DOCSLIB.ORG

Maximum Principles and Sharp Constants for Solutions of Elliptic and Parabolic Systems

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Maximum Principles and Sharp Constants for Solutions of Elliptic and Parabolic Systems Mathematical Surveys and Monographs Volume 183 Maximum Principles and Sharp Constants for Solutions of Elliptic and Parabolic Systems Gershon Kresin Vladimir Maz'ya American Mathematical Society http://dx.doi.org/10.1090/surv/183 Maximum Principles and Sharp Constants for Solutions of Elliptic and Parabolic Systems Mathematical Surveys and Monographs Volume 183 Maximum Principles and Sharp Constants for Solutions of Elliptic and Parabolic Systems Gershon Kresin Vladimir Maz'ya American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Ralph L. Cohen, Chair Benjamin Sudakov MichaelA.Singer MichaelI.Weinstein 2010 Mathematics Subject Classification. Primary 35A23, 35B50, 35J47, 35K40; Secondary 31B10, 35J30, 35Q35, 35Q74. The text was translated from Russian by Tatiana O. Shaposhnikova. For additional information and updates on this book, visit www.ams.org/bookpages/surv-183 Library of Congress Cataloging-in-Publication Data Kresin, Gershon, author. Maximum principles and sharp constants for solutions of elliptic and parabolic systems / Gershon Kresin, Vladimir Mazya p. cm. — (Mathematical surveys and monographs ; volume 183) Includes bibliographical references and index. ISBN 978-0-8218-8981-7 (alk. paper) 1. Inequalities (Mathematics) 2. Maximum principles (Mathematics) I. Mazya, V. G., author. II. Title. QA295.K85 2012 515.983–dc23 2012020950 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2012 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 171615141312 Contents Introduction 1 Part 1. Elliptic Equations and Systems 7 Chapter 1. Prerequisites on Operators Acting into Finite Dimensional Spaces 9 1.1. Introduction 9 1.2. Linear bounded operators defined on spaces of continuous vector- valued functions and acting into Rm or Cm 10 1.3. Linear bounded operators defined on Lebesgue spaces of vector-valued functions and acting into Rm or Cm 17 1.4. Comments to Chapter 1 20 Chapter 2. Maximum Modulus Principle for Second Order Strongly Elliptic Systems 21 2.1. Introduction 21 2.2. Systems with constant coefficients without lower order terms 23 2.3. General second order strongly elliptic systems 33 2.4. Comments to Chapter 2 52 Chapter 3. Sharp Constants in the Miranda-Agmon Inequalities for Solutions of Certain Systems of Mathematical Physics 55 3.1. Introduction 55 3.2. Best constants in the Miranda-Agmon inequalities for solutions of strongly elliptic systems in a half-space 58 3.3. The Lam´e and Stokes systems in a half-space 64 3.4. Planar deformed state 69 3.5. The system of quasistatic viscoelasticity 71 3.6. Comments to Chapter 3 75 Chapter 4. Sharp Pointwise Estimates for Solutions of Elliptic Systems with Boundary Data from Lp 77 4.1. Introduction 77 4.2. Best constants in pointwise estimates for solutions of strongly elliptic systems with boundary data from Lp 79 4.3. The Stokes system in a half-space 83 4.4. The Stokes system in a ball 85 4.5. The Lam´e system in a half-space 87 4.6. The Lam´esysteminaball 91 4.7. Comments to Chapter 4 92 v vi CONTENTS Chapter 5. Sharp Constant in the Miranda-Agmon Type Inequality for Derivatives of Solutions to Higher Order Elliptic Equations 93 5.1. Introduction 93 5.2. Weak form of the Miranda-Agmon inequality with the sharp constant 94 5.3. Sharp