DOCSLIB.ORG

Preface: a Beautiful Walk in the Way of the Understanding

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read 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. The trust of De Giorgi on the possibility to reach the truth was based on his belief that mathematics, as the other sciences, is a branch of the tree of the wisdom which \has built her house, has hewn out its seven pillars, has prepared her meat and mixed her wine, has set her table, has sent out her maids and calls from the highest point of the city: let all who are simple come in here, come, eat my food and drink the wine I have mixed, walk in the way of understanding " Prov. 9, 1-6. This serene and convivial nature of the wisdom agrees perfectly with the be- haviour of the whole life of Ennio De Giorgi who always liked to share with his students and colleagues all the steps of the research process, namely choice of the problem, formulation of the model, achievement of mathematical solutions and its validation. Developing this research process usually is extremely demanding and without guarantee of success. To give an idea of the challenge that this result rep- resents is worth to provide the statement of the theorem on which the resolution of the 19th problem by D. Hilbert is based (we update the language) (see[1], [2]). Theorem 0.1. Let w(x) 2 H1(E) a function and for each y 2 E and for each 0 < ρ < δ(y) = d(y n Rn n E), let us denote by I(ρ, y) the ball of radius ρ centered in y, by A(k) the subset of E in which w(x) > k and by B(k) the subset of E in which w(x) < k. Let us assume that there exists a constant γ such that, for each y 2 E and for all real numbers k; ρ1; ρ2 with 0 < ρ1 < ρ2 < δ(y) we have i ii PREFACE Z Z 2 γ 2 jgrad wj dx ≤ 2 (w(x) − k) dx A(k)\I(ρ1;y) (ρ1 − ρ2) A(k)\I(ρ1;y) Z Z 2 γ 2 jgrad wj dx ≤ 2 (w(x) − k) dx: B(k)\I(ρ1;y) (ρ1 − ρ2) B(k)\I(ρ1;y) Then the function w(x) is uniformly H¨older-continuous on each compact subset of E. It is worth mentioning the deep difference between the De Giorgi approach to the solution of the 19th problem and the J.F. Nash one, (see [4]), and that, as O.A. Ladyzhenskaya et al. say in [5], page IX of the preface, \a new stage in the study of many-dimensional linear equations started with the articles by De Giorgi and Nash". However, the same authors of [5], say at page XIV of the preface that \the only work that had a direct influence on the authors is the article of De Giorgi from which they used the idea of treating solutions u(x) on the sets A(k) \ I(ρ, y) and one simple though very useful inequality". As it is well-known, the attempts to extend to systems the H¨oldercontinuity results were vain. In this regard De Giorgi himself provided a counterexample which shows that, in the case of systems of elliptic equations with L1 coefficients, the solution may not be regular. Here we can appreciate De Giorgi's thinking about the failures in the research work. According to him, the failures, even if painful and wearing, are not less useful than the successes, because they can address to new paths and enrich our knowledge. Exactly this happens with De Giorgi's counterexample which is the following. De Giorgi's Counterexample. Let Ω be the ball B(0; σ) in Rn, n ≥ 3 and let us consider the strongly elliptic system n X hk k DiAij Dju = 0 k = 1;:::;NN = n = 3 i;j=1 where hk h k Aij (x) = Bi (x)Bj (x) + δijδhk; x x Bh(x) = (n − 2)δ + n i k : i ik kxk2 Then the function " # x n 1 u = with α = 1 − kxkα 2 p(2n − 2)2 + 1 belongs to H1(Ω; Rn) and is a solution of the above system. The above counterexample shows that the solution is neither continuous for x = 0 nor bounded in a neighbourhood of x = 0. Starting from this remark the deep theory of quasi-regularity for elliptic and parabolic systems arose. The core of this theory is that for a multidimensional system a singular set might exist and only for low values of the dimension it is necessarily empty. Under mild assumption, however, it is always Lebesgue negligible, regardless of the dimension. For the main results on partial regularity theory we refer for instance to the papers [6]- [11], to the references quoted in [11] and also we point out the results contained in [6], [7], [8], [10] for what concerns the estimate of the Hausdorff measure of suitable PREFACE iii dimension of the singular set. As we can see from the references, De Giorgi was not only concerned with the study of partial regularity, but he was interested in a problem - at that time poorly understood at least in high dimensions - the problem of minimal surfaces. In the cartesian form, if Ω is a domain