Nonlinear Partial Differential Equations and Their Applications: College de France Seminar Volume XIV

By Doina Cioranescu and Jaques-Louis Lions

This book contains the written versions of lectures delivered since 1997 in the well-known weekly seminar on Applied Mathematics at the Collège de France in Paris, directed by Jacques-Louis Lions. It is the 14th and last of the series, due to the recent and untimely death of Professor Lions.

The texts in this volume deal mostly with various aspects of the theory of nonlinear partial differential equations. They present both theoretical and applied results in many fields of growing importance such as Calculus of variations and optimal control, optimization, system theory and control, operations research, fluids and continuum mechanics, nonlinear dynamics, meteorology and climate, homogenization and material science, numerical analysis and scientific computations

The book is of interest to everyone from postgraduate, who wishes to follow the most recent progress in these fields.

• Language: English
• Publisher: Elsevier Science
• Release date: Jun 21, 2002
• ISBN: 9780080537672

Book preview

Nonlinear Partial Differential Equations and their Applications

Collège de France Seminar Volume XIV

First edition

Doina Cioranescu

Centre National de la Recherche Scientifique Laboratoire J.L. Lions Université Pierre et Marie Curie Paris, France

Jacques-Louis Lions

Collège de France Paris, France

N·H

2002

ELSEVIER

AMSTERDAM  –  BOSTON  –  LONDON  –  NEW YORK  –  OXFORD  –  PARIS

SAN DIEGO  –  SAN FRANCISCO  –  SINGAPORE  –  SYDNEY  –  TOKYO

Table of Contents

Cover image

Title page

Copyright page

Dedication

Preface

Chapter 1: An Introduction to Critical Points for Intergral Functionals

1 Introduction

2 A mountain pass theorem for nondifferentiable functionals

3 A simple model

4 Main examples

Acknowledgment

Chapter 2: A Semigroup Formulation for Electrmagnetic Waves in Dispersive Dielectric Media

1 Introduction

2 Modeling of dispersiveness in dielectric media

3 A semigroup formulation

4 Concluding remarks

Acknowledgment

Chapter 3: Limite Non Visqueuse Pour Les Fluides Incompressibles Axisymetriques

Résumé

Abstract

Introduction

1 Résultats de convergence pour la régularité stratifiée

2 Un résultat de décroissance exponentielle

3 Preuve du résultat de convergence pour la régularité stratifiée ou conormale

4 Convergence pour des données initiales sans régularité stratifiée

Appendice

Chapter 4: Global Properties of Some Nonlinear Parabolic Equations

1 Introduction

2 Existence of global solutions

3 Further extensions. The case μ = −1

4 Decay as t → + ∞

Acknowledgement

Chapter 5: A model for two coupled turbulent fluids part I: Analysis of the system

1 Introduction

2 Transformation of the system

3 The equations on the velocities

4 The equations on the turbulent kinetic energies (TKE)

5 The global system

6 A uniqueness result for smooth solutions

7 Some further regularity properties in dimension 2

Chapter 6: Determination de Conditions Aux Limites En Mer Ouverte Par Une Methode de Controle Optimal

Résumé

Abstract

1 Introduction

2 Equations du modèle

3 Le modèle adjoint

4 Conditions d’existence et d’unicité du contrôle optimal dans le cas linéaire

5 Principe de la méthode de résolution

6 Mise en œuvre numérique

7 Résultats numériques

Conclusion

Chapter 7: Effective diffusion in vanishing viscosity

1 Introduction

2 The setup

3 Effective diffusion

4 Hypothesis 1: Sufficient conditions

Chapter 8: Vibration of a Thin Plate With a Rough Surface

1 Introduction

1 Statement of the problem

2 Preliminary results

3 Limit problems

Chapter 9: Anisotropy and Dispersion in Rotating Fluids

1 Introduction

2 Convergence of Leray’s solutions

3 Anisotropic Strichartz estimates

4 Global existence and convergence of smooth solutions

Chapter 10: Integral equations and saddle point formulation for scattering problems

1 Introduction

2 Some definitions

3 Derivation of the system

4 Variational formulation and well-posedness. The penalized systems

5 System for general impedance boundary conditions

6 Calderon projectors

Chapter 11: Existence and uniqueness of a strong solution for nonhomogeneous micropolar fluids

1 Introduction

2 Preliminaries

3 A priori estimates

4 The convergence of the sequence

Chapter 12: Homogenization of dirichlet minimum problems with conductor type periodically distributed constraints

0 Introduction

1 Notations and preliminary results

2 Estimates from above

3 Estimates from below

4 Representation results for homogenized functionals

5 The convergence of infima and of minimizing sequences

6 An example

Chapter 13: Transport of trapped particles in a surface potential

1 Introduction

2 A Kinetic Model for the Transport of Trapped Particles in a Surface Potential

3 The Boundary Collision Operator

4 The Macroscopic Limit

5 Examples

6 Extensions

Chapter 14: Diffusive energy balance models in climatology

1 Introduction

2 The transient model

3 On the stationary problem

Chapter 15: Uniqueness and Stability in the Cauchy Problem for Maxwell and Elasticity Systems