constants for biharmonic functions 98 5.4. Comments to Chapter 5 104 Chapter 6. Sharp Pointwise Estimates for Directional Derivatives and Khavinson’s Type Extremal Problems for Harmonic Functions 105 6.1. Introduction 105 6.2. Khavinson’s type extremal problem for bounded or semibounded harmonic functions in a ball and a half-space 110 6.3. Sharp estimates for directional derivatives and Khavinson’s type extremal problem in a half-space with boundary data from Lp 117 6.4. Sharp estimates for directional derivatives and Khavinson’s type extremal problem in a ball with boundary data from Lp 131 6.5. Sharp estimates for the gradient of a solution of the Neumann problem in a half-space 145 6.6. Comments to Chapter 6 148 Chapter 7. The Norm and the Essential Norm for Double Layer Vector-Valued Potentials 151 7.1. Introduction 151 7.2. Definition and certain properties of a solid angle 154 7.3. Matrix-valued integral operators of the double layer potential type 161 7.4. Boundary integral operators of elasticity and hydrodynamics 173 7.5. Comments to Chapter 7 197 Part 2. Parabolic Systems 201 Chapter 8. Maximum Modulus Principle for Parabolic Systems 203 8.1. Introduction 203 8.2. The Cauchy problem for systems of order 2 205 8.3. Second order systems 217 8.4. The parabolic Lam´e system 230 8.5. Comments to Chapter 8 235 Chapter 9. Maximum Modulus Principle for Parabolic Systems with Zero Boundary Data 237 9.1. Introduction 237 9.2. The case of real coefficients 238 9.3. The case of complex coefficients 246 9.4. Comments to Chapter 9 249 Chapter 10. Maximum Norm Principle for Parabolic Systems without Lower Order Terms 251 10.1. Introduction 251 10.2. Some notation 255 CONTENTS vii 10.3. Representation of the constant K(Rn,T) 256 10.4. Necessary condition for validity of the maximum norm principle for the system ∂u/∂t − A0(x, t, Dx)u = 0 259 10.5. Sufficient condition for validity of the maximum norm principle for the system ∂u/∂t − A0(x, t, Dx)u = 0 262 10.6. Necessary and sufficient condition for validity of the maximum norm principle for the system ∂u/∂t − A0(x, Dx)u = 0 264 10.7. Certain particular cases and examples 269 10.8. Comments to Chapter 10 275 Chapter 11. Maximum Norm Principle with Respect to Smooth Norms for Parabolic Systems 277 11.1. Introduction 277 11.2. Representation for the constant K(Rn,T) 280 11.3. Necessary condition for validity of the maximum norm principle for the system ∂u/∂t − A(x, t, Dx)u = 0 284 11.4. Sufficient condition for validity of the maximum norm principle for the system ∂u/∂t − A(x, t, Dx)u = 0 with scalar principal part 288 11.5. Criteria for validity of the maximum norm principle for the system ∂u/∂t − A(x, Dx)u = 0. Certain particular cases 291 11.6. Example: criterion for validity of the maximum p-norm principle, 2 <p<∞ 294 11.7. Comments to Chapter 11 296 Bibliography 297 List of Symbols 307 Index 313 Bibliography [Ag1] S. Agmon, Multiple layer potentials and the Dirichlet problem for higher order elliptic equations in the plane, I, Comm. Pure Appl. Math., 10 (1957), 179–239. [Ag2] S. Agmon, Maximum theorems for solutions of higher order elliptic equations, Bull. Amer. Math. Soc., 66 (1960), 77–80. [ADN1] S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I., Comm. Pure Appl. Math., 12 (1959), 623–727. [ADN2] S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, II., Comm. Pure Appl. Math., 17 (1964), 35–92. [AAD] L. Aizenberg, A. Aytuna, and P. Djakov, An abstract approach to Bohr’s