of Rn, the functional connected to the problem of the minimal surfaces is Z A(u) = p1 + jDuj2dx Ω and the equation of the minimal surfaces is n X Diu Di = 0: p 2 i=1 1 + jDuj The first open problem was to prove an existence theorem for the n-dimensional minimal surfaces problem and, preliminary to this study, the first question to solve was to find a suitable class of measurable surfaces where to set the problem. Here De Giorgi took advantage of the pioneering contributions to this problem given by Caccioppoli in [12] and of his own researches (see [27]-[17]), which led him to intro- duce the class of \sets of finite perimeter". In this broad class it is not difficult to show the existence of a minimal surface, but a priori this surface can be irregular and De Giorgi himself showed that the singular sets of minimal n-dimensional sur- faces has zero surface measure. The problem to solve was if and when such singular points do exist. De Giorgi and W. Fleming realize that the problem was closely related to the existence of singular (i.e. non flat) minimal cones. As a matter of fact the tangent plane at the regular points of the minimal surface is flat, whereas at the singular points it is a singular n-dimensional minimal cone. In this study De Giorgi involved some younger collaborators, E. Bombieri, E. Giusti, M. Miranda and they obtained the first significant result for the problem in cartesian form, showing an a priori estimate for the gradient of the function u(x) and hence the impossibility of the existence of a tangent cone (see [19]). The impossibility of the existence of the tangent cone remains in the fact that the tangent cone in cartesian form must be vertical and hence the gradient should be unbounded. Further the existence of minimal cone was crucial not only for the regularity of minimal surfaces but also to decide whether the Bernstein theorem holds true in dimension n > 2, namely if a minimal surface in the whole Rn is a hyperplane. In fact in the year 1962 W. Fleming showed that the existence of a minimal surface on Rn different from a hy- perplane has, as a consequence, the existence of an n-dimensional minimal cone. De Giorgi improved this result showing the existence of a (n − 1)-dimensional minimal cone. In the paper [20] J. Simons showed that minimal cones of dimension less than 7 cannot exist and gave a suggestion about a 7-dimensional cone in 8 dimensions which he conjectured could be minimal. In the paper [18] Bombieri, De Giorgi and Giusti solved the question showing that the Simons cone is minimal and provided a counterexample to Bernstein's theorem in dimension 8.
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]
  • Meetings & Conferences of the AMS, Volume 49, Number 5
    Meetings & Conferences of the AMS IMPORTANT INFORMATION REGARDING MEETINGS PROGRAMS: AMS Sectional Meeting programs do not appear in the print version of the Notices. However, comprehensive and continually updated meeting and program information with links to the abstract for each talk can be found on the AMS website.See http://www.ams.org/meetings/. Programs and abstracts will continue to be displayed on the AMS website in the Meetings and Conferences section until about three weeks after the meeting is over. Final programs for Sectional Meetings will be archived on the AMS website in an electronic issue of the Notices as noted below for each meeting. Niky Kamran, McGill University, Wave equations in Kerr Montréal, Quebec geometry. Rafael de la Llave, University of Texas at Austin, Title to Canada be announced. Centre de Recherches Mathématiques, Special Sessions Université de Montréal Asymptotics for Random Matrix Models and Their Appli- cations, Nicholas M. Ercolani, University of Arizona, and May 3–5, 2002 Kenneth T.-R. McLaughlin, University of North Carolina at Chapel Hill and University of Arizona. Meeting #976 Combinatorial Hopf Algebras, Marcelo Aguiar, Texas A&M Eastern Section University, and François Bergeron and Christophe Associate secretary: Lesley M. Sibner Reutenauer, Université du Québec á Montréal. Announcement issue of Notices: March 2002 Combinatorial and Geometric Group Theory, Olga G. Khar- Program first available on AMS website: March 21, 2002 lampovich, McGill University, Alexei Myasnikov and Program issue of electronic Notices: May 2002 Vladimir Shpilrain, City College, New York, and Daniel Issue of Abstracts: Volume 23, Issue 3 Wise, McGill University. Commutative Algebra and Algebraic Geometry, Irena Deadlines Peeva, Cornell University, and Hema Srinivasan, Univer- For organizers: Expired sity of Missouri-Columbia.
