partial differential equations on multistructunes proceedings of the conference held in Luminy, France

edited by Felix AN Mehmeti Universite de Valenciennes et du Hainaut Cambresis Valenciennes, France

Joachim von Below Universite du Littoral Cote d'Opale Calais, France

Serge Nicaise Universite de Valenciennes et du Hainaut Cambresis Valenciennes, France

MARCEL DEKKER, INC. Copyright © 2001 IOP Publishing Ltd

NEW YORK • BASEL

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ISBN: 0-8247-0565-3

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Copyright © 2001 IOP Publishing Ltd

Preface

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The international conference on Partial Differential Equations on Multistructures was held at the Centre International de Rencontres Mathematiques (CIRM), Luminy, France. The conference brought together specialists from rather different research domains in partial differential equations who all investigate problems bearing a multistructure aspect, such as networks, ramified spaces, diffractions, or interfaces.

The analysis of multistructures arises in many applications, e.g. in neurobiology, electronics and continuum mechanics. The modelling of these problems is governed by partial differential equations in each of the components and by transition conditions between these. In the classical context, one considers a partial differential equation or a system of these in one domain subject to boundary conditions. The aim of the conference and of its proceedings was the investigation of partial differential equations on several domains connected by interaction zones. For some of the recent research already done we refer to F. Ali Mehmeti, J. von Below, J. Lagnese, G. Leugering, G. Lumer, S. Nazarov, St. Miiller, S. Nicaise, J. P. Roth, E. Schmidt et al. Several groups are actually working in various aspects of this research field, which are developing in a very promising way not only in theory, but also in application. The contributions to the conference and its proceedings treat the general theme under the following aspects: specific modelling, spectral theory and operator theory, nonlinear analysis, variational calculus, integral equations, dynamical systems, control theory and numerical analysis.

The modelling of physical phenomena in heterogeneous media leads to the mathematical analysis on ramified spaces or, more generally, on multistructures that correspond to systems of partial differential equations under nonclassical boundary conditions. The latter ones play a key role since they describe the transition in the interaction zones and link the different domains to the whole multistructure. Many analogues to the classical domain case are now available for multistructure-like spaces. Especially the spectral theory reveals specific features of these spaces and, thereby, it helps to characterize and to describe the interaction models by means of their associated spectra. Correspondingly, the variational approach, the control theory,

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Preface

nonlinear dynamical systems and the numerical analysis, all reveal the specific character of the multistructure in question.

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The articles of this volume are based on 14 of the 24 lectures of the conference and present new results in various topics of the analysis on multistructures. They include generalizations of classical theories to multistructures as well as specific characterizations and modellings of these. The volume gives a good overview of the actual research in that field and constitutes a valuable source for the interested researcher in partial differential equations, functional analysis and control theory who wants to participate in this research field of growing interest. The organization of the conference was enabled and financed by the Centre International de Rencontres Mathematiques (CIRM), Luminy, France, the Universite du Littoral Cote d'Opale, France, the Universite de Valenciennes et du Hainaut Cambresis, France. The organizers would like to express their gratitude to all of them, in particular to the staff of the conference center CIRM, notably to its Director J. P. Labesse, Mrs. A. Zeller-Meier and Mrs. S. Arnaud.

Moreover, the editors of this volume would like to express their gratitude to the contributors, to the referees and to Marcel Dekker, Inc., especially to Ms. M. Allegra and Ms. J. Paizzi, for their kind and generous cooperation in the production of the volume.

Felix Ali Mehmeti Joachim von Below Serge Nicaise

Copyright © 2001 IOP Publishing Ltd

Contents Preface Contributors Conference Program Conference Participants

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1.

Transient Vibrations of Planar Networks of Beams: Interaction of Flexion,Transversal and Longitudinal Waves F. Ali Mehmeti and B. Dekoninck

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2.

Can One Hear the Shape of a Network? Joachim von Below

19

3.

Sensitivity Analysis of 2D Interface Cracks M. Bochniak and A.-M. Sdndig

37

4.

On the Asymptotic Expansion of the Solution of a Dirichlet-Ventcel Problem with a Small Parameter M. Bourlard, A. Maghnouji, S. Nicaise, and L. Paquet

49

On Instantaneous Control of Singularly Perturbed Hyperbolic Equations on Graphs T. Fischer and G. Leugering

69

5.

6.

Hadamard Formula in Nonsmooth Domains and Applications G. Fremiot and J. Sokolowski

7.

Singular Stress Field at the Tip of a Closed Interface Crack Dominique Leguillon

8.

On the Geometric and Algebraic Multiplicities for Eigenvalue Problems on Graphs

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121

135

Jose A. Lubary

9.

