DOCSLIB.ORG

Acoustics, Mechanics, Mathematical Flnalqsis

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Acoustics, Mechanics, and the Related Topics of Mathematical flnalqsis This page intentionally left blank Proceedings of the International Conference to Celebrate Robert P. Gilbert's 70th Birthday Acoustics, Mechanics, and the Related Topics of Mathema tical Analqsis CAES du CNRS, Frejus, France 18 - 22 June 2002 Editor Armand Wirgin Laboratoire de Mecanique et dAcoustique Marseille, France orld Scientific Jersey London Singapore Hong Kong Published by World Scientific Publishing Co. Re. Ltd. 5 Toh Tuck Link, Singapore 596224 USA ofice: Suite 202,1060 Main Street, River Edge, NJ 07661 UK ofice: 57 Shelton Street, Covent Garden, London WCZH 9HE British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library. ACOUSTICS, MECHANCIS, AND THE. RELATED TOPICS OF MATHEMATICAL ANALYSIS Proceedings of the International Conference to Celebrate Robert P. Gilbert’s 70th Birthday Copyright 0 2002 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, eZectronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 981-238-264-X This book is printed on acid-free paper. Printed in Singapore by Mainland Press PREFACE The international conference Acoustics, Mechanics, and the Related Topics of Mathematical Analysis (AMRTMA) was held on 18-22 June, 2002 at the Villa Clythia, CAES du CNRS, in Frbjus, France. This interdis- ciplinary meeting was in some respects a smaller-format copy of previous ISAAC conferences. AMRTMA was sponsored jointly by the Rbgion PACA (Provence Alpes CBte d’Azur), the CNRS (Centre National de la Recherche Scientifique) and ISAAC (International Society for Analysis, its Applica- tions and Computation). The objective of AMRTMA was two-fold. First, and above all, it was the occasion for friends and admirers of Robert P. Gilbert to celebrate his 70th birthday in a convivial atmosphere. Second, the conference was to Figure 1. Bob Gilbert on the boat to St. Tkopez during the AMRTMA conference provide a means for bringing together scientists from many different fields to discuss the intricacies and usefulness of applied mathematics, notably in the realm of mechanics. A group of over 60 persons, composed of scholars, scientists, engineers (some with their Spouses/companions and children) from the USA, Rance, Ger- many, Belgium, Italy, Greece, Austria, England, Sweden, Russia, Ukraine, Algeria, Tunisia and China, participated in this event. The quality of the (all half-hour) scientific communications (most of which appear in extenso V vi in this book) was high and the banquet in honor of R. Gilbert a memorable birthday celebration. As the participants were together (all lodged in the Villa Clythia, a vacation and conference facility of the CNRS) most of the time during the five days of the meeting, they had many occasions (notably during the meals, cocktails, and boat trip to St. Tropez) to socialize as well as exchange ideas and impressions. I would like to express my gratitude to the Centre National de la Recherche Scientifique, G. Berger, Directrice and E. Brun, Administrateur DkleguB, to the Conseil Wgional Provence-Alpes-C8te d’Azur, M. Hayot, Vice- PrBsident, to the International Society for Analysis, its Applications and Computation, H. Begehr, President of the Board, R.P. Gilbert, Member of the Board, and to the Laboratoire de MBcanique et d’Acoustique, M. Ritous, Directeur for their financial support and/or encouragement. The Scientific Committee members: A. Ben-Israel , Rutgers Univ., USA, A.-S. Bonnet Ben-Dhia, ENSTA, France, A. Bourgeat , Univ. St. Eti- enne, fiance, J. Buchanan, US Naval Acad., USA, C. Depollier, LAUM, fiance, R.P. Gilbert, Univ. Delaware, USA, G. Hile, Univ. Hawaii, USA, A. Jeffrey, Univ. Newcastle/Tyne, U.K., G. Maugin, LMM/CNRS/UPMC, fiance, F. Nicolosi, Univ. Catania, Italy, A. Panchenko, Penn State Univ., USA, A. Wirgin, LMA/CNRS, France, and Y. Xu, Univ. Tennessee, USA did a great job in selecting the communications and reviewing the papers for the proceedings book. The Organization Committee comprised of: G. Hile, Univ. Hawdi, USA, C. Tsogka, LMA/CNRS, France and A. Wirgin, LMA/CNRS, France acted efficiently in setting up the conference. I. Czyz, DBlCgation RBgionale du CNRS, Marseille, France provided pre- cious help on budgetary and financial issues, M. Morano, LMA/CNRS acted efficiently for the secretarial tasks, and E. Ogam, LMA/CNRS helped solve many problems connected with audiovisual presentations and informatics. My old friend R. Tanteri from Toulouse, France, made the web page that helped draw attention to the conference and A. DBlinibre LMA/CNRS, France installed this page on the LMA web site. The task of designing the logo of the conference was submitted