phenomenon, Proc. Amer. Math. Soc., 128:9 (2000), 2611–2619. [Alb1] G. Albinus, Uniform estimates in maximum norms and uniqueness of solutions of the Dirichlet problem,Math.Nachr.,89 (1979), 9–16. [Alb2] G. Albinus, Estimates of Miranda-Agmon type in plane domains with corners and existence theorem, in: Constructive Function Theory ’81, Sofia, 197–204. [Ale1] A.D. Alexandrov, Uniqueness conditions and estimates for the solution of the Dirichlet problem,VestnikLGU,18 (1963), 5–29. [Ale2] A.D. Alexandrov, Majorization of second order linear equations,VestnikLGU,21 (1966), 5–25. [Ali1] N. Alikakos, Remarks on invariance in reaction-diffusion equations, Nonlinear Analysis. Theory, Methods & Applications, 5:6 (1981), 593–614. [Ali2] N. Alikakos, Quantitative maximum principle and strongly coupled gradient-like reaction- diffusion systems, Proc. Royal Soc. Edinburg, Sect. A, 94:3-4 (1983), 265–286. [AB] C.D. Aliprantis and K.C. Border, Infinite Dimensional Analysis, 3rd Ed., Springer, New York etc., 2006. [Am] H. Amann, Invariant sets and existence theorems for semilinear parabolic and elliptic sys- tems, J. Math. Anal. Appl., 65 (1978), 432–467. [AKK] T.S. Angell, R.E. Kleinman, and J. Kr´al, Layer potentials on boundaries with corners and edges, Cas.ˇ Pˇest. Mat., 113 (1988), 387–402. [At] F.V. Atkinson, Normal solvability of linear equations in normed spaces,Mat.Sb.,Nov.Ser., 28(70) (1951), 3–14. [ABBO]P.Auscher,L.Barth´elemy, Ph. B´enilan, and E. M. Ouhabaz, Absence de la L∞- contractivit´e pour les semi-groupes associ´es aux op´erateurs elliptiques complexes sous forme divergence, Potential Analysis, 12 (2000), 169–189.
Load more

Recommended publications
  • Scarica Scarica
    Archivi, Biblioteche, Musei L’archivio e la biblioteca di Francesco G. Tricomi ∗ ERIKA LUCIANO - LUISA ROSSO 1. Francesco Giacomo Tricomi1 Nato a Napoli il 5 maggio 1897 in un’agiata famiglia borghese, Tricomi frequenta l’Istituto tecnico locale, dove concepisce una passione per gli studi scientifici grazie all’influenza del suo insegnante di Matematica Alfredo Perna. Conseguito il diploma all’età di appena sedici anni, s’iscrive al corso di laurea in Chimica dell’Università di Bologna. L’anno successivo passa a Fisica e infine, nel 1915-16, torna a Napoli e si immatricola al terz’anno del corso di studi in Matematica. Arruolato nell’autunno del 1916, segue un corso per allievi ufficiali di complemento presso l’Accademia militare di Torino e il 1 aprile 1917, fresco di nomina, è inviato al fronte, dapprima sul Carso e poi nella zona del monte Grappa e del Piave. Nonostante la guerra, riesce comunque a portare avanti gli studi scientifici e consegue la laurea in Matematica a Napoli il 16 aprile 1918, durante una licenza. Terminata la Grande Guerra, Tricomi torna a Napoli all’inizio del 1920, decorato di due croci al merito, e riallaccia i rapporti con alcuni suoi ex docenti, fra cui R. Marcolongo e G. Torelli che lo indirizzano nelle prime ricerche e lo mettono in contatto con G. Fubini. Nel febbraio del 1921, su suggerimento di U. Amaldi, è assunto da F. Severi quale assistente alla cattedra di Geometria analitica presso l’Università di Padova. L’esperienza alla ‘Scuola’ di Severi è tuttavia di breve durata, poiché nei primi mesi del 1922 Tricomi è chiamato a Roma dove consegue la libera docenza in Analisi algebrica e infinitesimale e dove ha modo di inserirsi in un ambiente accademico di eccezionale levatura, che vanta in quegli anni la presenza di illustri matematici, fra cui V.
