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Progress in Nonlinear Differential Equations and Their Applications Progress in Nonlinear Differential Equations and Their Applications Volume 95 Series Editor Haïm Brezis Université Pierre et Marie Curie, Paris, France Technion – Israel Institute of Technology, Haifa, Israel Rutgers University, New Brunswick, NJ, USA Editorial Board Members Antonio Ambrosetti, Scuola Internationale Superiore di Studi Avanzati, Trieste, Italy Luigi Ambrosio, Scuola Normale Superiore, Pisa, Italy Henri Berestycki, Ecoles des Hautes en Science Sociales, Paris France Luis Caffarelli, The University of Texas, Austin, TX, USA Sun-Yung Chang, Princeton University, NJ, USA Jean-Michel Coron, University Pierre et Marie Curie, Paris, France Manuel del Pino, University of Chile, Santiago, Chile Lawrence Evans, University of California, Berkeley, CA, USA Alessio Figalli, The University of Texas, Austin, TX, USA Rupert Frank, CalTech, CA, USA Nicola Fusco, Univeristy of Naples Federico II, Naples, Italy Sergiu Klainerman, Princeton University, NJ, USA Robert Kohn, New York University, NY, USA Pierre Louis Lions, Collège de France, Paris, France Andrea Malchiodi, Scuola Normale Superiore, Pisa, Italy Jean Mawhin, Université Catholique de Louvain, Louvain-la-Neuve, Belgium Frank Merle, U de Cergy-Ponoise and IHES, Paris France Giuseppe Mingione, Universita degli Studi di Parma, Parma, Italy Louis Nirenberg, New York University, NY, USA Felix Otto, Max Planck Institute, Leipzig, Germany Paul Rabinowitz, University of Wisconsin, Madison, WI, USA John Toland, Isaac Newton Institute, Cambridge, UK Michael Vogelius, Rutgers, NJ, USA More information about this series at http://www.springer.com/series/4889 George Dinca • Jean Mawhin Brouwer Degree The Core of Nonlinear Analysis George Dinca Jean Mawhin Faculty of Mathematics & Computer IRMP Science Université Catholique de Louvain University of Bucharest Louvain-la-Neuve, Belgium Bucharest, Romania ISSN 1421-1750 ISSN 2374-0280 (electronic) Progress in Nonlinear Differential Equations and Their Applications ISBN 978-3-030-63229-8 ISBN 978-3-030-63230-4 (eBook) https://doi.org/10.1007/978-3-030-63230-4 Mathematics Subject Classification: 58E95, 55M25, 47H05, 47H11, 47J15, 39A23, 35J60, 34B15, 34C25 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Preface Solving equations and widening the concept of solutions has been a driving force of the progress and evolution of mathematics. After the study of linear scalar algebraic equations led to the concept of rational numbers, the algebraic equations of the second degree forced mathematicians to introduce the irrational numbers. The complex numbers, introduced to give a meaning to a formula for the solutions of algebraic equations of the third order, have shown, through the existing formulas for solutions, that the algebraic equations of order smaller or equal to four with complex coefficients always have exactly four complex solutions (counted with their multiplicity). Before the vain quest of formulas for solutions in the case of degree five or more (justified later by Abel and Galois), it was conjectured that an algebraic equation of any order n with complex coefficients should always have at least one complex solution, and hence n complex solutions (counting multiplicity), a result first proved by Gauss in 1799 after the less fruitful efforts of Laplace, Lagrange and others. In between, the ‘geometrically evident’ fact that an algebraic equation of odd degree with real coefficients always had a real solution, already used by Cardano and Stevin to approximate the roots of some equations of the third degree, received several unsatisfactory proofs by Lagrange and others until Bolzano in 1817 and Cauchy in 1821 showed rigorously that this ‘evident’ fact was a consequence of a deep feature (the Bolzano intermediate value theorem) of the wider class of continuous real functions, a concept that they had defined clearly for the first time. Great efforts had also been made, starting with Descartes, to find conditions on the coefficients of a real algebraic equation to evaluate the number of its real solutions on a given interval, culminating, after the works of Budan and Fourier, to the elegant final solution by Sturm in 1829. In the complex case, the same problem was solved by Cauchy between 1831 and 1837 through the introduction of a suitable contour