to P. de Sentenac, Paris, France and to V. Quesnel, Rouen, fiance (whose design was finally retained). vii The success of the conference was largely due to the efforts of the kind and efficient personnel of the Villa Clythia: its Director, M. Chevalier, its Assistant Director, E. Porcu, and all the other members of the staff who made our stay an enjoyable experience. I wholeheartedly thank these many persons for their precious help. Next to last, but not least, I thank my wife Nicole and my daughter Zo6 for their encouragements and efficient help at the reception desk of the confer- ence. As concerns this book, I acknowledge the helpful aid and initiatives of Daniel Cartin and Chelsea Chin from World Scientific. Armand Wirgin This page intentionally left blank Contents Preface V A. Wirgin Paen R.P. Gilbert 1 R. Carroll About Bob Gilbert 4 A. Wirgin 8 Representation formulas in Clifford analysis H. Begehr Imaging methods in random media 14 J. Berryman, L. Borcea, G. Papanicolaou and C. Tsogka Resonances of an elastic plate in a duct, in the presence of a uniform 21 flow AS. Bonnet-Ben Dhia and J.-F. Mercier First order asymptotic modeling of a nuclear waste repository 28 A. Bourgeat, I. Boursier, 0. Gipoulowc and E. Marusic-Paloka Exact axisymmetric solution for temperature-dependent compressible 34 Navier-Stokes Equations P. Broadridge and T.M. Barrett Recovery of the poroelastic parameters of cancellous bone using low 41 frequency acoustic interrogation J.L. Buchanan, R.P. Gilbert and K. Khashanah Mathematical model of the interaction problem between 48 electromagnetic field and elastic body F. Cakoni and G.C. Hsiao Bore evolution in inhomogeneous channels 55 J. G. Caputo and Y.A. Stepanyants An inverse spectral problem for a Schrodinger operator with an 64 unbounded potential L. Cardoulis, M. Cristofol and P. Gaitan ix X Trapping regions for discontinuously coupled dynamic systems 71 S. Carl and J. W. Jerome Differential calculi 78 R. Carroll Reconstruction problem for a periodic boundary between two media 85 J. Chandezon, A. Ye. Poyedinchuk and N.P. Yashina. A note on generalized Cesho operators 92 D.C. Chang, R. Gilbert and G. Wang On the boundedness of functions from an anisotropic weighted space 100 satisfying some integral inequalities P. Cianci Homogenizing a flow of an incompressible inviscid fluid through 108 an elastic porous media T. Clopeau and A. Mickelic Approximation of a dynamic unilateral contact problem for a cracked 116 viscoelastic body M Cocou and G. Scarella Principles of signal based ray tracing for 2D and 3D complex tectonics 123 P. Cristini and E. De Bazelaire Regularity up to the boundary for a class of solutions of a functional- 130 differential system S. D 'Asero On the Hardy spaces of harmonic and monogenic functions in the unit 137 ball of R~+~ R. Delanghe Time domain wave equations for lossy media obeying a frequency 143 power law: application to the porous materials Z.E.A. Fellah, S. Berger, W. Lauriks and C. Depollier xi A model for porous ductile viscoplastic solids including void shape 150 effects L. Flandi and J. B. Leblond Acoustic wave propagation in a composite of two different poroelastic 157 materials with a very rough periodic interface: a homogenization approach R. Gilbert and MJ. Ou Effective acoustic equations for a nonconsolidated medium with 164 microstructure R.P. Gilbert andA. Panchenko A domain decomposition method for the Helmholtz equation in an 171 unbounded waveguide N. Gmati and N. Zrelli Support function method for inverse obstacle scattering problems 178 S. Gutman andA.G. Ramm Heat polynomial analogs 185 G.N. Hile and A. Stanoyevitch Blow-up, shock formation, and acceleration waves in hyperelastic 192 media A. Jeffrey Summability of solutions of Dirichlet problem 199 A. Kovalevsky and F. Nicolosi On isophonic surfaces 207 R. Magnanini A survey of pointwise interpolation inequalities for integer and 212 fractional derivatives V. Maz )a and T. Shaposhnikova Non-uniqueness in connection with methods for the reconstruction of 222 the shape of cylindrical bodies from acoustic scattering data E. Ogam, T. Scotti and A. Wirgin xii Dispersion identification using the Fourier analysis of resonances in 229 elastic and viscoelastic rods R. Othman, G. Gary, R. Blanc, M.N. Bussac and P. Collet Application of the likelihood method to the analysis of waves in elastic 236 and viscoelastic rods R. Othman, G. Gary, M.N. Bussac and P. CoIlet On the controlled evolution of level sets and like methods: the shape 243 and contrast reconstruction C. Ramananjaona, M. Lambert, D. Lesselier and J-P. Zol&o Recent progress in the theoretical and numerical modeling of thin-layer 25 1 flow L. Schwartz Seismic response in a city 258 C.
Recommended publications
  • Futureacademy.Org.UK