1 Preliminaries

2 Pseudoconvexity and Carleman estimates

3 Uniqueness and stability for principally scalar systems

4 Applications to the Maxwell system

5 Applications to the classical elasticity system

Appendix. A diagonalization of the classical elasticity system

Chapter 16: On the Unstable Spectrum of the Euler Equation

Introduction

1 The Euler equations for fluid motion

2 The essential spectral radius

3 Examples of instability in the essential spectrum

4 Examples of instability in the discrete spectrum

5 Nonlinear instability

Chapter 17: Decomposition en Profils Des Solutions de L’equation Des Ondes Semi Lineaire Critique a L’exterieur D’un Obstacle Strictement Convexe

Résumé

Abstract

1 Introduction

2 Etude préliminaire

3 Le problème linéaire: démonstration du théorème 1

4 Le problème non linéaire: démonstration du théorème 2

Chapter 18: Upwind Discretizations of a Steady Grade-Two Fluid Model in Two Dimensions

1 Introduction

2 The exact problem

3 A discontinuous upwind scheme

4 Error estimates

Chapter 19: Stability of Thin Layer Approximation of Electromagnetic Waves Scattering by Linear and Non Linear Coatings

1 Introduction

2 Thin layer approximation: the linear case

3 Thin layer approximation: the non linear case

Chapter 20: Remarques Sur La Limite α → 0 Pour Les Fluides de Grade 2

Résumé

Abstract

1 Introduction

2 Préliminaires

3 Preuve du théorème 1.1

Chapter 21: Remarks on the Kompaneets Equation, a Simplified Model of the Fokker-Planck Equation

1 Introduction

2 Preliminary results

3 Existence of local solutions

4 Qualitative properties, global existence or blow up

Chapter 22: Singular Perturbations Without Limit in the Energy Space. Convergence and Computation of the Associated Layers

1 Introduction

2 Singular perturbations

3 First example in dimension one

4 Convergence to the layer

5 Second example in dimension one

6 Convergence in the second example

7 An example in dimension two

8 Numerical approximation

Acknowledgements

Chapter 23: Optimal Design of Gradient Fields with Applications to Electrostatics

1 Introduction

2 Background and main theoretical results

3 The discrete problem

4 Minimizing sequences of configurations

5 A complete characterization of minimizing sequences

6 Numerical solution and a practical approach to design of graded materials

Acknowledgements

Chapter 24: A Blackbox Reduced-Basis Output Bound Method for Noncoercive Linear Problems

1 Introduction

2 Problem description

3 Reduced-basis output bound formulation

4 Error analysis

5 Computational procedure

6 The Helmholtz problem

Acknowledgements

Chapter 25: Simulation of Flow in a Glass Tank

1 Introduction

2 Mathematical model

3 Discretisation

4 PISO method

5 Local defect correction

6 Stirred flow

7 Bubbling

Chapter 26: Control Localized on thin Structures for Semilinear Parabolic Equations

1 Introduction

2 Preliminary estimates for some linear equations

3 State equation

4 Control problem (P1)

5 Derivative of the cost functional with respect to deformations of γ

Chapter 27: Stabilite Des Ondes de Choc de Viscosite Qui Peuvent Etre Caracteristiques

Résumé

Abstract

1 Introduction

2 Rappels ; réduction à un cas particulier

3 Preuve du lemme 2.1

Remerciements

Copyright

Dedication

Preface

This volume is the 14th and last one of the series Nonlinear Partial Differential Equations and their Applications. Collège de France Seminar, which published the texts of the lectures given at the seminars organized by Jacques-Louis Lions, from his election at the Collège de France in 1973 until his retirement in 1998. It was one of the foremost seminars in nonlinear PDE’s and their applications during that period.

It is unfortunate that because of his untimely death, on May 17, 2001, Jacques-Louis Lions will not see its publication. This volume is dedicated to his memory.

Chapter 1

An Introduction to Critical Points for Intergral Functionals

David Arcoya [e-mail]    Departamento de Análisis Matemático, Universidad de Granada, 18071-Granada, Spain

Lucio Boccardo [e-mail]    Dipartimento di Matematica, Università di Roma 1, Piazza A. Moro 2, 00185 Roma, Italy

1 Introduction

The study of minima of functionals defined in spaces of functions may be considered one of the keystones of the mathematical analysis. Remind the efforts by the great mathematicians of the last and present century to develop sufficient conditions on the functional for the existence of minimum. This theory is deeply related with the existence of solutions of boundary value problems. Indeed, this connection is estabilished by the well-known Euler-Lagrange equations associated to the functional.