    [Show full text]
  • Network Map of Knowledge And
    Humphry Davy George Grosz Patrick Galvin August Wilhelm von Hofmann Mervyn Gotsman Peter Blake Willa Cather Norman Vincent Peale Hans Holbein the Elder David Bomberg Hans Lewy Mark Ryden Juan Gris Ian Stevenson Charles Coleman (English painter) Mauritz de Haas David Drake Donald E. Westlake John Morton Blum Yehuda Amichai Stephen Smale Bernd and Hilla Becher Vitsentzos Kornaros Maxfield Parrish L. Sprague de Camp Derek Jarman Baron Carl von Rokitansky John LaFarge Richard Francis Burton Jamie Hewlett George Sterling Sergei Winogradsky Federico Halbherr Jean-Léon Gérôme William M. Bass Roy Lichtenstein Jacob Isaakszoon van Ruisdael Tony Cliff Julia Margaret Cameron Arnold Sommerfeld Adrian Willaert Olga Arsenievna Oleinik LeMoine Fitzgerald Christian Krohg Wilfred Thesiger Jean-Joseph Benjamin-Constant Eva Hesse `Abd Allah ibn `Abbas Him Mark Lai Clark Ashton Smith Clint Eastwood Therkel Mathiassen Bettie Page Frank DuMond Peter Whittle Salvador Espriu Gaetano Fichera William Cubley Jean Tinguely Amado Nervo Sarat Chandra Chattopadhyay Ferdinand Hodler Françoise Sagan Dave Meltzer Anton Julius Carlson Bela Cikoš Sesija John Cleese Kan Nyunt Charlotte Lamb Benjamin Silliman Howard Hendricks Jim Russell (cartoonist) Kate Chopin Gary Becker Harvey Kurtzman Michel Tapié John C. Maxwell Stan Pitt Henry Lawson Gustave Boulanger Wayne Shorter Irshad Kamil Joseph Greenberg Dungeons & Dragons Serbian epic poetry Adrian Ludwig Richter Eliseu Visconti Albert Maignan Syed Nazeer Husain Hakushu Kitahara Lim Cheng Hoe David Brin Bernard Ogilvie Dodge Star Wars Karel Capek Hudson River School Alfred Hitchcock Vladimir Colin Robert Kroetsch Shah Abdul Latif Bhittai Stephen Sondheim Robert Ludlum Frank Frazetta Walter Tevis Sax Rohmer Rafael Sabatini Ralph Nader Manon Gropius Aristide Maillol Ed Roth Jonathan Dordick Abdur Razzaq (Professor) John W.
    [Show full text]
  • Curriculum Vitae
    Umberto Mosco WPI Harold J. Gay Professor of Mathematics May 18, 2021 Department of Mathematical Sciences Phone: (508) 831-5074, Worcester Polytechnic Institute Fax: (508) 831-5824, Worcester, MA 01609 Email: [email protected] Curriculum Vitae Current position: Harold J. Gay Professor of Mathematics, Worcester Polytechnic Institute, Worcester MA, U.S.A. Languages: English, French, German, Italian (mother language) Specialization: Applied Mathematics Research Interests:: Fractal and Partial Differential Equations, Homog- enization, Finite Elements Methods, Stochastic Optimal Control, Variational Inequalities, Potential Theory, Convex Analysis, Functional Convergence. Twelve Most Relevant Research Articles 1. Time, Space, Similarity. Chapter of the book "New Trends in Differential Equations, Control Theory and Optimization, pp. 261-276, WSPC-World Scientific Publishing Company, Hackenseck, NJ, 2016. 2. Layered fractal fibers and potentials (with M.A.Vivaldi). J. Math. Pures Appl. 103 (2015) pp. 1198-1227. (Received 10.21.2013, Available online 11.4.2014). 3. Vanishing viscosity for fractal sets (with M.A.Vivaldi). Discrete and Con- tinuous Dynamical Systems - Special Volume dedicated to Louis Niren- berg, 28, N. 3, (2010) pp. 1207-1235. 4. Fractal reinforcement of elastic membranes (with M.A.Vivaldi). Arch. Rational Mech. Anal. 194, (2009) pp. 49-74. 5. Gauged Sobolev Inequalities. Applicable Analysis, 86, no. 3 (2007), 367- 402. 6. Invariant field metrics and dynamic scaling on fractals. Phys. Rev. Let- ters, 79, no. 21, Nov. (1997), pp. 4067-4070. 7. Variational fractals. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997) No. 3-4, pp. 683-712. 8. A Saint-Venant type principle for Dirichlet forms on discontinuous media (with M.