The Asymptotic Laplace Transform: New Results and Relation to Komatsu's Laplace Transform of Hyperfunctions Gtinter Lumer and Frank Neubrander

10.

Some Systems of PDE on Polygonal Networks D. Mercier

11.

About a Geometrical Approach to Multistructures and Some Qualitative Properties of Solutions O. M. Penkin

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183

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12.

13.

Study of a Vibration Problem for a Perforated Plate with Fourier Boundary Conditions J. -M. Sac-Epee and J. Saint Jean Paulin

193

Singular Perturbations with Nonsmooth Limit and Finite Element Approximation of Layers for Model Problems of Shells J. Sanchez-Hubert and E. Sanchez Palencia

207

Modelling of a Thin Piezoelectric Shell Coupled with a Distributed Electronic Circuit by Distributed Piezoelectric Transducers G. Senouci and M. Lenczner

227

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Copyright © 2001 IOP Publishing Ltd

Contributors

FELIX ALI MEHMETI Laboratoire de Mathematiques Appliquees et Calcul Scientifique, Universite de Valenciennes et du Hainaut - Cambresis, Le Mont Houy, F-59313 Valenciennes Cedex 9, France.

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JOACHIM VON BELOW Laboratoire de Mathematiques Pures et Appliquees Joseph Liouville, Universite du Littoral Cote d'Opale, B.P. 699, F-62228 Calais Cedex, France.

MARIUS BOCHNIAK Mathematische Institut A, Universitat Stuttgart, Pfaffenwaldring 57, D-70511 Stuttgart, Germany.

MARYSE BOURLARD Laboratoire de Mathematiques Appliquees et Calcul Scientifique, Universite de Valenciennes et du Hainaut - Cambresis, Le Mont Houy, F-59313 Valenciennes Cedex 9, France.

BERTRAND DEKONINCK Laboratoire de Mathematiques Appliquees et Calcul Scientifique, Universite de Valenciennes et du Hainaut - Cambresis, Le Mont Houy, F-59313 Valenciennes Cedex 9, France. TORSTEN FlSCHER

Fakultat fur Mathematik und Physik, Universitat Bayreuth, D-95440 Bayreuth, Germany.

GILLES FREMIOT Institut Elie Cartan, Universite de Nancy I, B.P. 239, F-54506 Vandceuvre-les-Nancy Cedex, France.

DOMINIQUE LEGUILLON Laboratoire de Modelisation en Mecanique, Universite de Pierre et Marie Curie, F-75252 Paris Cedex 05, France.

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MICHEL LENCZNER Equipe de Mathematiques, Universite de Franche Comte, Route de Gray, F-25030 Besangon Cedex, France.

GUNTHER LEUGERING Fachbereich Mathematik, Technische Universitat Darmstadt, SchlofigartenstraBe 7, D-64289 Darmstadt, Germany.

JOSE A. LUBARY Departement de Matematica Aplicata II, Universitat Polytecnica de Downloaded by [37.44.207.135] at 05:46 19 January 2017

Catalunya, 5 Pau Gargallo, ES-08028 Barcelona, Spain. GUNTER LUMER

Institut de Mathematiques et d'lnformatique, Universite de MonsHainaut, 6, Ave. du Champ de Mars, B-7000 Mons, Belgium.

ABDERRAHMAN MAGHNOUJI Laboratoire de Mathematiques Appliquees et Calcul Scientifique, Universite de Valenciennes et du Hainaut - Cambresis, Le Mont Houy, F-59313 Valenciennes Cedex 9, France.

DENIS MERCIER Laboratoire de Mathematiques Appliquees et Calcul Scientifique, Universite de Valenciennes et du Hainaut - Cambresis, Le Mont Houy, F-59313 Valenciennes Cedex 9, France.

FRANK NEUBRANDER Departement of Mathematics, Lousiana State University, Baton Rouge, La. 70803-4918 U.S.A.

SERGE NICAISE Laboratoire de Mathematiques Appliquees et Calcul Scientifique, Universite de Valenciennes et du Hainaut - Cambresis, Le Mont Houy, F-59313 Valenciennes Cedex 9, France.

Luc PAQUET Laboratoire de Mathematiques Appliquees et Calcul Scientifique, Universite de Valenciennes et du Hainaut - Cambresis, Le Mont Houy, F-59313 Valenciennes Cedex 9, France.

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Contributors

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OLEG M. PENKIN Voronezh State University, 1 Universitatskaya PL, 304693 Voronezh, Russia.

JEAN-MARC SAC-EPEE Departement de Mathematiques, Universite de Metz, He du Saulcy, F-57045 Metz Cedex 01, France.

