The Queen of all Sciences Reveals her Beauty via Nonlinear Elasticity Problems Tadeusz Iwaniec (Syracuse, USA) Thank You All So Much Rector Gaetano MANFREDI Magnifico Rettore Universit`adegli Studi di Napoli Federico II President Piero SALATINO Presidente della Scuola Politecnica e delle Scienze di Base Universit`adegli Studi di Napoli Federico II Chair Gioconda MOSCARIELLO Direttore del Dipartimento di Matematica e Applicazioni Renato Caccioppoli Universit`adegli Studi di Napoli Federico II Professor Carlo SBORDONE Professore di Analisi Matematica Universit`adegli Studi di Napoli Federico II Distinguished GUESTS and all my FRIENDS 2 Your Magnificence Rector Gaetano Manfredi This award is a great honor. Today, in such a nostalgic feeling I would like to thank you for the highest award offered by the historic and triumphant University of Naples in recognition of my modest contribution. It signifies my lasting research commitments to the faculty, and the desire of helping young scholars and PhD students in the Mathematics Department. Passing knowledge to a new generation of scholars and helping them to pursue their passion and dream to become successful researchers is not a small job. This is a noble commitment and should be the highest priority of every scientist. For me it was an experience of a lifetime that I cannot express in a few words, except for these ones (quoting Queen Rania of Jordan): "If you are too big for the small jobs you are too small for the big ones" 3 Carlo and I in the cage of mathematical lions Roma 1974, Istituto Nazionale Di Alta Matematica This photo breaks me down in tears. It is not the time to describe details of my emotions. At this conference Carlo Sbordone and I did not realize that about two decades later we would become "mathematical brothers" and close friends. 4 Thanks to Professor Carlo Sbordone It gives me a special pleasure and honor to speak on this occasion about Professor Carlo Sbordone. Since the early 1990s, I have been engaged in research and collaborative efforts to reach out to graduate students at the University of Naples. Carlo had wide research interests and a history of active cooperation with young scholars in developing modern Geometric Analysis and PDEs. Nowadays his students participate in such efforts, often becoming co-authors. Our engagement in research and his passion to interact with young scholars is tremendously inspiring to me. I am extremely grateful to Carlo for a fantastic cooperation which resulted in numerous joint publications recognized internationally. 5 If I had to choose, here are my memorable early papers with Carlo On the integrability of the Jacobian under minimal hypotheses, Arch. Rat.Mech.Anal. (1992) Weak minima of variational integrals, J. fur Reine Angew. Mat. (1994) Quasiharmonic Fields, Annales Inst. H. Poincar´eAnalyse Nonlineair´e(2001) Poincar´eMedal, Paris 2002 Our adventure would not be possible without enthusiasm, energy and the long-term friendship with junior scholars who loved me and whom I love. 6 Of course, Carlo was the heart of this endeavor. He brought a young Neapolitan generation to mathematics, especially to nonlinear analysis and PDEs. I am proud to be a part of Carlo's program, because I know it really helped young scholars, directly or indirectly. This is what matters most. Here is my first co-advisee (drawing by Gra_zyna) Luigi Greco and Carlo 7 La Compagnia Quasi Completa 8 Funding mathematical projects is superior to scientific expansion in research As you know I am from Syracuse (Upstate New York, a sister city of Siracusa). Many Neapolitan young scholars visited Syracuse University over an extended period of time to resume or continue our joint research already initiated in Naples. Guided in this spirit, the University of Naples assumed a key role in sharing costs with the NSF to support those visits. It paid off for all our scientific projects. Carlo and his associates; Gioconda Moscariello and Nicola Fusco have been unfailingly supportive. THANKS! 9 Sometimes when Neapolitan scholars visited Syracuse University the city was buried under snow. What a contrast with Naples 10 Renato Caccioppoli 1904 { 1959 The architect of the methods of functional analysis in the calculus of variations He was born in Naples and lived in Naples. Caccioppoli's work and ideas became a crucial part of the flourishing and internationally acclaimed theory of PDEs. Among many of his profound concepts was what we call the Caccioppoli inequalities nowadays. These are the fundamental estimates which allow us to get the regularity theory of PDE off the ground. His perceptions, rich in concepts and technical novelties, remain permanently on the stage of his legacy. Did you know that an asteroid discovered on October 20, 1985 by Ted Bowell was named Asteroid 9934 Caccioppoli ? 