However, there exist boundary value problems for which the associated functional is indefinite, i.e. it is unbounded from below and from above. This means that it has not global extrema and so we have to search the solutions of the problem among the critical points, i.e. the points for which the derivative of the functional vanishes.

From the abstract point of view there is a difference between the study of minima and of critical points. Indeed, for the existence of minima we need only assumptions on the functional. On the contrary, we point out that the results of existence of critical points involve additional hypotheses on the regularity of the functional to assure the existence of a derivative in some sense. This may explain why the theory of mimima handles classes of functionals with more general hypotheses of smoothness than the critical point theory.

In some papers [4], [5], [6], we overcame this difference by developing a critical point theory for nondifferentiable functionals. We observe explicitely that our model functionals does not involve similar functions to the modulus. In fact, the nondifferentiability of the considered functionals is due to the introduction of some smooth Carathéodory function A(x, u) (as smooth as you want). Specifically, we consider here functionals J N open, N > 1) by

   (1)

and f(x, z) ≡ F′(x, z)(derivative respect to z) a subcritical Carathéodory function. Observe that J , even for smooth functions A (see [11]).

This note is devoted to present the critical point theory developed in [5] (see also [?, ?, ?, ?, ?, ?, ?]) for functionals which are not differentiable in all directions.

2 A mountain pass theorem for nondifferentiable functionals

Our abstract setting for the functionals J that we study is given by the following asumption:

(H)(X, || · ||X) is a Banach space and Y ⊂ X is a subspace which is a normed space endowed with a norm || · ||Y. Moreover, J : X is a functional on X such that it is continuous in (Y, || · ||X + || · ||Y) and satisfies the following hypotheses:

a) J has a directional derivative 〈J′(u), v〉 at each u ∈ X through any direction v ∈ Y.

b) For fixed u ∈ X, the function 〈J′(u), v〉 is linear in v ∈ Y, and for fixed v ∈ Y, the function 〈J′(u), v〉 is continuous in u ∈ X.

Due to the lack of regularity of the functional, some words are needed in order to establish our definition of critical points.

Definition 2.1

A function u ∈ X is called a critical point of J if

In this framework a suitable version of the Ambrosetti-Rabinowitz Theorem has been proved in [5]. Specifically,

Theorem 2.2

Assume (H) and that for e ∈ Y,

with Γ the set of the continuous paths γ : [0, 1] → (Y, || · ||X + || · ||Y), such that γ(0) = 0 and γ(1) = e. Suppose, in addition, that J satisñes the condition

(C) Any sequence {un} in the Banach space Y satisfying for some {Kn+ and {εn} → 0 the conditions

   (2)

   (3)

   (4)

possesses a convergent subsequence in X.

Then there exists a nonzero critical point u ∈ Y of J such that J(u) = c.

Remarks 2.3

1. The proof of this theorem is done by dividing it into two steps. In the first one, only the geometric hypotheses are used to deduce the existence of a sequence {un} in Y satisfying for some {Kn+ and {εn} → 0 the conditions (2)–(4). The proof is then concluded by using condition (C).

2. In this way, condition (C) can be considered as a compactness condition on the functional J, which substitutes in the nondifferentiable case the role done by the well-known Palais-Smale condition in the regular case Y = X.

3. This compactness condition is connected with the coercivity of J extending the previous results for C¹ functionals in [11]. To be specifìcal in [7] we prove

Theorem 2.4

In addition to (H), assume that Y is dense in X and that J is continuous in X and bounded from below. If J satisfies condition (C) then J is coercive, i.e.,

3 A simple model

The application of the abstract result quoted in the previous section to the study of the functional J defined in (1) is very technical. In particular, the verification of the condition (C, but for which the verification of condition (C) has not technical difficulties like for functionals studied in [5], [6]. Specifically, we consider the functional J by setting

   (5)

where 1 ≤ q < 2 < m < 2* (2* = 2N/(N – 2), if 2 < N; 2* = ∞ if N ≤ 2) and a is a function which is measurable respect to x ∈ Ω, C¹ with respect to z ≠ 0 and such that

(a1) There exist β > α > 0 such that

for almost every x ∈ Ω and z .

(a2) There exists γ > 0 such that

and a′(x, z) = 0, z < 0.

(a3) Either

   (6)

or

   (7)

, endowed with the norm || · ||Y = || · ||2/(2–q). By (a1) and (a2)the functional J is continuous on X and satisfies (H). We point out that X = Y .

Theorem 3.1 [5]

Assume (a1 — a3) and 1 < q < 2 < m < 2*. Then the functional J defined in (5) satisfìes (C).

Proof. Let {un} be a sequence in Y satisfying (2), (3) and (4) for some {Kn+ and let {εn} → 0. We prove that {un} is bounded in X. Indeed, taking v = un as test function in (4), multiplying (2) by m and adding, we obtain

Hence, if a(x,z) satisfies (6), taking into account (a1), it follows for some C2 > 0 that

and thus we deduce

which implies, since q < 2, that the sequence {un.