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Mathematisches Institut A, Universitat Stuttgart, PfafFenwaldring 57, D-70511 Stuttgart, Germany.

JEANNINE SAINT JEAN-PAULIN Departement de Mathematiques, Universite de Metz, lie du Saulcy, F-57045 Metz Cedex 01, France.

JACQUELINE SANCHEZ-HUBERT UFR Sciences, Mecanique, Universite de Caen, Boulevard du Marechal Juin, Campus 2, F-14032 Caen, France.

EVARISTE SANCHEZ PALENCIA Laboratoire de Modelisation en Mecanique, Universite de Pierre et Marie Curie, F-75252 Paris Cedex 05, France.

GHOUTI SENOUCI Equipe de Mathematiques, Universite de Franche Comte, Route de Gray, F-25030 Besangon Cedex, France.

JAN SOKOLOWSKI Institut Elie Cartan, Universite de Nancy I, B.P. 239, F-54506 Vandceuvre-les-Nancy Cedex, France.

Copyright © 2001 IOP Publishing Ltd

Conference Program Chairman: J. von Below

10:30 Opening session

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11:00 J. Saint Jean Paulin, Metz (France) Study of optimal control with rapidly oscillating coefficients Chairman: F. Ali Mehmeti

14:00 A. Sandig, Stuttgart (Germany) Sensitivity analysis for elastic structures in presence of stress singularities 15:00 J. Sokolowski, Nancy (France) Shape sensitivity analysis for optimal control problems

16:30 P. Ferreira, Polytechnique, Palaiseau (France) Diffraction at plane and curved gratings 17:30 T. Abboud, Polytechnique, Palaiseau (France) Diffraction by gratings at low and high frequencies

Chairman: G. Lumer

8:30 S. Nicaise, Valenciennes (France) Asymptotic expansion of the solution of a mixed Dirichlet-Ventcel problem with a small parameter 10:00 Y. Dermanjian, Marseille (France) Courbes et relations de dispersion en elasticite 11:00 O. Poisson, Marseille (France) Principe d'amplitude limite pour la propagation d'ondes acoustiques dans une bande multistratifiee, et prolongement meromorphe de la resolvante de 1'operateur associe

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Conference Program

Chairman: S. Nicaise 15:00 M. Lenczner, Besangon (Prance) Modelisation et strategic de controle pour des systemes avec actionneurs et capteurs distribues

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16:30 D. Leguillon, Paris (France) Three dimensional interface crack tip singularities with contact and friction 17:30 D. Mercier, Valenciennes (France) Some systems of partial differential equations on two-dimensional networks

Chairman: G. Leugering 8:30 J. von Below, Littoral (France) Can one hear the shape of a network?

10:00 B. Dekoninck, Valenciennes (France) Spectrum and controllability of networks of beams 11:00 J. Lubary, Barcelona (Spain) On the geometric and algebraic multiplicities for eigenvalue problems on graphs Chairman: E. Schmidt

8:30 G. Leugering, Bayreuth (Germany) On optimal control problems and domain decomposition for evolution problems on graphs 10:00 C. Cattaneo, Milano (Italy) The spectrum of the continuous Laplacian on a graph 11:00 L. Fontana, Milano (Italy) The Cauchy problem for the wave equation on finite, weighted one dimensional networks Copyright © 2001 IOP Publishing Ltd

Conference Program

Chairman: M. Costabel 14:00 D. Cioranescu, Paris (France) On some homogenization problems

15:00 G. Lumer, Mons (Belgium) Singular dynamics in general and for ramified systems

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16:30 J. Sanchez Palencia, Paris (France) Generalities on singular perturbations without limit in the space of finite energy

17:30 J. Sanchez-Hubert, Caen (France) Model problem for boundary and internal layers in parabolic shells

Chairman: J. von Below

8:30 M. Costabel, Rennes (France) Singularities of Maxwell interface problems 10:00 F. Ali Mehmeti, Valenciennes (France) Various interaction problems 11:00 O. Penkin, Voronezh (Russia) Some qualitative properties of the solutions of elliptic inequalities on stratified sets

Copyright © 2001 IOP Publishing Ltd

Conference Participants T. ABBOUD, Centre de Mathematiques Appliquees, Ecole Polytechnique, 91128 Palaiseau Cedex (France) [email protected]