11 Carlo Miranda 1912 { 1982 Obviously, details do not fit into my brief speech today. Let me only bring on stage another renowned Neapolitan mathematician, Carlo Miranda and his famous influential book on "Partial Differential Equations of Elliptic Type" (1970)". Miranda and Caccioppoli both served as Editors in Chief of the famous "Giornale di Matematiche". They both taught at the Engineering Department of the University of Naples. Their successful students (Angelo Alvino, Carlo Ciliberto, Renato Fiorenza, Donato Greco, Guido Stampacchia, Guido Trombetti, ... ) and their descendants (Ciliberto's student Luciano Carbone,... ) are the living messengers that Caccioppoli and Miranda sent to the present time. 12 My affection to applied mathematics It goes back to the greatest Greek mathematician, physicist, engineer, inventor and astronomer of antiquity. Archimedes of Siracusa anticipated modern calculus and geometric analysis using the concept of infinitesimals to rigorously prove geometrical statements. I feel humbled to be a part of historical developments of Archimedes' approach to science. I simply benefit and gain insights from recent advances in mathematics. It leads to rigorous proofs of the existing experimental solutions and to the discovery of new phenomena in the applied fields. 13 Archimedes of Siracusa (287BC - 212 BC) Father of the application of scientific knowledge For me, learning and creating something mathematical out of practical problems is an intellectual joy. - Taddeo of SiracUSA 14 Engineering Mathematics (Technomath) Engineering research seeks improvements in both theory and practice via rigorous mathematical arguments. I have always been fascinated with Quasiconformal Geometry and the mathematical models of Nonlinear Elasticity. Both theories share a compelling beauty through challenging problems in the Calculus of Variations and nonlinear PDEs. They both rely on geometric intuition and an in-depth analysis of the energy-minimal deformations of Euclidean domains (flat plates) or, more generally, curved Riemannian manifolds (thin films). Let me briefly outline some of my results and new phenomena coming from an advanced mathematical approach. This is largely my joint work with Jani Onninen. 15 Coauthor Jani Onninen 16 No Coffee ≡ No Theorems Coffee is proof that God loves mathematicians, wants us to prove theorems and be happy (A paraphrase of Benjamin Franklin) For us, KIMBO Espresso was the best for proving Theorems, other coffees were OK, but only for Lemmas or Propositions 17 Nonlinear Hyperelasticity (A thumbnail description) onto We enquire into homeomorphisms h : X −! Y of smallest stored energy [h] = R E(x; h; Dh) dx ; E : × × n×n ! E X X Y R R where E characterizes the mechanical properties of the materials in X and Y . The Dirichlet energy-integral and the p -harmonic energy-integrals are ideal examples. R 2 R p 2[h] = jDh(x)j dx p[h] = jDh(x)j dx E X E X 18 The Principle of Non-Interpenetration of Matter It is axiomatic in the theory of elasticity that the energy-minimal displacement field h : X ! Y should be a homeomorphism. However, from the mathematical point of view, this is highly oversimplified precondition. One quickly runs into a serious difficulty when passing to the limit of an energy-minimizing sequence of homeomorphisms; injectivity is lost. In search for mathematical models of hyperelasticity, we must accept and explore the limits of Sobolev homeomorphisms. This approach turns out to be particularly effective in the 2D-theory of plates and thin films (surfaces). 19 Limits of Sobolev Homeomorphisms (Onninen & I. JEMS , 2017) 2 onto THEOREM. Let X ; Y 2 R be Lipschitz domains and hj : X −! Y homeomorphisms of Sobolev class W 1;p(X; Y) , converging weakly to 1;p h 2 W (X; Y) ; 2 6 p < 1 . Then there exists a better sequence 1 onto 1;p of C -diffeomorphisms fj : X −! Y; fj 2 h + W◦ (X; Y) , which onto converges to h: X −! Y strongly in W 1;p(X; Y). In our application fhj g is a sequence of homeomorphisms whose weak limit minimizes the energy integral. This gives us the existence of hyper-elastic deformations 20 Existence of Traction Free Energy-Minimal Deformations Z Z p E [h] = E(x; h; Dh)dx (confine yourselve to) jDh(x)j dx ; p > 2 X X The class Hfp(X; Y) of weak limits of Sobolev homeomorphisms onto f 2 Hp(X; Y) contains a deformation h◦ : X −! Y of smallest energy, Z Z p p jDh◦(x)j dx = min jDh(x)j dx = X h2Hfp(X;Y) X Z = inf jDf(x)jp dx f2 ( ; ) Hp X Y X 21 Approximation of a W 1;p- homeomorphism with Diffeomorphisms Iwaniec, Kovalev, Onninen, Arch. Rat. Mech. (2011) THEOREM (solution of Ball-Evans Conjecture) onto Every planar homeomorphism h : X −! Y that belongs to W 1; p(X; Y) ; 1 < p < 1 , can be approximated in the Sobolev norm with C 1-smooth onto diffeomorphisms hj : X −! Y : 1 1recently extended to the borderline case p = 1 by A. Pratelli and S.