On the other hand, if instead of (6), it holds (7), then that the sequence {un. Therefore the sequence {un. Then there exist u ∈ X and a subsequence (still denoted un) such that un converges weakly to u.

Now let

and

To conclude the proof, it suffices to prove

Step 1. {Tk(un)} → Tk(u, as n → ∞, for every k ≥ R1.

Step 2. For every δ > 0, there exist k0 ≥ R1 and nsuch that ||Gk(un)|| < δ for every k ≥ k0 and n ≥ n0.

Indeed, Steps 1 and 2 imply that, given δ > 0, there exist nand k1 ≥ R1 such that

i.e. {un.

Step 1. Putting wn,k = Tk(un) – Tk(u) as test function in (4), we deduce

. Remark that

and the right hand side converges to zero.

Moreover,

Thus, it follows that the sequence Tk(unto Tk(u) for every k > 0.

Step 2. The assertion is easily proved by taking Gk(un) as test function in (4) and using (a

Thanks to the previous lemma, we can prove existence of a nontrivial critical point for the functional J. That is, the existence of a weak solution of the quasilinear Dirichlet problem

Theorem 3.2

Assume (a1 – a3) and 1 < q < 2 < m < 2*. Then the functional J given by (5) has at least a positive critical point.

Proof. We point out that every nonzero critical point of J is positive. In fact, it is deduced taking Tk(u−) as test function (note that u may not belong to Y, but Tk(u−) ∈ Y). In order to show the existence of a nonzero critical point, and following the ideas of Lemma 3.3 in [2], it is easy to check that u = 0 is a strict local minimum of J, that is, there exist ρsuch that

   (8)

Moreover, lim|t|→∞ J(tφ1) = − ∞, being φ1 > 0 an eigenfunction associated to the first eigenvalue λ1 of the homogeneous Dirichlet problem for the laplacian operator with Lsuch that J(t0φ1) < 0. Thus, letting e = t0φ1 and considering the set Γ of the (continuous) paths

which join 0 and e, i.e. such that γ(0) = 0 and γ(1) = t0φ, so that, by such that

Hence

Then, taking into account Lemma 3.1 and applying , of J with J(u)= c> 0 and thus u

4 Main examples

of the defined by

   (9)

i.e. nonnegative solutions of the boundary value problem:

   (P)

where f is a Carathéodory function with subcritical growth. It is clear that for a solution u of (P) we are meaning

.

The hypotheses that we assume on the Carathéodory coefficient A are the following:

(A1) There exists α > 0 such that

for almost every x ∈ Ω and z ≥ 0.

(A2) There exists Rfor almost every x ∈ Ω, for every z ≥ R1.

(A3) There exist m > 2 and α1 > 0 such that

for almost every x ∈ Ω, z ≥ 0.

Notice that all assumptions on A(x, z) are for z ≥ 0. In fact, since we are looking for nonnegative solutions of (P) we can assume without loss of generality that A(x, z) is even on z.

On the other hand, we will assume the following conditions on f(x, z):

(f1) There exist C1, C2 > 0 such that

with σ + 1 < 2*, (2* = 2N/(N − 2) if 2 < N, and 2* = ∞ if 2 ≥ N).

(f2) There exists R2 > 0 such that

for almost x ∈ Ω and every z ≥ R2(m is the same as in (A3)).

(f3) f(x, |z|) = o(|z|) at z = 0, uniformly in x ∈ Ω.

Theorem 4.1

Assume (A1–3), (f1–3) and that

   (10)

Then the problem (P) has, at least, one nonnegative and nontrivial solution.

Remarks 4.2

1. The above theorem is essentially contained in [5]. However, in that paper it is assumed in addition that A(x, zwith respect to z is also bounded. In [7], we have seen that these additional hypotheses are not necessary for the existence.

2. The general case of functionals

could be also handled as in [5]. For simplicity reasons, we just present here the case p .

3. Some remarks about the meaning of (A3) and (f

Proof of , let hn be a nondecreasing C¹function in [0, ∞) satisfying

Consider the coefficients An(x, z) ≡ hn(A(x, z)), x . Clearly, An satisfies (A(with respect to zby setting

then using (f1–2) and (A3), it can be seen in a similar way to the one in Section 2 that Jn satisfies (C). Indeed, we have

Lemma 4.3

(Compactness condition) Assume (A1–3) and (f1_2). Then the functional Jn satisfies (C).

Using in addition (f3) and following the ideas of [2], it is easily seen that Jn satisfies the geometrical hypotheses of Theorem 2.2. Consequently, by it, there exists a nontrivial and nonnegative solution un of the problem

   (11)

with critical level

where

is such that Jn(en) < 0. Taking into account that An(x, z) ≤ A(x, z) and (10), we observe that

for all t ∈ [t0, ∞) if t0 > 0 is large enough. This allows us to choose en = t0φ). On the other hand, by the Mountain Pass geometry of J1 there exist δ, r > 0 such that

(i.e., roughly speaking, υ = 0 is a strict local minimum of Jn, uniformly in n ). This implies that

   (12)

We claim that {un. Indeed, using again that An(x, z) ≤ A(x, z), we deduce

we derive

which, by (A3), implies that ||un|| is bounded, thus proving the claim.