integral in the complex case in the frame of holomorphic functions, other proofs being given by Sturm and Liouville. Through a more algebraic approach, Hermite obtained in 1853 similar results for systems of algebraic equations. The way was paved for Kronecker to introduce in 1869 his famous index and integral, providing a bold generalization and synthesis of the work of Cauchy, v vi Preface Sturm, Liouville and Hermite mentioned above. In this epoch-making but not widely read memoir, Kronecker showed among many other things that, in order to obtain a count of the number of solutions of a system of equations insensitive to small perturbations, it was necessary to replace the naive count by an ‘algebraic’ one, attributing in some way a sign to each solution in counting it. For example, the = R = 2 − equation fε(x) 0, in with fε(x) x √ ε, has no solution for ε<0, one solution for√ ε = 0 and two√ solutions ± ε for ε>0. In this elementary example, f (− ε) < 0 and f ( ε) > 0, so that the difference between the number of solutions with positive derivative and the number of solutions with negative derivative is zero. Hence, this algebraic count of the number of zeros of fε remains thesameforε<0 and ε>0. One can even say that it is zero for ε = 0, as f (0) = 0 for the solution 0. In the case of a system f(x) = 0ofn equations of class C1 with n unknowns, Kronecker replaced the sign of the derivative by that of the Jacobian and showed that the corresponding algebraic count of solutions in a given bounded domain D of Rn with smooth boundary could be expressed by an integral on ∂D of some expression depending upon f and its partial derivatives. For the two-dimensional system associated to the complex equation f(z)= 0inC,the Kronecker integral reduces to the Cauchy integral mentioned above. Interestingly, Kronecker borrowed most of its technical tools to potential theory in constructing his theory. The memoir of Kronecker did not have many immediate followers, but had illustrious ones. In 1883, Poincaré used it to obtain an n-dimensional generalization of the Bolzano intermediate value theorem and applied it in various questions of celestial mechanics and automorphic functions. Between 1891 and 1894, Picard used it to the study of the number of solutions of systems of two or three equations, including his results in his famous Traité d’analyse. From 1888 till 1898, Dyck developed interesting geometrical and topological consequences of the Kronecker index in a series of papers. In 1904, Bohl applied Kronecker’s integral to some questions of mechanics and topology. An important step forward in the topological development of the Kronecker index was made in 1910 by Hadamard, in extending Kronecker’s definition from smooth to continuous mappings and including in his paper an unpublished fixed point theorem of Brouwer, known to Hadamard before its publication through a letter of the Dutch mathematician. The contribution of Hadamard makes the link between the Kronecker integral and a closely related concept introduced implicitly in 1911 by Brouwer in his proof of the invariance of dimension and made explicit in 1912 in a paper ending by the fixed point theorem sent to Hadamard. This concept is the degree of a continuous mapping between two oriented n-dimensional boundaryless topological manifolds. Brouwer’s construction, completely different from Kronecker-Hadamard’s one, was based upon approximations of the manifolds by polyhedra and of the mappings by simplicial ones. Other important topological consequences of his degree were given by Brouwer between 1912 and 1921. From those considerations, a Brouwer degree for a continuous mapping f on the closure of an open bounded set D ⊂ Rn into Rn such that 0 ∈ f(∂D) was constructed in 1935 by Alexandroff and Hopf in their book Topologie, generalizing the algebraic count of the number of solutions of Preface vii equation f(x)= 0inD to the situations where the solutions are not isolated and/or the Jacobians are not defined. This degree is insensitive to small perturbations of f . The richness of the concept of Brouwer degree is already revealed by its history and the quality of its protagonists. It is linked to fundamental problems coming from many different parts of mathematics, like algebra, analysis, topology and geometry and is associated to the names of the greatest mathematicians.
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