    [Show full text]
  • Sobolev Spaces on Gelfand Pairs
    Sobolev spaces on Gelfand pairs Mateusz Krukowski Institute of Mathematics, Łódź University of Technology, Wólczańska 215, 90-924 Łódź, Poland e-mail: [email protected] March 20, 2020 Abstract The primary aim of the paper is the study of Sobolev spaces in the context of Gelfand pairs. The article commences with providing a historical overview and motivation for the researched subject together with a summary of the current state of the literature. What follows is a general outline of harmonic analysis on Gelfand pairs, starting with a concept of positive-semidefinite functions, through spherical functions and spherical transform and concluding with the Hausdorff-Young inequality. The main part of the paper introduces the notion of Sobolev spaces on Gelfand pairs and studies the prop- erties of these spaces. It turns out that Sobolev embedding theorems and Rellich-Kondrachov theorem still hold true in this generalized context (if certain technical caveats are taken into consideration). Keywords : Gelfand pairs, spherical transform, Hausdorff-Young inequality, Sobolev spaces, Sobolev embedding theorems, Rellich-Kondrachov theorem, bosonic string equation AMS Mathematics Subject Classification : 43A15, 43A32, 43A35, 43A40, 43A90 1 Introduction It is difficult to imagine a mathematician working in a field of differential equations, who has not heard of Sobolev spaces. These spaces are named after Sergei Sobolev (1908 ´ 1989), although they were known before the rise of the Russian mathematician to academic stardom. In 1977 Gaetano Fichera wrote arXiv:2003.08519v1 [math.FA] 19 Mar 2020 (comp. [32]): “These spaces, at least in the particular case p “ 2, were known since the very beginning of this century, to the Italian mathematicians Beppo Levi and Guido Fubini who investigated the Dirichlet minimum principle for elliptic equations.” According to Fichera, at the beginning of the fifties, a group of French mathematicians decided to dub the spaces in question and they came up with the name “Beppo Levi spaces”.
    [Show full text]
  • Unione Matematica Italiana 1921 - 2012
    Unione Matematica Italiana 1921 - 2012 Unione Matematica Italiana Soggetto produttore Unione Matematica Italiana Estremi cronologici 1921 - Tipologia Ente Tipologia ente ente di cultura, ricreativo, sportivo, turistico Profilo storico / Biografia La fondazione dell’Unione Matematica Italiana è strettamente legata alla costituzione del Consiglio Internazionale delle Ricerche e al voto da questo formulato a Bruxelles nel luglio 1919 nel quale si auspicava il sorgere di «Unioni Internazionali» per settori scientifici, ai quali avrebbero dovuto aderire dei «Comitati nazionali» costituiti «ad iniziativa delle Accademie nazionali scientifiche o dei Consigli Nazionali delle Ricerche». In Italia, non essendo ancora stato costituito il C.N.R., fu l’Accademia dei Lincei, nella persona del suo presidente Vito Volterra, a farsi promotrice della costituzione dei Comitati nazionali. Per il Comitato della matematica Vito Volterra propose nel 1920, insieme a un gruppo di matematici, fra cui Luigi Bianchi, Pietro Burgatti, Roberto Marcolongo, Carlo Somigliana e Giovanni Vacca, la costituzione di una Unione Matematica Italiana, redigendo un primo schema di programma che poneva fra gli scopi dell’Unione l’incoraggiamento alla scienza pura, il ravvicinamento tra la matematica pura e le altre scienze, l’orientamento ed il progresso dell’insegnamento e l’organizzazione, la preparazione e la partecipazione a congressi nazionali ed internazionali. La proposta fu accolta dall’Accademia dei Lincei nel marzo 1921 e quale presidente dell’Unione venne designato Salvatore Pincherle, illustre matematico dell'Università di Bologna. L’Unione Matematica Italiana (UMI) nacque ufficialmente il 31 marzo 1922 con la diffusione da parte di Salvatore Pincherle di una lettera con la quale presentava il programma della Società invitando i destinatari ad aderire all’iniziativa.
