Document Zbl 1251.49001

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C.</a>; <a href="/authors/zolesio.jean-paul" title="Author Profile">Zolésio, J.-P.</a></div> <h2 class="title"> <strong>Shapes and geometries. Metrics, analysis, differential calculus, and optimization. 2nd ed.</strong> <i>(English)</i> <a class="label nowrap" href="/1251.49001">Zbl 1251.49001</a> </h2> <div class="source"> <a href="/serials/4632" title="Series Profile">Advances in Design and Control</a> 22. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (ISBN 978-0-898719-36-9/hbk; 978-0-89871-982-6/ebook). xxiii, 622 p. (2011). </div> <div class="abstract">The book was written by two famous mathematicians in applied mathematics area (especially in optimal shape design branch). It was also published as a series in “Advances in Design and Control” by SIAM, which indicates the important rate of the book.<br class="zbmathjax-paragraph">The main goal of the book is “to give a comprehensive presentation of mathematical construction and tools that can be used to study the problems where the modeling, optimization, or control variable is no longer a set of parameters or functions, but that shape or the structure of a geometric object.” It coveres the most basic and some new mathematical ideas, constructions, and more methods that come from different fields of mathematics and areas of applications. An encyclopedic investment to bring together the basic theories and materials for study of geometric objects was beyond the scope of the authors. Therefore, the book contains most of important fundamentals at this field. <br class="zbmathjax-paragraph"> I have never seen the first edition, but the authors believed that all chapters have been updated and considerably enriched with the new material in the second edition. This book went from 9 to 20 chapters with more elaborate section of each chapter. In the introductory chapter, chapter 1, a series of illustrative generic examples have been added (and were used in other chapter) to motivate the reader and illustrate the basic dilemma (the details of changing in chapters were mentioned in pages xx- xxii). <br class="zbmathjax-paragraph"> This book is suitable for a broad audience of mathematicians, specially applied mathematicians and also advanced engineers and scientists, especially mechanical engineers. The material of the beginning of each chapter is accessible to a broad audience, while the later sections sometimes may require more mathematical experiences. Even some parts of this book had been used as lecture notes in graduate courses at the Université de Montreal since 1986-1987, 1993-1994, 1995-1996, 1997-1998 (as authors said), I believe that this book is suitable as a reference book and also post-graduate text’s book in optimal shape area. <br class="zbmathjax-paragraph"> The concepts of chapters:<br class="zbmathjax-paragraph">Chapter 1) <span class="zbmathjax-textit">Introduction: Examples, background and perspectives</span>. Since the central objective of the book is the analysis of geometric stuctures, we need a differential calculus, spaces, evolution equations and other familiar concepts in analysis related to this geometry. In this chapter, first a series of generic examples are given which they will be used in the subsequent chapters. The first example is the celebrated problem of the optimal shape of a column formulated by Lagrange in 1770. The next illustrates the fact that even simple problems can be no differentiable with respect to the geometry. This is generic of all eigenvalue problems when the eigenvalue is not simple. Optimal triangular meshing, minimal surfaces, design of a thermal radiator and examples of image processing are some of these examples. Moreover, in section 9 some related methods are discussed. In section 10, more than the background of geometries, some fundamental subjects in this area, like parameterizing by functions, shape analysis, characteristic functions, distance of functions, shape optimization and shape derivatives are introduced. Moreover, changes of these concepts in comparison to the first edition are explained in section 11.<br class="zbmathjax-paragraph">Chapter 2) <span class="zbmathjax-textit">Classical description of geometries and their properties</span>. The chapter is devoted to the classical description of nonempty subsets of the finite-dimensional Euclidean space that are characterized by the smoothness of properties of their boundary, in three ways: 1) to assume that we can associate with each point of the boundary a diffeomorphism; 2) to assume that the set is the union of the positive level sets of a continuous function where its boundary is zero level; 3) to assume that in each point of the boundary, the set is locally the epigraph of function. Then, basic definitions and constructions for sets are locally described by isomorphism in section 3, by the level sets of a function in section 