�4:2�

It is reasonable to assume that g�x; �� 2 L1�R�

�, which makes (4.2) a modelof material with fading memory. Moreover, we let

Z

1

0

g�x; u� du � ÿg0�x�:�4:3�

Letting g0 � dg�x; u�=du, for later convenience we assume also thatg0 2 L1

�R�

�.If Es and Hs are time-harmonic, then (4.2) reduces to (1.1) with

k�x;x� � g0�x� �Z

1

0

g�x; s� exp�ixs� ds:


Indeed, application of the Riemann-Lebesgue lemma yields

limx!1

k�x;x� � g0�x�:�4:4�

The limit (4.4) is the analogue of Graffi’s condition (3.9). The assumption(4.3) yields

k�x;x� ! 0 as x ! 0:�4:5�

Schelkunoff’s relation (3.10) corresponds to a memory functional of theform

Hs�x; t� � n�x� �Z

1

0

d�x; s�Ets�x; s� ds;

with d�x; �� 2j L2�R�

� but d�x; �� 2 L1�R�

�: For,

kÿ1�x;x� �

Z

1

0

d�x; s�exp�ixs� ds;

and hence the condition (4.5) still holds.

We return to real-valued fields and regard (2.10) as the expression of thedissipativity of the boundary. For generality we assume that the behavior ofthe boundary is represented by (4.2). Then we regard the boundary as a two-dimensional continuum whose state is the history of the magnetic fieldH t�x; �� up to the present time t. A process _H�x; �� on �t; t � d� determines

the state evolution from the initial state H t�x; �� to the final state H t�d

�x; ��: IfH t

� H t�d; then the one-parameter family of states H t�n

; n 2 �0;d�; is acycle. Based on the result (2.10) we state

Definition 1. A material boundary @X is said to be locally dissipative if, forevery x 2 @X; and every non-trivial cycle in �t; t � d�; the inequality

Z

d

0

�Es�x; t � n� � Hs�x; t � n�� n�x� dn > 0�4:6�

holds.

If the (tangential) magnetic field Hs depends on time in the form

Hs�x; t � n� � H0�x� cos x�t � n�; x 2 R��

;�4:7�

then H t�ns �x; ��; as n 2 �0; 2p=x�; is a cycle. Accordingly we examine whether

the constitutive relation (4.2) meets the condition of local dissipativity.By analogy with [16] and [17], we can show the following

Theorem 1. The constitutive relation (4.2) is locally dissipative if and only if

Re k > 0; x 2 R��

:�4:8�

Proof. To show that (4.8) is necessary, consider the time dependence (4.7),with d � 2p=x; and substitute in (4.6) to obtain


0 <

Z

2p=x

0

�

Hs�x; t � n� � n�x��

� Es�x; t � n�dn

� �H0�x� � n�x��2Z

2p=x

0

�

g0�x� cos2 x�t � n�

�

Z

1

0

g�x; s� cos x�t � n� cos x�t � nÿ s�ds�

dn:

Integration first on n and then on s yields

0 <px�H0�x� � n�x��2 g0 �

Z

1

0

g�x; s� cos xs � ds

2

4

3

5

and hence the necessity is proved.To show that (4.7) is also sufficient we consider a cycle in �t; t � d�; viz.,

H t� H t�d

: Hence H�t ÿ s� � H�t � d ÿ s�; for all s 2 R� and this impliesthat, for every n 2 �0;d�;H t�n

�s� is periodic with period d. Letting x � 2p=dwe represent H�t � n� and H t�n

�s� through the Fourier seires

H�t � n� �X

1

k�0

Ak cos kx�t � n� � Bk sin kx�t � n�;

H t�n�s� �

X

1

k�0

�Ak cos kx�t � nÿ s� � Bk sin kx�t � nÿ s��:

Substitution in (4.6) and integration with respect to n yields

Z

2p=x

0

�Hs�x; t � n� � n�x�� Es�x; t � n�dn

�

px

X

1

k�1

�A2k � B2

k� g0�x� �

Z

1

0

g�x; s� cos kxs ds

2

4

3

5;

which completes the proof. u

Application of the Riemann-Lebesgue lemma to (4.8) yields

g030:�4:9�

5. Boundary free energy

Also with a view to investigations of stability properties of the solution to(2.1), (2.2) along with suitable initial and boundary conditions, it is worth


looking for a functional that with each history H ts associates a function of t,

say w�t�; such that

_w�t�2Es�t� � Hs�t� � n:�5:1�

The desired functional formally plays a role similar to that of the free energyand hence we call it boundary free energy (cf. [16]). As we shall see in amoment, more than one functional exists with the required property.