Then, passing to a subsequence, if necessary, we can assume that {un. Now, we prove that the sequence {un} is bounded in L∞(Ω). Indeed, we can use v = Gk(un), k > R1, as test function in (11) to deduce that {un} satisfies

for any r ≥ 2* such that u ∈ Lr(Ω).

Hence there is a constant C4>0 such that ||un||∞ ≤ C4 (see [14]). Since the sequence {unand L∞(Ω) and (f1), we yield that {unby using the results of [9]. Therefore, {un} → u is a nonnegative critical point of J. In addition, {J(un)} → J(u) and we get from (12) that J(u) ≥ δ so that u ≠ 0. Thus u

Remark 4.4

We conclude by noting that in [7] the reader can find more existence results for nonlinearities f(x, z) which are different combinations of concave and convex functions in the quasilinear spirit of [1], [8].

Acknowledgment

This paper was partially presented by the second author at the Collège de France Seminar (24.3.1995). Both authors would like to thank to the organizers of the Seminar for having given the opportunity of presenting their work.

References

[1] Ambrosetti A, Brezis H, Cerami G. Combined effects of concave and convex nonlinearities in some elliptic problems. J. Funct. Anal. 1994;122:519–543.

[2] Ambrosetti A, Rabinowitz PH. Dual variational methods in critical point theory and applications. J. Funct. Anal. 1973;14:349–381.

[3] Arcoya D, Boccardo L. Nontrivial solutions to some nonlinear equations via minimization. In: Ambrosetti A, Chang KC, eds. Variational Methods in Nonlinear Analysis. Gordon and Breach Publishers; 1995:49–53.

[4] Arcoya D, Boccardo L. A min-max theorem for multiple integrals of the Calculus of Variations and applications. Rend. Mat. Acc. Lincei. 1995;v. 6(s. 9):29–35.

[5] Arcoya D, Boccardo L. Critical points for multiple integrals of Calculus of Variations. Arch. Rat. Mech. Anal. 1996;134(3):249–274.

[6] Arcoya, D. and Boccardo, L. Some remarks on critical point theory for non differentiable functionals, to appear in NoDEA.

[7] Arcoya D, Orsina L. Landesman-Lazer conditions and quasilinear elliptic equations. Nonlinear Anal. TMA. 1997;28:1623–1632.

[8] Boccardo L, Escobedo M, Peral I. A Dirichlet problem involving critical exponent. Nonlinear Anal. TMA. 1995;24:1639–1648.

[9] Boccardo L, Murat F, Puel JP. Existence de solutions faibles pour pour des équations quasilinéaires à croissance quadratique. In: Pitman; 19–73. Res. Notes in Mathematics. 1983;84.

[10] Canino A, Degiovanni M. Nonsmooth critical point theory and quasilinear elliptic equations. In: Top. methods in differential equations and inclusions. Kluwer Academic Publisher; 1995.

[11] Dacorogna B. In: Springer-Verlag; 1989. Direct Methods in the Calculus of Vairations.

[12] Degiovanni M, Marzocchi M. A critical point theory for nons-mooth functionals. Ann. Mat. Pura Appl. 1994;167:73–100.

[13] Pellacci B. Critical points for non differentiable functionals. Boll. U.M.I. 1997;11-B:733–749.

[14] Stampacchia G. Equations elliptiques du second ordre à coefficients discontinus. Les Presses de L’Université du Montréal; 1966.

Chapter 2

A Semigroup Formulation for Electrmagnetic Waves in Dispersive Dielectric Media

H.T. Banks [e-mail]; M.W. Buksas [e-mail]    Center for Research in Scientific Computation, NC State University, Raleigh, NC. 27695–8205, USA

Los Alamos National Laboratory, T-CNLS MS B258, Los Alamos, NM. 87545, USA

1 Introduction

In a forthcoming monograph [2] we have developed a theoretical and computational framework for electromagnetic interrogation of dispersive dielectric media. In that work we show that one can take a time domain variational or weak formulation of Maxwell’s equations in dispersive materials and, in the context of inverse problems, use partially reflected polarized microwave pulses to determine both dielectric material properties and geometry of bodies (specifically for plane waves inpinging on slab geometries in paradyms which approximate far field interrogation). This is done in configurations involving either supraconductive reflecting back boundaries or acoustically generated virtual reflectors.