    Futureacademy.Org.UK

    The European Proceedings of Social & Behavioural Sciences EpSBS Future Academy ISSN: 2357-1330 http://dx.doi.org/10.15405/epsbs.2017.08.55 EEIA-2017 2017 International conference "Education Environment for the Information Age" PROBLEMS OF AN INTERDISCIPLINARITY IN COMPARATIVE EDUCATION IN THE INFORMATION AGE Vladimir Myasnikov (a), Irina Naydenova (b), Natalia Naydenova (c)*, Tatyana Shaposhnikova (d) *Corresponding author (a) Institute for Strategy of Education Development of the Russian Academy of Education, 5/16, Makarenko St., 105062 Moscow, Russia (b) Institute for Strategy of Education Development of the Russian Academy of Education, 5/16, Makarenko St., 105062 Moscow, Russia (c) Institute for Strategy of Education Development of the Russian Academy of Education, 5/16, Makarenko St., 105062 Moscow, Russia, [email protected]* (d) Institute for Strategy of Education Development of the Russian Academy of Education, 5/16, Makarenko St., 105062 Moscow, Russia Abstract The article is devoted to the topical issues for the methodology of modern comparative research in education, which at the moment in the information age should be interdisciplinary in describing and explaining the educational phenomena and practices; build on the results of research in other areas of knowledge and carried out by specialists from different scientific fields. In modern conditions, the interdisciplinary of comparative research requires an expansion of the framework of this approach, their orientations to results of new disciplines taking into account electronic means of the analysis and assessment of data with the prospect of moving from interdisciplinarity to transdisciplinarity. The interdisciplinarity in comparative research is studied in the works of foreign and domestic scientists.
  • The Futurist Moment : Avant-Garde, Avant Guerre, and the Language of Rupture