4, by the epigraph of a function in section 5, by geometric segment properties in section 6 (the stronger uniform segment property is further explained in chapter 7). Also, the important Sobolev spaces are surveyed in subsection 2.5, classes $C^k$ and $C^{k,l}$ are considered in subsection 3.1 and enriched by Hausdorff measure and H”olderian sets properties. <br class="zbmathjax-paragraph"> Chapter 3) <span class="zbmathjax-textit">Courant metric on images of asset</span>. A natural way to construct a family of variable domain is to consider the images of a fixed subset of $\mathbb R^n$ by some family of transformations of $\mathbb R^n$. In section 2, generic construction associated with the space $C_0^k (\mathbb R^n,\mathbb R^n)$ of mapping from $\mathbb R^n$ into $\mathbb R^n$ is extended to a large family of Banach spaces. Moreover, the geodesic character of construction as trajectories of bounded variation the group is emphasized. Next, the choice of the closed subgroup of transformation of $\mathbb R^n$ is discussed. It is shown that, as long as the subgroup is closed, we get a complete Courant metric on the quotient group. Also, the tangent space to the group of transformations of $\mathbb R^n$ is characterized leading to the Courant metric. <br class="zbmathjax-paragraph"> Chapter 4) <span class="zbmathjax-textit">Transformations generated by velocities</span>. The results of previous chapter are specialized to spaces of transformations that are generated at time $t=1$ by the flow of a velocity field over a generic time interval [0,1]. Section 2 presents the results to transformations that implicitly use a notion of geodesic path with discontinuities. Section 3 motivates the definitions of Gateaux and Hadamard semi-derivatives in topological vector space to shape functionals defined on shape spaces. The following two sections give technical results that are used to characterize continuity and semi-differentiability of the results of section 2. Also, the constrained case where the family of domains is a subset of a fixed holdall is studied in section 5. Moreover, sharp theorems on the equivalence between transformations and velocities are given. Based on the results of section 4, in section 6 the continuity of a shape function with respect to the Courant metric is established. <br class="zbmathjax-paragraph"> Chapter 5) <span class="zbmathjax-textit">Metrics via characteristic functions</span>. The constructions of the metric topologies of chapter 3 are limited to families of sets which are the image of a fixed set by a family of homeomorphisms or diffeomorphisms. In this chapter, the family of available sets is considerably enlarged by relaxing the smoothness assumption to the mere Lebesgue measurability and even just measurability to include Hausdorff measures. First, an Abelian group structure on characteristic functions of measurable subsets of a fixed holdall is introduced. Then, the metric spaces of equivalence classes of measurable characteristic functions via the $L^p$ norms are constructed in different toplogies (strong topology, $L^p$-topology). A nice representative in the equivalence class of sets and convex subset of a fixed bounded holdall is given in section 3. The use of the $L^p$-topologies is illustrated in section 4 by revisiting the optimal design problem. Another problem amenable to the formulation is the buckling of columns is illustrated in section 5. The Caccioppoli or finite perimeter sets of the celebrated plateau problem are revisited in section 6 with the family of Lipschitzian domains in a fixed bounded holdall. Section 7 gives an example of the use of the perimeter in the Bernoulli free boundary problem and in particular for water wave. <br class="zbmathjax-paragraph"> Chapter 6) <span class="zbmathjax-textit">Metrics via distance functions</span>. The volume and perimeter are not continuous with respect to the Hausdorff topology. This can be fixed by changing the space $C(D)$ to the space $W^{(l,p)} (D)$ since distance functions also belong to the space. The price to pay is the loss of compactness when $D$ is bounded. But other sequentially compact families can easily be constructed. In this chapter, general compactness theorems are obtained for such families under global or local conditions. Also, the family of open sets are characterized by the distance function to their complement. The properties of distance functions and Hausdorff complementary metric topologies are studied in section 2. The projections, skeletons, cracks and differentiability properties of these functions are discussed in section 3. $W$-topologies and related characteristic functions are introduced in section 4. Compact families of sets of bounded and locally bounded curvature are characterized in section 5 which accompany with two useful examples. The notion of Reach and Federer sets of positive reach are studied in section 6. Approximation of distance functions and their critical points are presented in section 7 and convex sets are characterized in section 8. Finally, section 9 gives several compactness theorems under global and local conditions on the Hessian matrix of the distance function.