    [Show full text]
  • GUIDO STAMPACCHIA Silvia Mazzone
    1 GUIDO STAMPACCHIA Silvia Mazzone 1. Formazione scientifica e prima attività di ricerca alla Scuola Normale Superiore di Pisa e all’Università di Napoli. Guido Stampacchia nasce a Napoli, nel quartiere Chiaia, il 26 marzo 1922 da Emanuele e Giulia Campagnano. Giulia, di religione ebraica,1 apparteneva ad una famiglia di origini fiorentine che aveva un laboratorio di biancheria ricamata a mano; gli Stampacchia, invece, erano una famiglia di origine leccese ed osservavano la religione valdese. Il papà, Emanuele, gestiva una fabbrica di ferramenta che sarà costretto a vendere al tempo della guerra in Etiopia, come conseguenza del suo rifiuto a prendere la tessera del partito fascista. Il giovane Guido riceve una educazione essenzialmente laica anche se, da bambino, insieme alle due sorelle frequenta la chiesa valdese. Egli consegue la maturità classica a 18 anni, nel 1940, al Liceo-Ginnasio Gian Battista Vico di Napoli riportando come unico voto di eccellenza 9 in matematica e fisica. Nonostante gli studi classici, aveva chiara l'intenzione di dedicarsi alla matematica e perciò aveva approfondito per suo conto la preparazione di matematica e di fisica studiando “i capisaldi del programma di Liceo Scientifico, cercando … di intravederne il processo logico”2. Nell’autunno del 1940 è ammesso come alunno interno alla Scuola Normale Superiore di Pisa, classe di Scienze, corso di laurea in matematica pura, essendo riuscito quinto al concorso3 e, nei tre anni successivi, supera brillantemente tutti gli esami previsti dal piano di studio assolvendo agli obblighi cui sono tenuti i normalisti. In particolare all’università ha come docenti Francesco Cecioni e Salvatore Cherubino per gli insegnamenti del primo biennio mentre il terzo anno frequenta il corso di Analisi superiore di Leonida Tonelli e quello di Teoria delle funzioni di Lamberto Cesari.
    [Show full text]
  • In Memory of Gaetano Fichera Wolfgang L. Wendland
    LE MATEMATICHE Vol. LXII (2007) – Fasc. II, pp. 7–9 IN MEMORY OF GAETANO FICHERA WOLFGANG L. WENDLAND Dear Dr. Matelda Fichera, Prof. Dr. Maria Pia Colautti, Dr. Anna Maria Fichera, Dr. Massimo Fichera, Gaetano Fichera passed away at the age of 74 June 1st just 10 years ago. He left behind his dear wife Dr. Mathelda Fichera after 44 years of happy to- getherness - and the world of mathematics. I feel very honored to be asked to say a few words at this beautiful place not too far from his birth place. Personally I had had the privilege to meet the famous Gaetano Fichera the first time at the Oberwolfach conference on partial differential equations in 1969 where he gave a lecture on Orthogonal invariants of elliptic operators in the variety of Hilbert spaces and gave estimates on eigenvalues. He was an extraor- dinary, real gentleman, very handsome and impressive, being polite and kind to us youngsters; caring even for my work and giving advice in the discussions. His lectures were tastefully prepared and delivered in elegance with brilliant clarity. The Sicilian gentleman! In 1970 at the Dundee conference on ordinary and partial differential equations organized by Norrie Everitt I could listen to Gaetano’s four lectures in a series on “Some topics in the theory of partial dif- ferential equations” and looking back, this seems like becoming a guideline of my own work. At the next Dundee conference in 1772 I have met him again; vividly I re- member his lectures on “Abstract unilateral problems” corresponding to his fa- Entrato in redazione 1 gennaio 2007 8 WOLFGANG L.
    [Show full text]
  • Alfio Quarteroni
    An interview with Alfio Quarteroni Conducted by Philip Davis on 22 June, 2004 Brown University, Providence, Rhode Island Interview conducted by the Society for Industrial and Applied Mathematics, as part of grant # DE-FG02-01ER25547 awarded by the US Department of Energy. Transcript and original tapes donated to the Computer History Museum by the Society for Industrial and Applied Mathematics © Computer History Museum Mountain View, California ABSTRACT: Professor Alfio Quarteroni discusses a range of topics in this interview with Phil Davis, including his work on finite elements, spectral methods, and domain decomposition. He began studying economic issues in a more professional school before switching over to study mathematics. Quarteroni received his Ph.D. from University of Pavia in Italy, completing a thesis on finite methods under Franco Brezzi. He finished it up in Paris, at the Laboratoire d'Analyse Numérique (now called the Laboratorie Jacques-Louis Lions), where he was encouraged to begin studying spectral methods. Professor Enrico Magenes, who dominated mathematics at Pavia, also exerted a strong influence on Quarteroni. Quarteroni observes how numerical methods have gone from obscure, rarely-used tools to common ones, although they are still not popular in some quarters of the engineering community. Quarteroni notes that low-order methods such as finite elements and higher- order methods such as spectral methods seem to be merging. Numerical methods are necessarily tied to experimental data. One of the challenges, especially in the airplane industry, is multi-objective optimization involving various fields of mathematics and physics. Quarteroni later became interested in domain decomposition, an area that became important with the rise of parallel computing but which are now important in a variety of fields, including the design of racing yachts.