    The Futurist Moment : Avant-Garde, Avant Guerre, and the Language of Rupture

    MARJORIE PERLOFF Avant-Garde, Avant Guerre, and the Language of Rupture THE UNIVERSITY OF CHICAGO PRESS CHICAGO AND LONDON FUTURIST Marjorie Perloff is professor of English and comparative literature at Stanford University. She is the author of many articles and books, including The Dance of the Intellect: Studies in the Poetry of the Pound Tradition and The Poetics of Indeterminacy: Rimbaud to Cage. Published with the assistance of the J. Paul Getty Trust Permission to quote from the following sources is gratefully acknowledged: Ezra Pound, Personae. Copyright 1926 by Ezra Pound. Used by permission of New Directions Publishing Corp. Ezra Pound, Collected Early Poems. Copyright 1976 by the Trustees of the Ezra Pound Literary Property Trust. All rights reserved. Used by permission of New Directions Publishing Corp. Ezra Pound, The Cantos of Ezra Pound. Copyright 1934, 1948, 1956 by Ezra Pound. Used by permission of New Directions Publishing Corp. Blaise Cendrars, Selected Writings. Copyright 1962, 1966 by Walter Albert. Used by permission of New Directions Publishing Corp. The University of Chicago Press, Chicago 60637 The University of Chicago Press, Ltd., London © 1986 by The University of Chicago All rights reserved. Published 1986 Printed in the United States of America 95 94 93 92 91 90 89 88 87 86 54321 Library of Congress Cataloging-in-Publication Data Perloff, Marjorie. The futurist moment. Bibliography: p. Includes index. 1. Futurism. 2. Arts, Modern—20th century. I. Title. NX600.F8P46 1986 700'. 94 86-3147 ISBN 0-226-65731-0 For DAVID ANTIN CONTENTS List of Illustrations ix Abbreviations xiii Preface xvii 1.
  • Geologic Map of the Victoria Quadrangle (H02), Mercury

    Geologic Map of the Victoria Quadrangle (H02), Mercury

    H01 - Borealis Geologic Map of the Victoria Quadrangle (H02), Mercury 60° Geologic Units Borea 65° Smooth plains material 1 1 2 3 4 1,5 sp H05 - Hokusai H04 - Raditladi H03 - Shakespeare H02 - Victoria Smooth and sparsely cratered planar surfaces confined to pools found within crater materials. Galluzzi V. , Guzzetta L. , Ferranti L. , Di Achille G. , Rothery D. A. , Palumbo P. 30° Apollonia Liguria Caduceata Aurora Smooth plains material–northern spn Smooth and sparsely cratered planar surfaces confined to the high-northern latitudes. 1 INAF, Istituto di Astrofisica e Planetologia Spaziali, Rome, Italy; 22.5° Intermediate plains material 2 H10 - Derain H09 - Eminescu H08 - Tolstoj H07 - Beethoven H06 - Kuiper imp DiSTAR, Università degli Studi di Napoli "Federico II", Naples, Italy; 0° Pieria Solitudo Criophori Phoethontas Solitudo Lycaonis Tricrena Smooth undulating to planar surfaces, more densely cratered than the smooth plains. 3 INAF, Osservatorio Astronomico di Teramo, Teramo, Italy; -22.5° Intercrater plains material 4 72° 144° 216° 288° icp 2 Department of Physical Sciences, The Open University, Milton Keynes, UK; ° Rough or gently rolling, densely cratered surfaces, encompassing also distal crater materials. 70 60 H14 - Debussy H13 - Neruda H12 - Michelangelo H11 - Discovery ° 5 3 270° 300° 330° 0° 30° spn Dipartimento di Scienze e Tecnologie, Università degli Studi di Napoli "Parthenope", Naples, Italy. Cyllene Solitudo Persephones Solitudo Promethei Solitudo Hermae -30° Trismegisti -65° 90° 270° Crater Materials icp H15 - Bach Australia Crater material–well preserved cfs -60° c3 180° Fresh craters with a sharp rim, textured ejecta blanket and pristine or sparsely cratered floor. 2 1:3,000,000 ° c2 80° 350 Crater material–degraded c2 spn M c3 Degraded craters with a subdued rim and a moderately cratered smooth to hummocky floor.
  • EUROPEAN MATHEMATICAL SOCIETY EDITOR-IN-CHIEF ROBIN WILSON Department of Pure Mathematics the Open University Milton Keynes MK7 6AA, UK E-Mail: R.J.Wilson@Open.Ac.Uk

