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Aldo Pratelli Günter Leugering Editors New Trends in Shape Optimization International Series of Numerical Mathematics 166 Aldo Pratelli Günter Leugering Editors New Trends in Shape Optimization ISNM International Series of Numerical Mathematics Volume 166 Managing Editor G. Leugering, Erlangen-Nürnberg, Germany Associate Editors Z. Chen, Beijing, China R.H.W. Hoppe, Augsburg, Germany; Houston, USA N. Kenmochi, Chiba, Japan V. Starovoitov, Novosibirsk, Russia Honorary Editor K.-H. Hoffmann, München, Germany More information about this series at www.birkhauser-science.com/series/4819 Aldo Pratelli • Günter Leugering Editors New Trends in Shape Optimization Editors Aldo Pratelli Günter Leugering Department Mathematik Department Mathematik Universität Erlangen‐Nürnberg Universität Erlangen-Nürnberg Erlangen Erlangen Germany Germany ISSN 0373-3149 ISSN 2296-6072 (electronic) International Series of Numerical Mathematics ISBN 978-3-319-17562-1 ISBN 978-3-319-17563-8 (eBook) DOI 10.1007/978-3-319-17563-8 Library of Congress Control Number: 2015950019 Mathematics Subject Classification (2010): 35B20, 35B40, 35J25, 35J40, 35J57, 35P15, 35R11, 35R35, 35Q61, 45C05, 47A75, 49J45, 49R05, 49R50, 49Q10, 49Q12, 49Q20, 53A10, 65D15, 65N30, 74K20, 76D07, 76Q20, 78A50, 81Q05, 82B10 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved 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, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.birkhauser-science.com) Contents On the Minimization of Area Among Chord-Convex Sets ........... 1 Beatrice Acciaio and Aldo Pratelli Optimization Problems Involving the First Dirichlet Eigenvalue and the Torsional Rigidity.................................. 19 Michiel van den Berg, Giuseppe Buttazzo and Bozhidar Velichkov On a Classical Spectral Optimization Problem in Linear Elasticity .... 43 Davide Buoso and Pier Domenico Lamberti Metric Spaces of Shapes and Geometries Constructed from Set Parametrized Functions.................................... 57 Michel C. Delfour A Phase Field Approach for Shape and Topology Optimization in Stokes Flow .......................................... 103 Harald Garcke and Claudia Hecht An Overview on the Cheeger Problem ......................... 117 Gian Paolo Leonardi Shape- and Topology Optimization for Passive Control of Crack Propagation ..................................... 141 Günter Leugering, Jan Sokołowski and Antoni Zochowski Recent Existence Results for Spectral Problems .................. 199 Dario Mazzoleni Approximate Shape Gradients for Interface Problems ............. 217 A. Paganini Towards a Lagrange–Newton Approach for PDE Constrained Shape Optimization....................................... 229 Volker H. Schulz, Martin Siebenborn and Kathrin Welker v vi Contents Shape Optimization in Electromagnetic Applications .............. 251 Johannes Semmler, Lukas Pflug, Michael Stingl and Günter Leugering Shape Differentiability Under Non-linear PDE Constraints .......... 271 Kevin Sturm Optimal Regularity Results Related to a Partition Problem Involving the Half-Laplacian ................................ 301 Alessandro Zilio Introduction Shape optimization has emerged from calculus of variations and from the theory of mathematical optimization in topological vector spaces together with the field of numerical methods for optimization and can be said to have established itself as one of the most fruitful applications of mathematics to problems in science and engi- neering. The particular mathematical challenge that distinguishes this field from standard calculus of variations or mathematical optimization is the fact the optimization variables are now represented by shapes or sets. One, thus, has to topologize these sets in a way such that the convergence of sets is consistent with the topological properties of the performance criterion and possibly side constraints. Indeed, a typical shape optimization problem consists of a so-called cost function depending on the shape of a body and state variables. The state variables are governed by a partial differential equation, the state equation, and specific equality or inequality constraints, depending also on the shape of the body. The problem then is to choose a shape from the set of admissible shapes such that the cost function is minimized over all admissible states and shapes. While one branch of the field is concerned with set convergence, proper topologies, and existence results for the optimization problems together with optimality conditions, another branch investigates the sensitivity of the above-mentioned quantities with respect to shape changes. Here shape differen- tiability of first and second order comes into play and shape gradient descent as well as shape Newton methods are investigated and numerically implemented. A third development emerges into the direction of applications in science, engineering, and industry, where the discretization plays a major role along with modern tools in supercomputing. The goal of the workshop on “Trends in shape optimization,” held in Erlangen in the fall of 2013, was to bring together experts in these different areas of shape optimization in order to provide a platform for intense mathematical discussions. It was felt that in order to further develop and foster this field at the frontiers of modern mathematics, an intensified interaction and exchange of methods and tools would be indispensable. vii viii Introduction As a result, we brought together scientists from Canada, France, Germany, Italy, Poland, and USA focusing on general minimization problems, spectral problems, Cheeger problems, metric spaces of shapes and geometries, phase field approaches and crack propagation control for shape and topology problems, approximate shape gradients for interface problems, Lagrange–Newton methods for PDE-constrained problems, electromagnetic applications, nonlinear problems and regularity results for the half-Laplacian, and many other topics not covered in this volume. Most speakers of the workshop provided survey articles, research articles. as well as notes on works in progress. The present volume very much reflects the atmosphere of the workshop, where purely theoretical results and novel approaches interfered with more application-oriented developments and industrial applications. The editors sincerely hope that this book on shape optimization will have an impact on this exciting field of mathematics. We are grateful to Lukas Pflug for his administrative support during the process of editing this volume. Erlangen Prof. Aldo Pratelli July 2015 Prof. Günter Leugering On the Minimization of Area Among Chord-Convex Sets Beatrice Acciaio and Aldo Pratelli Abstract In this paper we study the problem of minimizing the area for the chord- convex sets of given size, that is, the sets for which each bisecting chord is a segment of length at least 2. This problem has been already studied and solved in the framework of convex sets, though nothing has been said in the non-convex case. We introduce here the relevant concepts and show some first properties. Keywords Area-minimizing sets · Chord convex 1 Introduction and Setting of the Problem Consider the convex planar sets with the property that all the bisecting chords (i.e., the segments dividing the set in two parts of equal area) have length at least 2. A very simple question is which set in this class minimizes the area. Surprisingly enough, the answer is not the unit disk, as one would immediately guess, but the so-called “Auerbach triangle”, shown in Fig. 1 left. The story of this problem is quite old: already in the 1920s Zindler posed the question whether the disk is the unique planar convex set having all the bisecting chords of the same length (see [4]), while few years later Ulam asked if there are other planar convex sets, besides the disk, which have the floating property, that can be described as follows. Assume that the set has density 1/2 and it is immersed in the water (hence, half of the set remains immersed while the other half stays out of Both the authors have been supported through the ERC St.G. AnOpt- SetCon. B. Acciaio (B) Department of Statistics, London School of Economics, 10 Houghton Street, London WC2A 2AE, UK e-mail: [email protected] A. Pratelli Department Mathematik, Universität Erlangen, Cauerstrasse 11, 91056 Erlangen, Germany e-mail: [email protected] © Springer International Publishing Switzerland 2015 1 A. Pratelli and G.
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