<br class="zbmathjax-paragraph">Chapter 7) <span class="zbmathjax-textit">Metrics via oriented distance functions</span>. The role of oriented distance functions in the geometric properties and smoothness of domains and their boundary are purpose in this chapter. They constitute a special family, algebraic or signed distance functions. The terminology emphasizes the fact that for a smooth open domain, the associated oriented distance functions specifics the orientation of the normal to the boundary of the underling set which enjoy many interesting properties. The first part of the chapter deals with the basic definitions and constructions and the main results. The second part specializes to specific subfamilies of oriented distance functions. The last part concentrates on compact families of subsets of oriented distance functions. Section 2 presents the basic properties of uniform metric topology and its connection with the Hausdorff and complementary Hausdorff’s topology of chapter 6. Section 3, which is the analogue white section in chapter 6 to the differentiability properties and the associated set of projections onto the boundary. Section 4 deals with $W^{(l,p)}$-topology on the set of oriented distance function. Section 5 studies the subfamily of sets for which the gradient of the oriented distance of function is a vector of functions of bounded variation and examples are given to illustrate the behavior of the norm in tubular neighborhoods in which with approximation by dilated sets are discussed in section6. Section 7 shows that sets of positive reach are locally bounded curvature and the boundary of their closure has zero volume. Section 8 gives the equivalence of the smoothness of a set and the smoothness of its oriented distance function. Sobolev domains are introduced in section 9 and the characterization of closed convex sets and their gradients are extended in section 10. Section 11 gives some new compactness theorems for sets of global and local bounded curvatures. Section 12 introduces a compactness theorem for a family of subsets of a bounded hold all and their compactness theorem is given in section 13. Section 14 combines the compactness under the uniform fat segment property with a bound on the De Giorgi perimeter of Caccioppoli sets. Section 15 introduces the families of cracked sets and they are used in section 16 to provide an original solution to a variation of the image segmentation problem.<br class="zbmathjax-paragraph">Chapter 8) <span class="zbmathjax-textit">Shape continuity and optimization</span>. This chapter is concentrated on continuity issues related to shape optimization problems under state equation constraints. It is nice that no adjoint system is necessary to characterize the minimizing function. Other problems have the structure of optimal control theory. In that case, the characterization of the optimal control involves an adjoint state equation coupled with the state equation, where the control will be the underlying geometry; the objective functional will depend on the domain and the state that itself depends on the domain. As in control theory, we need continuity of the objective functional and the state with respect to the geometry. Section 1 is devoted to the optimization of the first eigenvalue. The minimization of the Raleigh quotient problem is considered in section 2. The continuity of the transmission problem is discussed in section 3, and the continuity of the homogeneous Dirichlet boundary value problem is discussed in section 4, where the Neumann boundary value problems are considered in section 5. Section 6 introduces the basic elements and results from capacity theorem. After introducing the Crack-Free sets in section 7, any Lipschitz continuous transformation of $\mathbb R^n$ which have a Lipschitz continuous inverse transport set capacity, onto sets zero capacity is presented in section 8. Compactness of two families of functions on $D$ is discussed in section 9. <br class="zbmathjax-paragraph"> Chapter 9) <span class="zbmathjax-textit">Shape and tangential differential calculus</span>. It is more difficult to fully characterize tangent spaces for the spaces of characteristic functions or distance functions. In the absence of sharper results, the authors are concentrated on the notion of semi derivatives or derivatives of a shape function. Moreover, the velocity approach readily extends to the constrained case. This chapter has been structured along these general directions. In section 2, a review of semi-derivatives and derivatives in topological vector space is given, moreover, the chain rule is repeated. Section 3 gives the main properties of first order shape semi-derivatives