    EUROPEAN MATHEMATICAL SOCIETY EDITOR-IN-CHIEF ROBIN WILSON Department of Pure Mathematics the Open University Milton Keynes MK7 6AA, UK E-Mail: [email protected]

    CONTENTS EDITORIAL TEAM EUROPEAN MATHEMATICAL SOCIETY EDITOR-IN-CHIEF ROBIN WILSON Department of Pure Mathematics The Open University Milton Keynes MK7 6AA, UK e-mail: [email protected] ASSOCIATE EDITORS VASILE BERINDE Department of Mathematics, University of Baia Mare, Romania e-mail: [email protected] NEWSLETTER No. 47 KRZYSZTOF CIESIELSKI Mathematics Institute March 2003 Jagiellonian University Reymonta 4 EMS Agenda ................................................................................................. 2 30-059 Kraków, Poland e-mail: [email protected] Editorial by Sir John Kingman .................................................................... 3 STEEN MARKVORSEN Department of Mathematics Executive Committee Meeting ....................................................................... 4 Technical University of Denmark Building 303 Introducing the Committee ............................................................................ 7 DK-2800 Kgs. Lyngby, Denmark e-mail: [email protected] An Answer to the Growth of Mathematical Knowledge? ............................... 9 SPECIALIST EDITORS Interview with Vagn Lundsgaard Hansen .................................................. 15 INTERVIEWS Steen Markvorsen [address as above] Interview with D V Anosov .......................................................................... 20 SOCIETIES Krzysztof Ciesielski [address as above] Israel Mathematical Union ......................................................................... 25 EDUCATION Tony Gardiner
  • L P and Sobolev Spaces

    L P and Sobolev Spaces

    NOTES ON Lp AND SOBOLEV SPACES STEVE SHKOLLER 1. Lp spaces 1.1. Definitions and basic properties. Definition 1.1. Let 0 < p < 1 and let (X; M; µ) denote a measure space. If f : X ! R is a measurable function, then we define 1 Z p p kfkLp(X) := jfj dx and kfkL1(X) := ess supx2X jf(x)j : X Note that kfkLp(X) may take the value 1. Definition 1.2. The space Lp(X) is the set p L (X) = ff : X ! R j kfkLp(X) < 1g : The space Lp(X) satisfies the following vector space properties: (1) For each α 2 R, if f 2 Lp(X) then αf 2 Lp(X); (2) If f; g 2 Lp(X), then jf + gjp ≤ 2p−1(jfjp + jgjp) ; so that f + g 2 Lp(X). (3) The triangle inequality is valid if p ≥ 1. The most interesting cases are p = 1; 2; 1, while all of the Lp arise often in nonlinear estimates. Definition 1.3. The space lp, called \little Lp", will be useful when we introduce Sobolev spaces on the torus and the Fourier series. For 1 ≤ p < 1, we set ( 1 ) p 1 X p l = fxngn=1 j jxnj < 1 : n=1 1.2. Basic inequalities. Lemma 1.4. For λ 2 (0; 1), xλ ≤ (1 − λ) + λx. Proof. Set f(x) = (1 − λ) + λx − xλ; hence, f 0(x) = λ − λxλ−1 = 0 if and only if λ(1 − xλ−1) = 0 so that x = 1 is the critical point of f. In particular, the minimum occurs at x = 1 with value f(1) = 0 ≤ (1 − λ) + λx − xλ : Lemma 1.5.
  • Characterization of the Derain (H-10) Quadrangle Intercrater Plains