    [Show full text]
  • NBS-INA-The Institute for Numerical Analysis
    t PUBUCATiONS fl^ United States Department of Commerce I I^^^V" I ^1 I National Institute of Standards and Tectinology NAT L INST. OF STAND 4 TECH R.I.C. A111D3 733115 NIST Special Publication 730 NBS-INA — The Institute for Numerical Analysis - UCLA 1947-1954 Magnus R, Hestenes and John Todd -QC- 100 .U57 #730 1991 C.2 i I NIST Special Publication 730 NBS-INA — The Institute for Numerical Analysis - UCLA 1947-1954 Magnus R. Hestenes John Todd Department of Mathematics Department of Mathematics University of California California Institute of Technology at Los Angeles Pasadena, CA 91109 Los Angeles, CA 90078 Sponsored in part by: The Mathematical Association of America 1529 Eighteenth Street, N.W. Washington, DC 20036 August 1991 U.S. Department of Commerce Robert A. Mosbacher, Secretary National Institute of Standards and Technology John W. Lyons, Director National Institute of Standards U.S. Government Printing Office For sale by the Superintendent and Technology Washington: 1991 of Documents Special Publication 730 U.S. Government Printing Office Natl. Inst. Stand. Technol. Washington, DC 20402 Spec. Publ. 730 182 pages (Aug. 1991) CODEN: NSPUE2 ABSTRACT This is a history of the Institute for Numerical Analysis (INA) with special emphasis on its research program during the period 1947 to 1956. The Institute for Numerical Analysis was located on the campus of the University of California, Los Angeles. It was a section of the National Applied Mathematics Laboratories, which formed the Applied Mathematics Division of the National Bureau of Standards (now the National Institute of Standards and Technology), under the U.S.
    [Show full text]
  • Analysis, Partial Differential Equations and Applications: The
    Operator Theory: Advances and Applications Vol. 193 Editor: I. Gohberg Editorial Office: School of Mathematical Sciences V. Olshevski (Storrs, CT, USA) Tel Aviv University M. Putinar (Santa Barbara, CA, USA) Ramat Aviv A.C.M. Ran (Amsterdam, The Netherlands) Israel L. Rodman (Williamsburg, VA, USA) J. Rovnyak (Charlottesville, VA, USA) B.-W. Schulze (Potsdam, Germany) F. Speck (Lisboa, Portugal) Editorial Board: I.M. Spitkovsky (Williamsburg, VA, USA) D. Alpay (Beer Sheva, Israel) S. Treil (Providence, RI, USA) J. Arazy (Haifa, Israel) C. Tretter (Bern, Switzerland) A. Atzmon (Tel Aviv, Israel) H. Upmeier (Marburg, Germany) J.A. Ball (Blacksburg, VA, USA) N. Vasilevski (Mexico, D.F., Mexico) H. Bart (Rotterdam, The Netherlands) S. Verduyn Lunel (Leiden, The Netherlands) A. Ben-Artzi (Tel Aviv, Israel) D. Voiculescu (Berkeley, CA, USA) H. Bercovici (Bloomington, IN, USA) D. Xia (Nashville, TN, USA) A. Böttcher (Chemnitz, Germany) D. Yafaev (Rennes, France) K. Clancey (Athens, GA, USA) R. Curto (Iowa, IA, USA) K. R. Davidson (Waterloo, ON, Canada) Honorary and Advisory Editorial Board: M. Demuth (Clausthal-Zellerfeld, Germany) L.A. Coburn (Buffalo, NY, USA) A. Dijksma (Groningen, The Netherlands) H. Dym (Rehovot, Israel) R. G. Douglas (College Station, TX, USA) C. Foias (College Station, TX, USA) R. Duduchava (Tbilisi, Georgia) J.W. Helton (San Diego, CA, USA) A. Ferreira dos Santos (Lisboa, Portugal) T. Kailath (Stanford, CA, USA) A.E. Frazho (West Lafayette, IN, USA) M.A. Kaashoek (Amsterdam, The Netherlands) P.A. Fuhrmann (Beer Sheva, Israel) P. Lancaster (Calgary, AB, Canada) B. Gramsch (Mainz, Germany) H. Langer (Vienna, Austria) H.G. Kaper (Argonne, IL, USA) P.D.