The propagation and reflection of electromagnetic waves in dispersive dielectric media is, of itself, an interesting topic of investigation. As we point out in the next section (and demonstrate computationally in [2]), the underlying dynamical systems are not typical of either standard parabolic or standard hyperbolic (even with the usual dissipation) systems and are hence of mathematical interest. In this short note, we consider the Maxwell system for rather general dispersive dielectric media and show that such systems, under appropriate conditions on the polarization law, generate C0 semigroup solutions. These results are presented in the context of the 1-dimensional interrogating systems developed in detail in [2] and we invite interested readers to consult that reference for more detailed discussions and development of the underlying model employed here.

2 Modeling of dispersiveness in dielectric media

We begin with time domain Maxwell’s equations in second order form (e.g., see [2]) for the electric field E = E(t, z) of 1-dimensional polarized waves

   (1)

is the speed of light in vacuum, Jc is the conduction current density, Js is a source current density and P is the electric polarization of the dielectric medium. We assume very general constitutive material laws for the polarization and conductivity given by

   (2)

   (3)

where we have tacitly assumed that E(t, z) = 0 for t < 0 and that both gp(ξ, z) and gc(ξ, z) vanish for ξ < 0. With these assumptions, the integrals in equations (2), (3) are equivalent to integration over all of (–∞, ∞) and thus are indeed convolutions. The displacement susceptibility kernel gp (also referred to as the dielectric response function(DRF)) and the conductivity susceptibility kernel gc and conductivity σ when using a frequency domain approach. We assume that either P or Jc or both may contain instantaneous (local in time) components by introduction of δ distributions in the kernels gp and/or gc, respectively.

A medium is dispersive if the phase velocity of plane waves propagating through it depends on the frequency of the waves [16, Chap.7], [10, Chap.8], To determine the dispersive nature of a medium described by equations (1)-(3) we seek plane-wave solutions of the homogeneous analogue of (1) of the form E(t, z) = E0e–i(ωt±κz) which travel in the z direction and have wavelength λ = 2π/κ. The phase velocity υp of these waves is the speed at which planes of constant phase move through the medium. In this case the argument ωt – κz is constant and

   (4)

Seeking plane wave solutions of the form E0e–i(ωt±κz) in (1) is equivalent to seeking solutions of the form E0e±iκz in the frequency domain version of (1). Thus we use the Fourier transform in (1) and obtain

   (5)

where we have ignored the source term Js and where the overhat will represent the Fourier transform throughout. Since we see from and Ĵc = ĝcÊ, this can be written

   (6)

We note that (6) is the generalized Helmholtz equation [16, p. 271]

   (7)

with

   (8)

. It follows that the corresponding time domain solutions are our desired solutions of the form E(t, z) = E0e–i(ωt±iκz) where the wavenumber κ = κ(ω) will in general depend on the frequency ω. The equation (8) relating the frequency ω and the wavenumber κ of propagating waves is known as the dispersion equation for the medium. In the case of vacuum or free space where ĝp = ĝc = 0 so that κ = ω/c, we obtain the corresponding phase velocity υp = c = the speed of light as expected. More generally the phase velocity in a dielectric medium with conductivity and polarization is given by

   (9)

In light of (9) and the definition of a dispersive medium, we see that if either ĝc/ω or ĝp depend on ω, we will have dispersiveness. Several special cases are worthy of note.

For instantaneous conductivity, that is, gc(t, z) = σδ(t) so that (3) reduces to Ohm’s Law Jc = σE, we see that the term iĝc0ω becomes iσ0ω). For instantaneous polarization (often assumed in standard treatments of the Maxwell theory) we find gp(t, z0χδ(t), where χ is the dielectric susceptibility constant and hence ĝp0 = χ and the medium is not dispersive. One must turn to more complicated (and more realistic) models, such as those of Debye or Lorentz, to have a polarization based contribution to dispersiveness in a medium. For the usual Debye polarization model [11, p.386] one has

   (10)

where τ s∞ are familiar dielectric constants. In this case one finds

For the Lorentz model [16, p.496] we have

   (11)

. In the frequency domain this yields

and again we have a polarization based dispersive medium. Higher order (the Debye and Lorentz models correspond to first and second order, respectively, differential equation models for the polarization P - see [2] and the references therein) models, as well as combinations of such models also lead to dispersion in a medium.

Thus, in summary we see that instantaneous conductivity but not instantaneous polarization yields dispersiveness in a medium. For a polarization contribution to dispersiveness one must include first or higher order polarization models (instantaneous polarization can be correctly viewed as zero order polarization dynamics). For our semigroup presentation in the next section we shall therefore consider the model (1) with instantaneous conductivity and a general (higher order) polarization model given by (2) with gp = g where the DRF g is assumed smooth in time (i.e., without loss of generality we can assume no instantaneous component for g). Such distributed parameter systems are of interest since they are neither simple hyperbolic nor parabolic in nature.