We restrict attention to Hs�� 2 L1�R�

� and denote by Ht the history

Ht�u� �

Z

t

tÿu

Hs�n�dn:

The function Ht�u� is differentiable with respect to t and u;

dHt�u�

dt� Hs�t� ÿ Hs�t ÿ u�;

dHt�u�

dt� Hs�t ÿ u�:

Let g 2 L1�R�

� \ H1�R�

�: Since g�1� � 0 and Ht�0� � 0; upon an in-

tegration by parts we can write (4.2) as

Es�t� � g0Hs�t� � n ÿZ

1

0

g0�u�Ht�u� � n du:�5:2�

By use of the Parseval-Plancherel theorem we can write (5.2) as

Es�t� � g0Hs�t� � n ÿ2p

Z

1

0

g0s�x�Hts�x� � n dx�5:3�

where the subscript s denotes the half-range Fourier sine transform.In view of (5.1) we look for a positive-definite functional W such that

dW�Ht�

dt2Es�t� � Hs�t� � n�5:4�

and claim that, under suitable requirements on g; (5.4) holds for

W1�Ht� �

12

Z

1

0

Z

1

0

g12�ju1 ÿ u2j��Ht�u1� � n� � �Ht

�u2� � n�du1 du2;

W2�Ht� � ÿ

12

Z

1

0

g0�u��Ht�u� � n�2 du;

where g12�ju1 ÿ u2j� � @2g�ju1 ÿ u2j�=@u1@u2:

First examine the positive-definiteness. It is apparent that the negativityof g0�u� for every u 2 R� is a sufficient condition for W2 to be positive-definite. In regard to W1 observe that


g12�ju1 ÿ u2j� � ÿ2d�u1 ÿ u2�g0

�ju1 ÿ u2j� ÿ g00�ju1 ÿ u2j�:

Hence we obtain

W1�Ht� � ÿ2

Z

1

0

Ht�u1� �

Z

u1

0

�g00�u1 ÿ u2�

� g0�0� d�u1 ÿ u2��Ht�u2�du2 du1:

Application of the Parseval-Plancherel theorem yields

�5:5� W1�Ht� � ÿ

1p

Z

1

ÿ1

�g00c �x� � g0�0��

�Htc�x� � n�2 � �Ht

s�x� � n�2�

dx:

Also, an integration by parts gives

g00c �x� � g0�0� � xg0s�x�:�5:6�

Hence the requirement that g0s�x� < 0 for all x 2 R�� makes W1 positive-definite.

We now examine the validity of (5.4). Observe that, in the distributionalsense,

dHtc�x�

dt� ÿxHt

s�x�;dHt

s�x�dt

�

1x

Hs�t� � xHtc�x�:

Accordingly, by means of (5.5) and (5.6), a direct evaluation yields

dW1�Ht�

dt� ÿ

2p

Z

1

ÿ1

g0s�x�Hts�x� � n dx

2

4

3

5

� �Hs�t� � n�:

Substitution for the integral from (5.3) gives

dW1�Ht�

dt� Es�t� � Hs�t� � n ÿ g0�Hs�t� � n�2:

Because of (4.9), the desired property of W1 follows. Finally, consider W2 toobtain

dW2�Ht�

dt� ÿ

Z

1

0

g0�u��Ht�u� � n� � Hs�t� ÿ

dHt�u�

du

� �

� n� �

du:

Comparison with (5.2) and an integration by parts yield

dW2�Ht�

dt� Es�t� � Hs�t� � n ÿ g0�Hs�t� � n�2 ÿ

12

Z

1

0g00�u��H t

s�u� � n�2du:

Since g0 > 0; the condition that g00�u� > 0 for all u 2 R�; is sufficient for W2

to satisfy (5.4).In terms of the free energy W we can write an estimate which is essential

in the application of energy methods. Integrate the inequality (5.4) over theboundary @X of the body, and apply (2.5) to obtain


ddt

Z

@XWda �

Z

X�E �

_D � H �

_B � E � J� dv20:

If, further, D�x; t� � ��x�E�x; t�;B�x; t� � l�x�H�x; t�; and J � 0, then

ddt

12

Z

X

��E2� lH2

�dv �

Z

@X

W da

2

4

3

520:

Accordingly, the expression in the square bracket may be viewed as the‘‘energy’’ of the system.

6. Existence and uniqueness

In this section we investigate a problem connected with the electro-magnetic field in a dielectric occupying a smooth region X � E. For formalsimplicity we let the free charge density q be zero.

We consider the Maxwell equations (2.1), (2.2) and let the constitutiveequations be given by

D�x; t� � ��x�E�x; t�; B�x; t� � l�x�H�x; t�;�6:1�

where � and l are positive-valued. The current density J � Jf is regarded as aknown function of x and t. The boundary condition is taken in the form (4.2)subject to the thermodynamic condition (4.8), namely,

Re k�x;x� � g0�x� �

Z

1

0

g�x; s� cos xs ds > 0; x 2 R��

:�6:2�

The initial-boundary conditions are

E�x; 0� � E0�x�; H�x; 0� � H0�x�; x 2 X;

Hs�x; t� � 0; t < 0; x 2 @X:

Compatibility of the boundary and initial conditions is assumed. For brevity,this set of equations and conditions is referred to as problem P.

The appropriate formulation of the problem requires the introduction ofsome function spaces. For any f : R�

! Rn; the Laplace transform

^f �L�f�; if it exists, is defined by

^f �p� �Z

1

0

exp�ÿpt�f �t�dt; p 2 C��

:

Hence we let V be the translation space of E and consider the spaces

B�X� �

�

e 2 L2�X; V �;

Z

Xe � r/dv � 0 8/ 2 C1

0 �X;R�

�

;

R�X� �

�

e; h 2 B�X�;r� e 2 B�X� r � h 2 B�X��

;


R0�X� �

�

e; h 2 R�X�; es � g0hs � n�

;

Hl�Q� ��

e; h 2 L2loc�R

�;R�X�� \ H1loc�R

�; L2�X; V ��;

es�t� � g0hs�t� � n �Z t

0g�s�ht

s�s� ds � n�

;

H�Q� ��

e; h 2H1�Q�; e; h 2 L2�R�; H1

�X; V �� \ H1�R�; L2

�X; V��

�

;

H�

�Q� ��

e; h 2 L2�R�

; H1�X; V�� \ H1

�R�

; L2�X; V��;

es�t� � g0hs�t� � n �

Z t

0g�s�ht

s�s� ds � n�

;

where the subscript loc signifies that the functions satisfy the property�L2 or H1

� locally and that their Laplace transform exists.

Definition 2. A pair E;H 2Hl�Q� is called a strong solution of the problem Pwith initial data E0;H0 2 R0�X� and given current densityJf 2 H1

loc�R�

; L2�X; V �� if

�@E@t

� r � H ÿ Jf ; l@H@t

� ÿr� E�6:3�

hold almost everywhere and

limt!0�

kE�x;t� ÿ E0�x�kL2 � 0; limt!0�

kH�x;t� ÿ H0�x�kL2 � 0:

Apply the Laplace transform to (6.3) to obtain

��pE�x;p� ÿ E0�x�� r � H�x;p� ÿ Jf �x;p�;�6:4�

l�p H�x;p� ÿ H0�x�� ÿr� E�x;p�:�6:5�

Application of the Laplace transform to the boundary condition (4.2) gives

Es�x;p� � ~k�x;p� Hs�x;p� � n�x�; x 2 @X;�6:6�

where ~k�x; p� � g0�x� � ^g�x; p�:The symbols B0

�X�; R0

�X�; denote the set of complex-valued functionswhose real and imaginary parts belong to B�X�;R�X�: Also, denote by^H�Q�; ^Hl�Q� the sets of functions �^e; ^h� which are the Laplace transforms of

functions in H�Q�;Hl�Q�: Quite naturally we denote by ^f�ix� the restric-tions to the imaginary axis of the Laplace transform ^f of a function f. Such arestriction coincides with the Fourier transform of the causal function

f c�t� �

0; t < 0;f �t�; t30:

�


Further, denote by HF�Q� the set of functions that are the Fourier trans-forms of the functions in H�Q�: Finally, for any fixed p 2 C��

; consider thespace

R0

p�X� � fE; H 2 R0

�X�; Es �~k�p� Hs � ng:

For any two elements u1; u2 2 R0

p�X� we let

hu1; u2i �

Z

X

�

E1 � E�

2 �H1 �H�

2 �1lr� E1 � r �E

�

2

�

1�r �H1 � r �H

�

2

�

dv

� ReZ

@X

~k�p��H1s � n� � �H�

2s � n� da:

This hu1; u2i is an inner product that makes R0

p�X� a Hilbert space.

Remark. Let �e; h� 2 R0

p�X�: Upon taking the inner product of the conjugateof (6.4) with r� h and of (6.5) with r� e� we obtain

Z

X

1lp

�

r� E � r � e� �1�p

r� H � r � h�

ÿ E � r � h� � H � r � e��

dv

�

Z

X

�

1�p

�Jf ÿ �E0� � r � h� �1p

H0 � r � e��

dv:

Hence, from (6.6) it follows thatZ

X

1lp

r� E � r � e� �1�p

r�H�

� r � h� �

dv�6:7�

�

Z

@X

~k�H � n� � �h� � n� da

�

Z

X

1�p�

�Jf ÿ �E0��

� r � h �1p

H0 � r � e��

dv

for every solution E; H to the problem (6.4)–(6.6).

Theorem 2. Let Jf 2 L2�X;C3

� and E0;H0 2 R0�X�: For every p 2 C�� thereis one and only one solution E; H� 2 R0

p to the problem (6.4)–(6.6). Further-more, the solution satisfies the inequality


Z

X

�jE�x;p�j2 � jH�x;p�j2 � jr � E�x;p�j2 � jr � H�x;p�j2�dv

0

@

�6:8�

�

Z

@X

�Re ~k�x;p��jHs�x;p�j2� da

1

A

1=2

2cpfk~J�p� kL2 � k

~K kL2g

where ~J � Jf ÿ �E0; ~K � lH0 and cp is a positive constant.

Proof. Let u stand for the pair E; H : Consider the bilinear formB : R0

p�X� �R0

p�X� ! R defined by

B�u1; u2� �

Z

X

�

1lp

r� E1 � r � E�

2

�

1�p�

r � H�

1 � r � H2 ÿ E�

1 � r � H2 � H1 � r � E�

2

�

dv:

The bilinear form B is bounded and, moreover, by the generalized Poincar�elemma is coercive in that

Re B�u; u� �ReZ

X

�

1lp

jr � E�x;p�j2 �1�p�

jr � H�x;p�j2�

dv

�

Z

@X

�Re ~k�x;p��jHs�x;p�j2da3ckuk2R0

p

where c is a positive constant. Letting w � �e; h� we can write (6.7) as

B�u;w� � L�w�

where L�w� is the functional

L�w� �Z

X

~J1�p�

� r � h �1

lp~K � r � e�

� �

dv

which is linear and bounded on R0

p�X�: Hence, by the Lax-Milgram theorem,for every p 2 C� and for given ~J �

~Jf ÿ �E0 2 B0

�X�; ~K � lH0 2 B0

�X�;there is a unique solution �

^E; ^H� 2 R0

p: Now we establish the estimate (6.8).Let

I�E; H� �

Z

X

�

~J�

�x;p� �1�p�

r � H�x;p� � ~K�x� �1

lpr�E

�

�x;p��

dv:

By the equations (6.4),(6.5) we have


I�E; H� �

Z

X

n 1�r� H�x;p� �

h 1p�

r � H�

�x;p� ÿ � E�

�x;p�i

�

1lr� E�

�x;p� �h 1

pr� E�x;p� � lH

�

�x;p�io

dv

�

Z

X

n 1lp

jr � E�x;p�j2 �1�p�

jr �H�x;p�j2o

dv

�

Z

@X

~k��x;p�jHs�x;p�j2da:

Let minx2X

f�; lg � m > 0. A simple application of the the Cauchy-Schwarz

inequality gives

jI�E;H�j �

1jpj

k

~J�p�kL2

h

mÿ1Z

X

jr � H�x;p�j2dvi1=2

�

1jpj

k

~KkL2

h

mÿ1Z

X

jr � E�x;p�j2dvi1=2

:

Let M � maxx2X

f�; lg and set ap = [Re�1=p��=M . Comparison yields

ap

h

Z

X

�jr � Ej2 � jr � H j

2�dv �

1ap

Z

@X

�Re~k�jHs2da

i

� np

h

�

Z

X

jr � Ej2dv�1=2� �

Z

X

jr � H j

2dv�1=2� �

1ap

Z

@X

�Re~k�jHsj2da�1=2

i

where np � f1=�mjpj�g�k ~J�p�kL2 � k

~KkL2 �: Repeated use of the Cauchy-Schwarz inequality, some rearrangement, and arguing as in [18] lead to thedesired estimate (6.8) where cp is a constant which depends also on thedomain X: u

Theorem 2 provides the existence of the solution to the given problem inthe Laplace transform domain. The next theorem guarantees the existence ofthe solution in the time domain.

Theorem 3. The solution to the problem P, in the sense of Definition 2, existsand is unique.

Proof. By virtue of Theorem 2, we only need to prove that the inverse La-place transform can be performed. To this end, consider (6.8) and letp 2 C��

: Because the inverse Laplace transform of the right-hand side exists,so does that of the left-hand side. Furthermore if f 2H1, the Laplacetransform ^f at p � p1 � ip2;p1 > 0, reads


^f �p1 � ip2� �

Z

1

0

�f �t� exp�ÿp1t�� exp�ÿip2t� dt

and hence ^f�p1 � ip2� is the Fourier transform of the causal functionf�t� exp�ÿp1t�. Application of the Parseval-Plancherel theorem to f; g 2Hl

yieldsZ

1

ÿ1

^f �p1 � ip2�^g�

�p1 � ip2� dp2 � 2pZ

1

0

f �t�g�t� exp�ÿ2p1t� dt:

By (6.8), this implies that the inverse Laplace transforms of E�x;p�; H�x;p�and r� E�x;p�;r� H�x;p� exists and

�E�x; t�; lH�x; t�; r� E�x; t�; r� H�x; t� 2 L2loc

ÿ

R�

;B�X��

:

Furthermore, if e; h 2H1�Q�, so that ê;h 2 ^Hl�Q�, by (6.4) and (6.5) we haveZ

X

��pE ÿ E0� � e dv �Z

X

��r � H� � e ÿ Jf � e� dv;

Z

X

l�pH ÿ H0� �h dv � ÿ

Z

X

r� E �h dv:

The inverse Laplace transform of the right-hand side exists for all�ê; ^h� 2 R0

p�X�. Hence, by the arbitrariness of �ê; lh, the functionspE ÿ E0; pH ÿ H0 are elements of L2

loc�R�

; L2�X;V��. Accordingly, their

inverse Laplace transforms E; _H exist and belong to L2loc�R

�

; L2�X;V��. This

means that there exists a pair �E;H� 2Hl�Q� that is a solution of the prob-lem P in the sense of Definition 2. u

7. Asymptotic stability

In this section we show that if Jf 2 H1�R�

; L2�X�� \ L1

�R�; L2�X��, then

the pair E, H eventually approaches zero. To this end we consider the in-equality (6.2) to get the following result.

Theorem 4. If Jf 2 H1�R�

; L2�X�� \ L1

�R�

; L2�X�� and E0;H0 2 R0�X�, then

there exists a unique strong solution E;H 2H�Q� in the sense of Definition 2.