    [Show full text]
  • HOMOTECIA No. 11-12 (Noviembre 2014
    HOMOTECIA Nº 11 – Año 12 Lunes, 3 de Noviembre de 2014 1 La mayoría de los docentes de matemática aparentemente se caracterizan por su deseo de obtener rápidamente buenos resultados en el aprendizaje de los estudiantes que atienden. Así, al trabajar un objetivo en función de una estrategia con anterioridad informada a los alumnos, si en los primeros momentos los resultados no son satisfactorios o son deficientes, muchos de ellos intentan dar un vuelco a su metodología en procura de mejores resultados. Esto posiblemente sea un error, sobre todo si es en un momento intermedio del periodo lectivo, puesto que significa cambios GOTTLOB FREGE en el orden previamente establecido, lo que posiblemente (1848 - 1925) creará confusión en el estudiante. En la relación docente- Nació el 8 de noviembre de 1848 en Wismar, Mecklemburgo- alumno, un cambio intempestivo en la metodología puede no Pomerania Occidental; y murió el 26 de julio de 1925 en Bad dar origen a mejores logros. Es decir, el docente quizás ayude Kleinen, ambas localidades en Alemania. a que se suceda un fallo después de otro fallo. La experiencia da evidencia que todo docente, ante la expectativa del inicio de Fue uno de los fundadores de la lógica simbólica moderna, un nuevo curso, sobre todo en el caso de matemática, revisa el planteando la posición en cuanto a que es reducible a la lógica contenido a trabajar, evalúa los resultados que obtuvo en el matemática. curso anterior con la finalidad de mejorar las estrategias utilizadas, se auto critica responsablemente en cuanto al Friedrich Ludwig Gottlob Frege. Los padres de Gottlob Frege desarrollo de la labor realizada y después de analizar otros fueron Alexander Frege y Augusta Bialloblotzky.
    [Show full text]
  • Preface: a Beautiful Walk in the Way of the Understanding
    DISCRETE AND CONTINUOUS doi:10.3934/dcds.2011.31.4i DYNAMICAL SYSTEMS Volume 31, Number 4, December 2011 pp. i{vi PREFACE: A BEAUTIFUL WALK IN THE WAY OF THE UNDERSTANDING The International Conference \Variational Analysis and Applications" devoted to the memory of Ennio De Giorgi and Guido Stampacchia was held at the Inter- national School of Mathematics in Erice from May 9 to May 17 of 2009. About thirty lecturers from every part of the world took part in the conference and the workshop was enriched by the award of the \Third Gold Medal G. Stampacchia" to the young researcher Camillo de Lellis. Some of the lecturers, together with their well-known students, presented a paper for this special issue in honour of Ennio De Giorgi and Guido Stampacchia. It is not a case that the names of these two mathematicians appear together. Guido Stampacchia, during a mountain excursion, discussed with Ennio De Giorgi about the 19th Hilbert Problem which at that time was unsolved, trying to focus the steps necessary in order to obtain its solution. Some weeks later De Giorgi was able to provide the solution of the problem. Ennio De Giorgi in this event gave a proof of his way of conceiving mathematics. He believed that the natural attitude of mathematicians must be based both on humility, namely on the awareness of the possibility of being wrong and thus on paying attention to the remarks, results and criticism of other people. At the same time he believed that even a modest person can give a great contribution to an universal truth.
    [Show full text]