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F. ALI MEHMETI, Universite de Valenciennes, MACS, Le Mont Houy, 59313 Valenciennes Cedex 9 (France) [email protected] J. von BELOW, Universite du Littoral Cote d'Opale, Departement de Mathematiques, Laboratoire de Mathematiques Pures et Appliquees Joseph Liouville, 50 rue F. Buisson, B.P. 699, 62228 Calais Cedex (France) [email protected] C. CATTANEO, Dipartimento di Matematica, Universita di Milano, Via Saldini 50, 20133 Milano (Italy) font ana@ vmimat. mat. unimi. it D. CIORANESCU, Analyse Numerique, Tour 55-65, 5eme etage, Universite Pierre et Marie Curie, 4 place Jussieu, 75252 PARIS Cedex 05 (France) [email protected] M. COSTABEL, Universite de Rennes I, IRMAR, Campus de Beaulieu, 35042 Rennes Cedex (France) cost abel@lie. uni v-rennes 1. fr E. CROC, Analyse Numerique, Tour 55-65, 5eme etage, Universite Pierre et Marie Curie, 4 place Jussieu, 75252 PARIS Cedex 05 (France) [email protected] C. DE COSTER, Universite du Littoral Cote d'Opale, Departement de Mathematiques, Laboratoire de Mathematiques Pures et Appliquees Joseph Liouville, 50 rue F. Buisson, B.P. 699, 62228 Calais Cedex (France) [email protected] XV

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B. DEKONINCK, Universite de Valenciennes, MACS, Le Mont Houy, 59313 Valenciennes Cedex 9 (France) Ber tr and. Dekoninck@univ- Valenciennes. fr

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Y. DERMENJIAN, Universite de Provence, Centre de Mathematiques et d'lnformatique, 39, rue Joliot-Curie, 13453 Marseille Cedex 13 (France) [email protected]

P. FERREIRA, Centre de Mathematiques Appliquees, Ecole Polytechnique, 91128 Palaiseau Cedex (France) [email protected] T. FISCHER, Universitat Bayreuth, Fakultat fur Mathematik und Physik, Lehrstuhl V Mathematik, 95440 Bayreuth (Germany) [email protected] L. FONTANA, Dipartimento di Matematica, Universita di Milano, Via Saldini 50, 20133 Milano (Italy) font ana@ vmimat. mat. unimi. it

D. LEGUILLON, Laboratoire de Modelisation en Mecanique, Universite Pierre et Marie Curie, 4, place Jussieu, 75252 Paris Cedex 05 (France) dol@ccr. j ussieu. fr

M. LENCZNER, Universite de Franche Comte, Equipe de Mathematiques, UMR 6623, Route de Gray, 25030 Besancon Cedex (France) [email protected] G. LEUGERING, Fachbereich Mathematik, Technische Universitat Darmstadt, Schlofigartenstrafie 7, D-64289 Darmstadt (Germany) [email protected]

J. LUBARY, Departament de Matematica Aplicata II, Universitat Polytecnia de Catalunya, 5 Pau Gargallo, 08028 Barcelona (Spain) [email protected] Copyright © 2001 IOP Publishing Ltd

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G. LUMER, Universite de Mons-Hainaut, Institut de Mathematique et d'lnformatique, Batiment Le Pentagone, 6 Avenue du Champ de Mars, 7000 Mons (Belgium) [email protected]

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D. MERCIER, Universite de Valenciennes, MACS, Le Mont Houy, 59313 Valenciennes Cedex 9 (France) [email protected] S. NICAISE, Universite de Valenciennes, MACS, Le Mont Houy, 59313 Valenciennes Cedex 9 (France) [email protected] O. PENKIN, Voronezh State University, 1 Universitatskaya PL, 394693 Voronezh (Russia) [email protected] O. POISSON, Universite de Provence, CMI, 39 rue F. Joliot-Curie, 13453 Marseille Cedex 13 (France) [email protected]

J. SAINT JEAN PAULIN, Universite de Metz, Departement de Mathematiques, He du Saulcy, 57045 Metz Cedex 01 (France) [email protected] J. SANCHEZ-HUBERT, Universite de Caen, UFR Sciences, Mecanique, Boulevard du Marechal Juin, Campus 2, 14032 Caen (France) [email protected] J. SANCHEZ PALENCIA, Laboratoire de Modelisation en Mecanique, Universite Pierre et Marie Curie, 4, place Jussieu, 75252 Paris Cedex 05 (France) [email protected]

A. SANDIG, Universitat Stuttgart, Math. Institut A, 57 Pfaffenwaldring, 70569 Stuttgart (Germany) [email protected] Copyright © 2001 IOP Publishing Ltd

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E. SCHMIDT, Mac Gill University, Department of Math, and Statistics, 805 Sherbrooke St. West, Montreal, Quebec H3A2K6 (Canada) [email protected]

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J. SOKOLOWSKI, Universite de Nancy I, Institut Elie Cartan, B.P. 239, 54506 Vandoeuvre-les-Nancy Cedex (France) [email protected]

Copyright © 2001 IOP Publishing Ltd