For simple Ohm’s Law conductivity and instantaneous polarization (or no polarization), the system (1) becomes a well understood dissipative or damped hyperbolic system for which a semigroup formulation can readily be found in the research literature on distributed parameter systems. However, for (1) with polarization based dispersiveness, we obtain a system with behavior of solutions that are neither standard hyperbolic (finite speed of wave propagation along characteristics) nor standard parabolic (infinite speed of propagation of disturbances). Indeed for (1) with either Debye or Lorentz polarization, rather fascinating solutions can be observed. These involve the formulation of so-called Brillouin and Sommerfeld precursors where a pulsed excitation (containing waves with a range of frequencies) evolves into waves propagated with different velocities which coalesce into wave packets (see Chapter 4 of [2] and [1] and the references therein for discussions of these phenomena).

It is of both mathematical and practical interest to know whether these interesting systems can be described in a semigroup context. The potential advantages afforded by a semigroup formulation are widespread since there is a tremendous literature for control, estimation and identification, and stabilization of systems in a semigroup setting. Results for both stochastic and deterministic control methodologies (in both time domain and frequency domain) including open loop and feedback formulations are abundant [12], [3], [4], [11], [19].

In the next section we present a semigroup formulation of the system (1) with simple Ohm’s Law conductivity along with general polarization based dispersiveness generated by polarization laws of the form (2). To be more precise, we take (1) for t > 0 and z ∈ (0, 1) with Jc(t, z) = σ(z)E(t, z) where σ(z) vanishes outside Ω ⊂ (0, 1]. The closed region Ω is some dielectric material region (e.g., a slab or several slab-like regions) containing instantaneous conductivity as well as non trivial polarization of the form (2) with gp(t, z) = g(t, z) vanishing outside z ∈ Ω. Using this form of conductivity and polarization in (1), we obtain the system

   (12)

With this system we take boundary conditions (see [2] for details) that represent a total absorbing boundary at z = 0 and a supraconductive boundary at z = 1. This can be expressed by

   (13)

   (14)

With the definitions

we can write equation (12) as

   (15)

where α * E is the usual convolution

   (16)

One can use the boundary conditions (13) -, (14) to write (15) in weak or variational form so as to seek solutions t → E(tin a Gelfand triple setting V H V* with pivot space H = L²(0, 1). Under modest regularity assumptions on α, β, γ , one can establish existence, uniqueness and continuous dependence (on initial conditions and input) of solutions. Details are given in Chapter 3 of [2].

3 A semigroup formulation

We turn in this section to a semigroup formulation for the dispersive system (12) - (14) or equivalently, (13) - (15), with instantaneous conductivity and general (non instantaneous) polarization.

For our development we assume that γ, β ∈ L∞(0, 1) while α ∈ L∞((0, T) × (0, 1)) and α, β, γ vanish outside Ω. We moreover assume that α can be written as α(t, z) = α1(t)α2(z) where 0 < αL < α2(z) ≤ αU on Ω ⊂ (0, 1] for positive constants αL, αU, with α2 vanishing outside Ω. We assume that t → α1(t) is positive, monotone nonincreasing, and in H¹ (0, T. This monotonicity assumption is typical of the usual assumptions in displacement susceptibility kernels (e.g., see [9, p.102] or [15]). We shall return to discuss this monotonicity requirement further after our semigroup arguments of this section.

We consider the term (16) given by

from (15) and note that it can be equivalently written

where G(ξ) = α(–ξ). We denote G1(ξ) = α1(–ξ) so that G(ξ) = G1(ξ)α2.

The approximation is valid for r sufficiently large (r = ∞ is permitted) so that α(t) ≈ 0 for t > r. We observe at this point that Ġ1(ξ) ≥ 0 with G1(ξ) > 0 on (–r, 0].

, H = L²(0, 1) with V H V, the space L²(Ω) with weighting function α2, which is readily seen to be equivalent to L²(Ω) due to the upper and lower bounds on α2 ∈ L∞(Ω). We shall denote the restriction of functions ɸ in L²(0, 1) to Ω again by ɸ and write ɸ ∈ Lwhenever no confusion will result.

Using the above definitions and approximating, we may write (15) as

   (17)

Using an approach given in [5], [6], [14] and [7] for viscoelastic systems, we define an auxiliary variable w(tby w(t)(θ) = E(t) – E(t + θ), –r ≤ θ ≤ 0. Since G(θ, z) > 0 for θ ∈ (–r, 0], z ∈ Ω we may take as an inner product for W the weighted L² inner product

under which W is a Hilbert Space. We note that by our notational convention explained above, we have w(t) ∈ W for any E(t, z) with E. Using a standard shift notation, we may write w(t) = E(t) – E(t + θ) = E(t) – Et(θ) where Et(θ) = E(t + θ) for –r < θ < 0. Adding and subtracting appropriate terms in (17), we find

or, equivalently

   (18)

and w(t)(ξ) = E(t) – Et(ξ). We observe that G11, like β, is in L²(Ω) and L²(0, 1).