and derivatives of shape functionals with the complete structural theorem (the Eulerian and Hadamard semi-derivatives and Gateaux and Frechat derivatives). The main elements of the shape calculus and the basic formulae for domain boundary integrals and their applications are given in section 4. Section 5 deals with the main eliminates of that calculus for a $C^2$-submanifold of $\mathbb R^n$ of codimension 1, including Stokes’s and Green’s formulae and the relationship between tangential covariant derivative. Section 6 extends definitions and structure theorems to second-order derivatives and the basic formula for this is given; also the shape Hessian is decomposed into a symmetrical term plus the gradient acting on the first half of the Lie bracket.<br class="zbmathjax-paragraph">Chapter 10) <span class="zbmathjax-textit">Shape gradients under a state equation constraint</span>. One of the important technical advantages of the control theory approach is to avoid the differentiation of the state with respect to the control. This could be relaxed and replaced by the pointwise maximization with respect to the control variable of the Hamiltonian (maximum principle). The domain will be identified with the control constrains. Regarding the above facilities of optimal control approach, this chapter concentrates on two generic examples often encountered in shape optimization. The first one is associated with the so-called compliance problems, where the shape functional is equal to the minimum of a domain-dependent energy functional. The second one deals with shape functionals that can be expressed as the saddle point of some appropriate Lagrangian. In addition, the theorem on the differentiation of an infimum with respect to a parameter is applied. For Euler’s buckling load, an explicit expression of the semi-derivative and a necessary and sufficient condition are given in section 3. The theory is further illustrated in section 4 by providing the semi-derivative of the first eigenvalue of several boundary value problems over a bounded open domain: Laplace and bi-Laplace equation and linear Elasticity. Saddle point formulation and function space parameterizations are discussed in section 5. A non-homogeneous Dirichlet problem by saddle point formulation is presented in section 6. <br class="zbmathjax-paragraph"> The reviewer’s opinion: In addition to the mentioned facilities, the very nice literature review and historical background in the beginning of each chapter and some necessary sections, especially about the methods, are other advantages of this book. The pure mathematical concepts of optimal shape theory are covered and illustrated enough for the reader, even in applications and also numerical works. The authors considere and cite the literature after the year 2000 and also their own works in writing the book, but I would like to see some other new methods as well like the measure theoretical based works in optimal shape design and also for instance Munch’s works. Maybe the book seems not suitable for under-graduated students, but it is perfectly very well to be used as a basic research book in post-graduate applications.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A11371">Alireza Fakharzadeh Jahromi (Shiraz)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A5816515">Cited in <strong>1</strong> Review</a></div><div class="clear"><a href="/?q=rf%3A5816515">Cited in <strong>381</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A49-02" title="MSC2020">49-02</a> </td> <td class="space"> Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A51-01" title="MSC2020">51-01</a> </td> <td class="space"> Introductory exposition (textbooks, tutorial papers, etc.) pertaining to geometry </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A49Q10" title="MSC2020">49Q10</a> </td> <td class="space"> Optimization of shapes other than minimal surfaces </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A28Cxx" title="MSC2020">28Cxx</a> </td> <td class="space"> Set functions and measures on spaces with additional structure </td> </tr><tr> <td class="mono"> 37J05 </td> <td class="space"> Relations of dynamical systems with symplectic geometry and topology (MSC2010) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A46B22" title="MSC2020">46B22</a> </td> <td class="space"> Radon-Nikodým, Kreĭn-Milman and related properties </td> </tr></table> </div><div class="keywords"> <h3>Citations:</h3><a href="/1002.49029">Zbl 1002.49029</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1251.49001" data-ciurl="/ci/05816515" data-biburl="/bibtex/05816515.bib" data-amsurl="/amsrefs/05816515.bib" data-xmlurl="/xml/05816515.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05816515.pdf" title="Zbl 1251.49001 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1137/1.9780898719826" aria-label="DOI for “Shapes and geometries. 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