To prove this theorem we first consider the system (6.4) along with theboundary condition (4.2). Application of the causal Fourier transform yields

��ixE�ix� ÿ E0� � r � H�ix� ÿJf �ix�;

l�ixH�ix� ÿ H0� � ÿr � E�ix� in X;

�7:1�

Es�ix� � �g0 � ^g�ix��Hs�ix� � n on @X:�7:2�


Hence we proceed through the following lemmas.

Lemma 1. Let �E0;H0� 2 R0�X� and, for every x 2 R; let Jf�ix� 2 B0

�X�. Asa consequence of (6.2), there is a unique solution E�ix�; H�ix� 2 R0

0�X� to theproblem (7.1), (7.2), which depends continuously on the parameter x.

Proof. Consider two solutions E1; H1 and E2; H2 to �7:1�ÿ�7:2�. The differ-ence E1 ÿ E2; H1 ÿ H2 satisfies

ix� E � r� H ; ixl H � ÿr� E

along with (7.2). Take the inner product of the complex conjugateix�^E

�

� ÿr� H�

with E and of ixlH � ÿr� E withH�

. Comparison andthe identity

r � �E � H�

� � H�

� r � E ÿ E � r � H�

yieldZ

@X

E �H�

� n da � ixZ

X

��E � E�

ÿ l H � H�

� dv:

Hence (7.2) givesZ

@X

�g0 � ^g�ix��jH�ix� � nj2da � ixZ

X

��jEj2 ÿ ljH j

2� dv:

The real part results inZ

@X

�g0 � gc�x��jH�ix� � nj2da � 0:

Because of (6.2) it follows that Hs � n � 0 a.e. on @X, whence Hs � 0 and,by (7.2), ^Es � 0 a.e. at @X. Hence, by arguing as in [19], the uniqueness of thesolution to (7.1), (7.2) follows.

Uniqueness means that no eigensolutions exist for the problem (7.1),(7.2). Hence, in view of the ellipticity of the system and Theorem 3.12 of [20],existence of the solution follows. The continuity of the solution ( E, H) withrespect to x follows from Lemma 44.1 of [21].

Lemma 2. Every strong solution �E;H� 2H�Q� of the problem P withJf 2 H1

�R�

; L2�X�� \ L1

�R�; L2�X�� and E0;H0 2 R0�X� satisfies the con-

ditionZ

1

0

Z

X

�E � r � h ÿ lH �

_h � H � r � e � �E � _e ÿ Jf � e� dv dt

�

Z

X

��E0 � e�0� ÿ lH0 � h�0��dv � 0�7:3�

for every �e; h� 2H�Q�. Moreover, every �E;H� 2H�

�Q� that satisfies (7.3)is a strong solution to (6.4).


Proof. Take the inner product of the first equation in (6.4) with e and of thesecond with h. Moreover integration over Q by parts yields (7.3). Also by(7.3), the inverse integration by parts and the arbitrariness of e, h shows thatE, H 2H�X� is a strong solution to (6.4). u

Incidentally, by use of the Parseval-Plancherel theorem, (7.3) isequivalent to

Z

1

ÿ1

Z

X

� E�

�ix� � r � h�ix� ÿ l H�

�ix� �ÿ

ix h�ix� ÿ h�0��

� H�

�ix� � r �e �ix� � � E�

�ix� �ÿ

ixe�ix� ÿ e�0��

ÿ J�

f �ix� �e�ix�� dv dx

� 2pZ

X

ÿ

�E0 � e�0� ÿ lH0 � h�0��

dv � 0

�7:4�

where �E�ix�; H�ix�� 2HF�Q� are the Fourier transforms of the causalfunctions E;H , and e�ix�; h�ix� are arbitrary elements of H�

F�Q�:Existence and uniqueness for the problem (7.1), (7.2) imply the existence

of the solution (tensor Green’s functions) P1�x; x1;x�;P2�x; x0;x� to thesystem

r�P2�x; x0;x� � ix�P1�x; x0;x� � d�x ÿ x0�1;�7:5�

r �P1�x; x0;x� ÿ ixlP2�x; x0;x� � 0�7:6�;

P1�x; x0;x� � k�x; ix�P2�x; x0;x� � n�x0� on @X:�7:7�:

Comparison of equations (7.1), (7.2) with (7.5)–(7.7) yields

E�x; ix� �Z

X

flH0�x0�P2�x; x0;x�

ÿ �Jf �x0;ix� ÿ �E0�x0��P1�x; x0;x�g dv0:

�7:8�

The analogous relation for the magnetic field H is obtained by con-sidering the solution N1;N2 to the system of equations

r� N2�x; x0;x� � ix�N1�x; x0;x� � 0;

r� N1�x; x0;x� ÿ ixlN2�x; x0;x� � d�x ÿ x0�1;

N1�x; x0;x� � k�x;ix�N2�x; x0;x� � n�x0� on @X:

Hence we obtain

H�x;x� �Z

X

flH0�x0�N1�x; x0;x�

ÿ � Jf �x0;x� ÿ �E0�x0��N2�x; x0;x�g dv0:

�7:9�


By the Riemann-Lebesgue lemma, Jf approaches zero as x !1. Hencewe have the following asymptotic property of P1;P2 and N1;N2.

Lemma 3. Green’s functions P1;P2 satisfy the asymptotic property

limx!�1

ixh

Z

X

Z

X

� f1�x� �P1�x; x0;x�/�x0;x� dv0dv�7:10�

�

Z

X

Z

X

l f2�x� �P2�x; x0;x�/�x0;x� dv0dvi

� limx!�1

Z

X

f1�x� � /�x;x� dv

for all vectors f1; f2 2 R0

0�X� and / 2

^HF�Q�:

Proof. By (7.5)–(7.7) we haveZ

X

Z

X

�/ �P1r� f2 ÿ / �P2r� f1 � ix �/ �P1f1 � ixl/ �P2f2� dv dv0

�

Z

X

f1 � / dv;

whenceZ

X

Z

X

�P1�r � f2 � i x�^f1�� /ÿ�P2�r � f1 ÿ ixl f2�� / dv dv0

�

Z

1

f1 � / dv:

Since f1; f2 are independent of x, taking the limit as x ! �1 gives thedesired property (7.10). u

By the same token we find that

limx!�1

ixh

Z

X

Z

X

� f1�x� � N1�x; x0;x�/�x0;x� dv0dv

�

Z

X

Z

X

l f2�x� � N2�x; x0;x�/�x0;x� dv0dvi

� limx!�1

Z

X

f2�x� � /�x;x� dv

�7:10�

We are now in a position to prove the main result.

Proof of Theorem 4. By taking the inner product of (7.8) with / � ix^e ÿ e0

and using (7.10) we obtain


limx!�1

Z

X

E�x;x� � �ixe�x;x� ÿ e0�x�� dv

� limx!�1

1ix

Z

X

E0�x� � �ix e�x;x� ÿ e0�x�� dv:�7:11�

A strictly analogous result holds for the field H�x;x�; viz.,

limx!�1

Z

X

H�x;x� � �ix h�x;x� ÿ h0�x�� dv

� limx!�1

1ix

Z

X

H0�x� � �ixh�x;x� ÿ h0�x�� dv:�7:12�

The continuity ofE�x;x� and H�x;x� with respect to x and the conditions(7.11), (7.12) imply that

E; H 2 L2ÿR; L2�X�

�

:

Moreover, letting f1 � Jf ÿ �E0 with Jf�x; 0� � 0 we have

limx!�1

Z

X

�ix ^E�x;x� ÿ E0�x�� ix e�x;x� ÿ e0�x�� dv

� limx!�1

ixZ

X

Jf �x;x� � �ix e�x;x� ÿ e0�x�� dv:

�7:13�

The result (7.13) and the analogue for H allow us to say that

�ixE ÿ E0�; �ixH ÿ H0� 2 L2ÿR; L2�X�

�

:

Moreover, (7.11) and (7.12) imply that limx!1

�ixH�x� ÿ H0� � 0;limx!1

�ix E�x� ÿ E0� � 0: Hence, because of the Paley-Wiener theorem,E;H are the Fourier transforms of causal functions. Accordingly,E; H 2HF�Q� and satisfy (7.4). The inverse Fourier transformsE;H 2H�Q� satisfy (7.3). By Lemma 2, E, H is a strong solution of theproblem P. u