For our semigroup formulation, we consider (18) in the state space

with states (ɸ, ψ, η) = (E(t), Ė(t), w(t)) = (E(t), Ė(t), E(t) – Et(·)). To define an infinitesimal generator, we begin by defining a fundamental set of component operators. Let à *) be defined by

   (19)

where δ0 is the Dirac operator δ0ψ = ψ(0). Then we find

   (20)

defined by

   (21)

is symmetric, V continuous and V for constants λ0 and c1 > 0).

We also define operators B by

   (22)

so that

   (23)

,

   (24)

Since G(ξ, z) = 0 for z ∈ [0, 1]\Ω, we abuse notation and write this as

even though, strictly speaking, η(ξ, z) is only defined for z ∈ Ω.

With these definitions and notations, equation (18) can then be written as

or

   (25)

We rewrite equation (25) as a first order system in the state ζ(t) = (E(t), Ė(t), w(t)) where w(t) = E(t) – Et. To aid in this we introduce another operator

by

We then observe that w(t) = E(t) – Et satisfies

Thus we may formally rewrite (25) as a first order system and adjoin to it the equation

   (26)

We then obtain the first order system for ζ(t) given by

   (27)

given by

   (28)

is defined on

   (29)

for Φ = (ɸ, ψ, ηin is the infinitesimal generator of a C0-semigroup, we actually consider the system (27) on an equivlaent space Z1 = V1 × H × W where V1 is the space V is symmetric, V continuous and V coercive so that it is topologically equivalent to the V inner product.

We are now ready to prove the following generation theorem.

Theorem

– Suppose that γ, β ∈ L∞(0, 1), α ∈ L∞((0, 1) × (0, 1)) with α, β, γ vanishing outside Ω. We further assume that α can be written α(t, z) = α1(t)α2(z) where α1 ∈ H¹(0, T) with α1(t, and 0 < αL ≤ α2(z) ≤ αU for positive constants αL, αU. Then the operator defined by (28), (29) is the infinitesimal generator of a C0-semigroup on Z1 and hence on the equivalent space Z.

Proof. To prove this theorem, we use the Lumer-Phillips theorem ([15, p. 14]). Since Z1 is a Hilbert space, it suffices to argue that for some λis dissipative in Zfor some λ . We first argue dissipativeness.

Let Φ = (ɸ, ψ, η. Then

   (30)

We consider estimates for the last two terms in (30) separately. From (24) we have

Moreover,

Finally, since G1(θ) ≥ 0, Ġ1 ≥ 0, and η ∈ dom D requires η(0) = 0, we may argue

Combining these estimates with

which yields the desired dissipativeness in Z1.

To establish the range statement, we must argue there exists some λ > 0 such that for any given Ψ = (μ, v, ξ) in Zsatisfying

   (31)

, the equation (31) is equivalent to the system

   (32)

for (ɸ, ψ, η, (μ, v, ξ. The first equation is the same as ψ = λɸ–μ while the third can be written as η = (λ – D)–1(ξ+ψ) = (λ – D)–1(ξ + λɸ – μ). These two equations can be substituted in the second to obtain an equation for ɸ. If this equation can be solved for ɸ ∈ V, then the first and third can then be solved for ψ and η, respectively. The equation for ɸ that must be solved is given by

or

   (33)

If we can invert (33) for ɸ ∈ V, then ψ = λɸ – μ is in V, η = (λ – D)–1 [ξ + λɸ – μ] is in dom D ⊂ W and

is in H so that (ɸ, ψ, ηand solves (32).

Thus the range statement reduces to solving (33) for ɸ G V.

We first observe that (λ – D)–1 = (1 – eλ.)/λ since (λ – D)(1 – eλθ) = λ satisfies η(0) = 0 and hence is in dom D.

Thus, for ɸ ∈ Hsatisfies

and

Hence for λ sufficiently large we have

Thus if we define the sesquilinear form

we see that for λ sufficiently large, σλ is V coercive and hence, by the Lax-Milgram lemma [20], it is invertible. It follows immediately that (33) is invertible for ɸ ∈ V.

Let S(tso that solutions to (27) are given by

   (34)

Solutions are clearly continuously dependent on initial data ζ. The first component of ζ(t) is a generalized solution E(t) of (17). One can now argue that the solution agrees with the unique weak solution obtained in Chapter 3 of [2], by using the arguments in Chapter 4.4 of [8]. Briefly, one argues equivalence for sufficiently regular initial data and nonhomogeneous perturbation. Then density along with continuous dependence is used to extend the equivalence to more general data (see [8] for details).

4 Concluding remarks

In the previous section we presented a semigroup generation theorem under general conditions on the coefficients α, β, γ of where it is required that α1 (t. We consider more closely the condition for some common polarization laws.

where gp is given in (10). That is,

so that

and

Thus Debye polarization satisfies the conditions of the generation theorem and the associated system generates a C0 semigroup.

For Lorentz polarization, we have (recall (11))

and hence

We therefore see that it is not