    [Show full text]
  • Partial Differential Equations
    Rend. Lincei Mat. Appl. 20 (2009), 195–205 Partial Di¤erential Equations — A Calderon-Zygmund theory for infinite energy minima of some integral functionals,byLucio Boccardo, communicated on 8 May 2009. ‘‘. onore e lume vagliami ’l lungo studio e ’l grande amore che m’ha fatto cercar lo tuo volume.’’ (Inferno I, 82–84)1 Abstract. — A Calderon-Zygmund theory in Lebesgue and Marcinkiewicz spaces for infinite energy minima of some integral functionals is proved. Key words: Calderon-Zygmund theory; Infinite energy minima. AMS 2000 Mathematics Subject Classification: 49.10, 35J60. Acknowledgements This paper contains the unpublished part of the lecture held at the Conference ‘‘Recent Trends in Nonlinear Partial Di¤erential Equations (6.11.2008)’’. I would like to express my thanks to Accademia dei Lincei for the invitation to give a talk2 as a ‘‘tribute to Guido Stampacchia on the 30th anniversary of his death’’. 1. Introduction 1.1. Finite energy solutions. We recall the following regularity theorem by G. Stampacchia concerning solutions of linear Dirichlet problems in W, bounded subset of RN , N > 2, with right hand side a measurable function f ðxÞ. Consider a bounded elliptic matrix MðxÞ with ellipticity constant a > 0 and the related boundary value problem ÀdivðMðxÞDuÞ¼ f ðxÞ in W; ð1Þ u ¼ 0onqW: 1Always remembering (after 40 years) what I have learned from my teacher of ‘‘Istituzioni di Ana- lisi Superiore’’ and ‘‘Analisi Superiore’’ ([7] pg. 1, [8] pg. 1) 2... e piu´ d’onore ancora assai mi fenno, ch’e’ sı´ mi fecer de la loro schiera, sı´ ch’io fui sesto tra cotanto senno.
    [Show full text]
  • Arxiv:0707.3327V1 [Math.AP] 23 Jul 2007 Some Connections Between
    Some connections between results and problems of De Giorgi, Moser and Bangert Hannes Junginger-Gestrich & Enrico Valdinoci∗ October 31, 2018 Abstract Using theorems of Bangert, we prove a rigidity result which shows how a question raised by Bangert for elliptic integrands of Moser type is connected, in the case of minimal solutions without self-intersections, to a famous conjecture of De Giorgi for phase transitions. 1 Introduction The purpose of this note is to relate some probelms posed by Moser [Mos86], Bangert [Ban89] and De Giorgi [DG79]. In particular, we point out that a rigidity result in a question raised by Bangert for the case of minimal solutions of elliptic integrands would imply a one-dimensional symmetry for minimal phase transitions connected to a famous conjecture of De Giorgi. Though the proofs we present here are mainly a straightening of the existing literature, we hope that our approach may clarify some points in these important problems and provide useful connections. 1.1 The De Giorgi setting arXiv:0707.3327v1 [math.AP] 23 Jul 2007 A classical phase transition model (known in the literature under the names of Allen, Cahn, Ginzburg, Landau, van der Vaals, etc.) consists in the study ∗Addresses: HJG, Mathematisches Institut, Albert-Ludwigs-Universit¨at Freiburg, Abteilung f¨ur Reine Mathematik, Eckerstraße 1, 79104 Freiburg im Breisgau (Germany), [email protected], EV, Dipartimento di Matemati- ca, Universit`adi Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma (Italy), [email protected]. The work of EV was supported by MIUR Variational Meth- ods and Nonlinear Differential Equations.