8. A counterexample to asymptotic stability

In this section we show that the condition (6.2) is essential to theasymptotic stability property given by Theorem 4. Indeed, we show thatasymptotic stability is contradicted if Re k � 0:

Consider a simple version of (6.4) with E; Jf in the y-direction, H in thez-direction. The components Ey;Hz; Jfy are functions of time t and the co-ordinate x 2 �0; a�. For simplicity, �, l are taken to be constant. Hence wewrite equation (6.4) and the boundary condition (4.2) as


�@Ey

@t� ÿ

@Hz

@xÿ Jfy ; l

@Hz

@t�

@Ey

@x; x 2 �0; a� t 2 R�

;�8:1�

Ey � g0Hz �

Z

1

0

g�s�Htz�s�ds; x � 0; a; t 2 R�

:�8:2�

The initial histories of E and H are supposed to be given, i.e.,

Ey�x; r� � E0y�x; r�; Hz�x; r� � H0

z�x; r�; 8r 2 Rÿ

:�8:3�

In place of (4.8) assume the weaker condition

g0 � gc�x�30 8x 2 R�

:�8:4�

Hence there is at least a frequency �x > 0 such that

g0 � gc� �x� � 0:�8:5�

As an example we provide a function g and a value g0 satisfying (8.4), (8.5),and (4.3). Consider g in the form

g�u� � exp�ÿmu��a � bu � u2�

where m; a; b 2 R; m > 0. We determine a; b by requiring that gc�x� has onlyone root, say �x, and is positive as x 4 �x. We find that this is true if

a �b2m2

� 2bm� 98m2 ; b4

3m

Also we define g0 as ÿ

R

1

0 g�u�du. Such a function g meets the thermo-dynamic requirements.Let Jf � 0 and consider the fields

Ey�x; t� � E�x� exp�i �xt�; Hz�x; t� � H�x� exp�i �xt�:�8:6�

Substitution in (8.1) to (8.3) shows that the functions E�x�;H�x� are requiredto satisfy

i� �xE � ÿ

@H@x

; il �xH � ÿ

@E@x

;�8:7�

E � k� �x�H ; x � 0; a:�8:8�

Differentiation of the second equation in (8.7) and comparison yields

d2Edx2 � ÿ�l �x2E;

E � ik� �x�l �x

dEdx

; x � 0; a:

Since Re k� �x� � 0; the boundary condition becomes

dEdx

� ÿ

l �xgs� �x�

E; x � 0; a:


Solutions are found to be

E � A cos�cx � a�; tan a �1

gs� �x�� x; �l �x2a2

� n2p2;

where c ��

�lp

�x and n is any integer. By the same token we have

d2Hdx2 � ÿ�l �x2H ;

dHdx

� � �xgs� �x�; x � 0; a:

Solutions are found to be

H � B cos�cx � b�; b � a�p2; �l �x2a2

� n2p2:

The same condition on �xa is required in either case. Accordingly, there areinfinitely many values of �xa such that eigensolutions exist. This in turnimplies that infinitely many solutions exist for the problem (8.1) to (8.3).Such solutions are evidently not asymptotically stable and the necessity ofthe strict inequality in (4.3) to obtain asymptotic stability is then proved.

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Pitman, London, 1977; p. 66.[19] MULLERULLER, C., Foundations of the Mathematical Theory of Electromagnetic Waves,

Springer, Berlin, 1969; Th. 60, p. 267.[20] MIZOHATAIZOHATA, S., The Theory of Partial Differential Equations, Cambridge Uni-

versity press, 1973.[21] TREVESREVES, F., Basic Linear Partial Differential Equations, Academic, New York,

1975.

Note added in proof. A boundary condition analogous to (1.2), though for amechanical system, has recently been investigated in

[22] PROPSTROPST G. & J. PRUSSRUSS, On wave equations with boundary dissipation of memorytype, J. Integral Equations Appl. 8, 99–123 (1996).

Department of MathematicsUniversity of Bologna40127 Bologna, Italy

and

DIBE, University of Genoa16145 Genoa, Italy

(Accepted August 15, 1995)


A boundary condition with memory in electromagnetism

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