    Characterization of the Derain (H-10) Quadrangle Intercrater Plains

    2019 Planetary Geologic Mappers 2019 (LPI Contrib. No. 2154) 7016.pdf CAN THE INTERCRATER PLAINS UNIT ON MERCURY BE MEANINGFULLY SUBDIVIDED?: CHARACTERIZATION OF THE DERAIN (H-10) QUADRANGLE INTERCRATER PLAINS. J. L. Whit- ten1, C. I. Fassett2, and L. R. Ostrach3, 1Department of Earth and Environmental Sciences, Tulane University, New Orleans, LA 70118, ([email protected]), 2NASA Marshall Space Flight Center, Huntsville, AL 35805, 3U.S. Ge- ological Survey, Astrogeology Science Center, 2255 N. Gemini Dr., Flagstaff, AZ 86001. IntroDuction: The intercrater plains are the most little agreement about the definition of intercrater and complex and extensive geologic unit on Mercury [1, 2]. intermediate plains. Generally, the intercrater plains are identified as gently It appears that previous researchers were looking for rolling plains with a high density of superposed craters a way to divide up the massive intercrater plains unit by <15 km in diameter [1]. Analyses of the current crater mapping an intermediate unit. This seems like a good population indicate that the intercrater plains experi- idea, however, there was no quantitative measure or de- enced a complex record of ancient resurfacing [3, 4] finitive characteristic used to divide the intercrater (i.e., craters 20–100 km in diameter are missing). This plains from the intermediate plains. Qualitatively, these dearth of larger impact craters could have been caused two geologic units differ in their density of secondary by volcanism or impact-related processes. Various for- craters and their morphology. Intermediate plains have mation mechanisms have been proposed for the inter- a more muted appearance and have been interpreted as crater plains, including volcanic eruptions and basin older smooth plains [13].
  • Layer Potentials for General Linear Elliptic Systems 1

    Layer Potentials for General Linear Elliptic Systems 1

    Electronic Journal of Differential Equations, Vol. 2017 (2017), No. 309, pp. 1{23. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu LAYER POTENTIALS FOR GENERAL LINEAR ELLIPTIC SYSTEMS ARIEL BARTON Abstract. In this article we construct layer potentials for elliptic differential operators using the Babuˇska-Lax-Milgram theorem, without recourse to the fundamental solution; this allows layer potentials to be constructed in very general settings. We then generalize several well known properties of layer potentials for harmonic and second order equations, in particular the Green's formula, jump relations, adjoint relations, and Verchota's equivalence between well-posedness of boundary value problems and invertibility of layer potentials. 1. Introduction There is by now a very rich theory of boundary value problems for the Laplace operator, and more generally for second order divergence form operators − div Ar. The Dirichlet problem − div Aru = 0 in Ω; u = f on @Ω; kukX ≤ CkfkD and the Neumann problem − div Aru = 0 in Ω; ν · Aru = g on @Ω; kukX ≤ CkgkN are known to be well-posed for many classes of coefficients A and domains Ω, and with solutions in many spaces X and boundary data in many boundary spaces D and N. A great deal of current research consists in extending these well posedness results to more general situations, such as operators of order 2m ≥ 4 (for example, [19, 25, 45, 47, 53, 54]; see also the survey paper [22]), operators with lower order terms (for example, [24, 30, 34, 55, 62]) and operators acting on functions defined on manifolds (for example, [46, 50, 51]).
  • Large Impact Basins on Mercury: Global Distribution, Characteristics, and Modification History from MESSENGER Orbital Data Caleb I

    Large Impact Basins on Mercury: Global Distribution, Characteristics, and Modification History from MESSENGER Orbital Data Caleb I

    JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117, E00L08, doi:10.1029/2012JE004154, 2012 Large impact basins on Mercury: Global distribution, characteristics, and modification history from MESSENGER orbital data Caleb I. Fassett,1 James W. Head,2 David M. H. Baker,2 Maria T. Zuber,3 David E. Smith,3,4 Gregory A. Neumann,4 Sean C. Solomon,5,6 Christian Klimczak,5 Robert G. Strom,7 Clark R. Chapman,8 Louise M. Prockter,9 Roger J. Phillips,8 Jürgen Oberst,10 and Frank Preusker10 Received 6 June 2012; revised 31 August 2012; accepted 5 September 2012; published 27 October 2012. [1] The formation of large impact basins (diameter D ≥ 300 km) was an important process in the early geological evolution of Mercury and influenced the planet’s topography, stratigraphy, and crustal structure. We catalog and characterize this basin population on Mercury from global observations by the MESSENGER spacecraft, and we use the new data to evaluate basins suggested on the basis of the Mariner 10 flybys. Forty-six certain or probable impact basins are recognized; a few additional basins that may have been degraded to the point of ambiguity are plausible on the basis of new data but are classified as uncertain. The spatial density of large basins (D ≥ 500 km) on Mercury is lower than that on the Moon. Morphological characteristics of basins on Mercury suggest that on average they are more degraded than lunar basins. These observations are consistent with more efficient modification, degradation, and obliteration of the largest basins on Mercury than on the Moon. This distinction may be a result of differences in the basin formation process (producing fewer rings), relaxation of topography after basin formation (subduing relief), or rates of volcanism (burying basin rings and interiors) during the period of heavy bombardment on Mercury from those on the Moon.
  • Degenerate Parabolic Initial-Boundary Value Problems*

    Degenerate Parabolic Initial-Boundary Value Problems*

    JOURNAL OF DIFFERENTIALEQUATIONS 31,296-312 (1979) Degenerate Parabolic Initial-Boundary Value Problems* R. E. SHOWALTER Department of Mathematics, RLM 8.100, The University of Texas, Austin, Texas 78712 Received September 22, 1977 l. INTRODUCTION We consider a class of implicit linear evolution equations of the form d d-t J[u(t) + 5flu(t) = f(t), t > O, (1.1) in Hilbert space and their realizations in function spaces as initial-boundary value problems for partial differential equations which may contain degenerate or singular coefficients. The Cauchy problem consists of solving (1.1) subject to the initial condition Jdu(0) = h. We are concerned with the case where the solution is given by an analytic semigroup; it is this sense in which the Canchy problem is parabolic. Sufficient conditions for this to be the case are given in Theorem 1; this is a refinement of previously known results [15] to the linear problem and it extends the related work [13] to the (possibly) degenerate situation under consideration. Specifically, we do not assume ~ is invertible, but only that it is symmetric and non-negative. Our primary motivation for considering the Cauchy problem for (1.1) is to show that certain classes of mixed initial-boundary value problems for partial differential equations are well-posed. Theorem 2 shows that if the operators ~/ and 5fl have additional structure which is typical of those operators arising from (possibly degenerate) parabolic problems then the evolution equation (1.1) is equivalent to a partial differential equation d d~ (Mu(t)) -}- Lu(t) -= F(t) (1.2) (obtained by restricting (1.1) to test functions) and a complementary boundary condition d d-~ (8,~u(t)) + 8zu(t) = g(t) (1.3) in an appropriate space of boundary values.
  • International Congress of Mathematics 2002 Satellite

    International Congress of Mathematics 2002 Satellite

    中 央 研 究 院 數 學 研 究 所 INSTITUTE OF MATHEMATICS ACADEMIA SINICA TAIPEI 10617, TAIWAN TEL : 886-2-23685999 FAX : 886-2-23689771 International Congress of Mathematics 2002 Satellite Conference – Taipei 2002 International Conference on Nonlinear Analysis 13 August – 17 August 2002 Institute of Mathematics, Academia Sinica Organized by Nai-Heng Chang 張乃珩 (Academia Sinica) Kin-Ming Hui 許健明 (Academia Sinica) Fon-Che Liu 劉豐哲 (Academia Sinica) Tai-Ping Liu 劉太平 (Academia Sinica) Organized by National Science Council 中 央 研 究 院 數 學 研 究 所 INSTITUTE OF MATHEMATICS ACADEMIA SINICA TAIPEI 10617, TAIWAN TEL : 886-2-23685999 FAX : 886-2-23689771 Table of Contents August 13, Tuesday, 2002 13:30 – 14:30 Kyuya Masuda (Himeji Institute of Technology) Analyticity of solutions for some evolution equations .......................................................................... 1 14:40 – 15:40 Vladimir Maz'ya (Linkoping University) The Schrödinger and the relativistic Schrödinger operators on the energy space: boundedness and compactness criteria ....................................................................................................................... 2 16:00 – 17:00 Mariano Giaquinta (Scuola Normale Superiore, Pisa) Graph currents and applications ............................................................................................................ 3 August 14, Wednesday, 2002 09:00 – 10:00 Claude Bardos (University of Paris 7) Mean field dynamics of N quantum particles, Hartree, Hartree Fock approximations and Coulomb Potential ................................................................................................................................
  • Math 246B - Partial Differential Equations