    [Show full text]
  • Luis Caffarelli Lecture Notes
    Luis Caffarelli Lecture Notes Broddie often melodizes stunningly when tautological Robert misforms robustiously and christens her irade. Stuffed Wojciech manoeuvre palatially, he overcrop his antibodies very sorrowfully. Sopping Giffy mystifies oracularly. The sponsors and was a review team, lecture notes that reflects exceptional mathematical analysis this item to seeing people around topics discussed are lipschitz Dirección de correo verificada de sissa. Communications on homeland and Applied Mathematics. The stamp problem it the biharmonic operator. The wreck of this lecture is void discuss three problems in homogenization and their interplay. Nirenberg should phone to study theoretical physics. Walter was a highly regarded researcher in analysis and control theory. Fundamental solutions of homogeneous fully nonlinear elliptic equations. Regularity of news free accident with application to the Pompeiu problem. Talks were uniformly excellent. Proceedings of the International Congress of Mathematicians. The nonlinear character while the equations is used in gravel essential way, whatsoever he obtains results because are the nonlinearity not slate it. Caffarelli, Luis; Silvestre, Luis. Don worked with luis caffarelli lecture notes in many years ago with students, avner variational problems in any book contains the error has had three stellar researchers. The smoothness of separate free surface tablet a filtration problem. Nike sneaker in above one week. The continuity of the temperature in the Stefan problem. Axially symmetric infinite cavities. The subtle problem every two fluids. On the regularity of reflector antennas. Nav start or be logged at this place although if instant is NOT progressively loaded. Laplacian and the water obstacle problem. Growing up with luis caffarelli lecture notes in.
    [Show full text]
  • L' Addio a Un Grande Matematico
    CAPITOLO 1 L' ADDIO A UN GRANDE MATEMATICO Si riportano i discorsi pronunciati il 27 ottobre 1996 nel cortile della Scuola Normale Superiore di Pisa, in occasione del commiato accademico. Nello stesso giorno, presso la Chiesa di S. Frediano (Pisa) si `e tenuto il fu- nerale, officiato dal teologo Severino Dianich; il giorno dopo presso la Basilica di S. Croce (Lecce) il funerale `e stato officiato dall' Arcivescovo di Lecce, Cosmo Francesco Ruppi. 1.1 DISCORSO DI L. MODICA Intervento di Luciano Modica, allievo di De Giorgi e Rettore dell' Universita` di Pisa. Confesso che quando Franco Bassani e Luigi Radicati mi hanno chiesto di prendere la parola oggi durante questo triste e solenne commiato acca- demico da Ennio De Giorgi, la mia prima reazione `e stata quella di tirarmi indietro, temendo che l' empito della commozione e dei ricordi dell' allie- vo sopraffacessero la partecipazione, certo commossa, ma necessariamente composta, di chi qui `e chiamato da Rettore a rappresentare l' Ateneo pisa- no e la sua comunita` di studenti e docenti. Se poi ho accettato, non `e stato perch´e, sono sicuro di superare questo timore, ma perch´e spero che tutti voi familiari, allievi, amici di Ennio, saprete comprendere e scusare l' emotivita` da cui forse non riusciro` ad evitare che sia pervaso il tono delle mie parole. Perch´e la vostra presenza in questo cortile, le cui soavi linee architettoniche tanto Ennio ha amato e che rimangono per tanti dei presenti indissolubil- mente legate alla loro giovinezza, non ha nulla del dovere accademico, se 2 L' ADDIO A UN GRANDE MATEMATICO non i suoi aspetti spirituali piu` alti, mentre invece vuole manifestare la ri- conoscenza e l' affetto tutti umani verso una persona accanto a cui abbiamo avuto il privilegio di trascorrere un periodo piu` o meno lungo, ma sempre indimenticabile, della nostra vita.
    [Show full text]
  • The Bernstein Theorem in Higher Dimensions
    The Bernstein Theorem in Higher Dimensions Umberto Massari,∗ Mario Mirandayand Michele Miranda Jr ∗ Dedicated to the memory of Guido Stampacchia Abstract. { In this work we have reconsidered the famous paper of Bombieri, De Gior- gi and Giusti [4] and, thanks to the software Mathematicar we made it possible for anybody to control the difficult computations. 1. { Introduction On April 27th, 1978, Guido Stampacchia died in Paris. His coffin arrived to Pisa for the Adieu Ceremonial, in the Courtyard of Scuola Normale. That day I was there, and I could not meet Ennio De Giorgi. The apparent absence of him seemed impossible to me, considering the lasting and close friendship between Ennio and Guido. Many years later, Sergio Spagnolo told me that Ennio remained, the whole duration of the Ceremonial, hidden in a corner of the building. The modesty did not permit Ennio to show around his sorrow. Guido had been a putative father for Ennio; he protected a younger and precious man, as long as he could. In March 1961, De Giorgi was invited to Paris for giving two talks. Stampacchia wanted to join him in the trip. There exists a beautiful picture of the two friends, on the Rive Gauche with N^otreDame behind them. A French mathematician told me that Stampacchia in Paris seemed to be the Assistant Professor of De Giorgi. He was wrong. Stampacchia had never been an academic servant. He could be said a devotee of Mathematics. In 1955 Guido pushed Ennio into studying the XIX Hilbert's Problem [5]. And, in 1968 Guido pushed Enrico Bombieri to work with De Giorgi.