    Math 246B - Partial Differential Equations

    Math 246B - Partial Differential Equations Viktor Grigoryan version 0.1 - 03/16/2011 Contents Chapter 1: Sobolev spaces 2 1.1 Hs spaces via the Fourier transform . 2 1.2 Weak derivatives . 3 1.3 Sobolev spaces W k;p .................................... 3 1.4 Smooth approximations of Sobolev functions . 4 1.5 Extensions and traces of Sobolev functions . 5 1.6 Sobolev embeddings and compactness results . 6 1.7 Difference quotients . 7 Chapter 2: Solvability of elliptic PDEs 9 2.1 Weak formulation . 9 2.2 Existence of weak solutions of the Dirichlet problem . 10 2.3 General linear elliptic PDEs . 12 2.4 Lax-Milgram theorem, solvability of general elliptic PDEs . 14 2.5 Fredholm operators on Hilbert spaces . 16 2.6 The Fredholm alternative for elliptic equations . 18 2.7 The spectrum of a self-adjoint elliptic operator . 19 Chapter 3: Elliptic regularity theory 21 3.1 Interior regularity . 21 3.2 Boundary regularity . 25 Chapter 4: Variational methods 26 4.1 The Derivative of a functional . 26 4.2 Solvability for the Dirichlet Laplacian . 27 4.3 Constrained optimization and application to eigenvalues . 29 1 1. Sobolev spaces In this chapter we define the Sobolev spaces Hs and W k;p and give their main properties that will be used in subsequent chapters without proof. The proofs of these properties can be found in Evans's\PDE". 1.1 Hs spaces via the Fourier transform Below all the derivatives are understood to be in the distributional sense. Definition 1.1. Let k be a non-negative integer. The Sobolev space HkpRnq is defined as k n 2 n α 2 H pR q tf P L pR q : B f P L for all |α| ¤ ku: k n 2 k p 2 n Theorem 1.2.
  • Six Units for Primary (K-2) Gifted/Talented Students. Self

    Six Units for Primary (K-2) Gifted/Talented Students. Self

    DOCUMENT RESUME ED 333 675 EC 300 431 AUTHOR McCallister, Corliss TITLE Six Units for Primary (K-2) Gifted/Talented Students. Se?f (Psychology), Plants (Botany), Animals (Zoology), Measurement (Mathematics), Space (Astronomy), Computers (Technology). INSTITUTION Education Service Center Region 7, Kilgore, Tex. PUB DATE 88 NOTE 403p. PUB TYPE Guides - Classroom Use - Teaching Guides (For Teacher) (052) EDRS PRICE MF01/PC17 Plus Postage. DESCRIPTORS Animals; Computers; *Curriculum; Diagnostic Teaching; Experiential Learning; *Gifted; Learning Activities; Measurement; Plants (Botany); Primary Education; Self Concept; Space Sciences; *Student Educational Objectives; *Talent; *Teaching Methods ABSTRACT This curriculum for gifted/talented students in kindergarten through grade 2 focuses on the cognitive, affective, and psychomotor domains in the areas of language arts, mathematics, music, physical education (dance), science, social studies, theatre, and visual arts. The curriculum is student centered, experientially based, exploratory, holistic/integrative, and individualized by diagnostic prescriptive teaching. An introductory section provides goals; long-term objectives; and information on adapting the curriculum by kind and degree of giftedness, minority subpopulation, and delivery system. The curriculum covers six units: self, plants, animals, measurement, space, and computers. For each unit, the curriculum contains background information, a chart depicting visual organization of the topics, short-term objectives, field trip ideas, speaker