    [Show full text]
  • Sets with Finite Perimeter in Wiener Spaces, Perimeter Measure and Boundary Rectifiability
    DISCRETE AND CONTINUOUS doi:10.3934/dcds.2010.28.591 DYNAMICAL SYSTEMS Volume 28, Number 2, October 2010 pp. 591–606 SETS WITH FINITE PERIMETER IN WIENER SPACES, PERIMETER MEASURE AND BOUNDARY RECTIFIABILITY Luigi Ambrosio Scuola Normale Superiore p.za dei Cavalieri 7 Pisa, I-56126, Italy Michele Miranda jr. Dipartimento di Matematica, Universit`adi Ferrara via Machiavelli 35 Ferrara, I-44121, Italy Diego Pallara Dipartimento di Matematica “Ennio De Giorgi”, Universit`a del Salento C.P. 193 Lecce, I-73100, Italy Abstract. We discuss some recent developments of the theory of BV func- tions and sets of finite perimeter in infinite-dimensional Gaussian spaces. In this context the concepts of Hausdorff measure, approximate continuity, recti- fiability have to be properly understood. After recalling the known facts, we prove a Sobolev-rectifiability result and we list some open problems. 1. Introduction. This paper is devoted to the theory of sets of finite perimeter in infinite-dimensional Gaussian spaces. We illustrate some recent results, we provide some new ones and eventually we discuss some open problems. We start first with a discussion of the finite-dimensional theory, referring to [13] and [3] for much more on this subject. Recall that a Borel set E Rm is said to be of ⊂ finite perimeter if there exists a vector valued measure DχE = (D1χE,...,DmχE) with finite total variation in Rm satisfying the integration by parts formula: ∂φ 1 m dx = φ dDiχE i =1,...,m, φ Cc (R ). (1) ∂x − Rm ∀ ∀ ∈ ZE i Z De Giorgi proved in [10] (see also [11]) a deep result on the structure of DχE , which could be considered as the starting point of modern Geometric Measure Theory.
    [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]
  • Opere Complete E Selecta)
    OPERE DI GRANDI MATEMATICI Da circa 45 anni, l’UMI cura la pubblicazione di Opere dei Grandi Matematici Italiani (Opere complete e Selecta). La pubblicazione viene decisa da parte della Commissione Scientifica dell’UMI, sentito il parere di una commissione da essa all’uopo costituita. (Ai Soci UMI sconto 20%) CESARE ARZELÀ (1847-1912) Opere Complete – Volume I 1992, pp.XXXIX+348, brossura f.to 17 cm x 24 cm Opere Complete – Volume II 1992, pp. VII+357, brossura f.to 17 cm x 24 cm Prezzo dei 2 volumi indivisibili € 49,90 Sono state raccolte, in due volumi, tutte le opere scientifiche di Cesare Arzelà. La ristampa degli scritti è preceduta da un saggio, dovuto a G. Letta, P.L. Papini e L. Pepe su «Cesare Arzelà e l’Analisi reale in Italia». I volumi sono corredati, oltre che dall’elenco delle pubblicazioni scientifiche, anche da una bibliografia scientifica riferentesi ad Arzelà, da una notizia di scritti su Cesare Arzelà e da un breve elenco delle onoranze che gli furono dedicate. L’indice dei nomi contenuto nel secondo volume completa l’opera. LUIGI BIANCHI (1856-1928) Vol. I -Aritmetica, Algebra e Analisi. Parte prima 1952, ristampa 2001, pp. 616, brossura f.to 17 cm x 24 cm € 33,90 Vol. II -Aritmetica, Algebra e Analisi. Parte seconda 1953, pp. 276, brossura f.to 17 cm x 24 cm € 25,10 Vol. II -Applicabilità e problemi di deformazioni 1953, pp. 338, brossura f.to 17 cm x 24 cm € 25,10 Vol. III -Sistemi tripli ed n-upli ortogonali e loro trasformazioni 1955, pp.
    [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]