Variational methods in shape optimization problems.

*(English)*Zbl 1117.49001
Progress in Nonlinear Differential Equations and their Applications 65. Basel: Birkhäuser (ISBN 0-8176-4359-1/hbk). viii, 216 p. (2005).

In this book shape optimization problems, variational ones in which the competing objects are domains of \({\mathbb R}^{N}\) instead of functions, are treated. The subject is of great interest not only to mathematicians due to its numerous applications. The typical problem is
\[
\min\{F(A) : A\in{\mathcal A}\},
\]
where \({\mathcal A}\) is the class of admissible domains and \(F\) is the cost function that will be minimized over \({\mathcal A}\). Working with shapes instead of functions usually introduces additional difficulties that lead to the lack of existence of solutions and to the introduction of relaxed formulations of the problem. The volume is based on lecture notes from two courses given by the authors and addressed mainly to Ph. D. students. As indicated in the preface “…the style of the work remains quite informal and follows, in large part, the lectures as given”. Some background on topics in functional analysis, usually covered in undergraduate courses, is required. A related monograph is the one by [J. Sokolowski and J.-P. Zolesio, Introduction to shape optimization: shape sensitivity analysis. Berlin etc.: Springer- Verlag (1992; Zbl 0761.73003)].

The work is organized in seven chapters. Chapter 1 is an introductory one and includes some examples of shape optimization problems: the isoperimetric problem (both the boundaryless and the free boundary one), the Newton problem of optimal aerodynamical profiles, the optimal distribution of two different media in a given region, and the optimal shape of a thin insulator around a given conductor.

In chapter 2 the additional constraint of convexity is assumed on the competing domains: this situation often provides the extra compactness necessary to prove the existence of an optimal shape. A prototype for this class is the Newton problem, where the convexity in the competing bodies permits the existence of an optimal shape, together with some necessary conditions of optimality.

Chapter 3 treats optimal control problems in which an admissible shape plays the role of an admissible control, and the corresponding state variable is usually the solution of a partial differential equation on the control domain. The corresponding relaxation theory is developed, which provides a general way to construct relaxed solutions through \(\Gamma\)-convergence methods. Some applications when the state equation is an ordinary differential equation are given in Section 3.5.

In chapter 4 shape optimization problems associated to elliptic operators are discussed. After the introduction of the notion of capacity, an explicit example where no optimal solution exist is given. The relaxed form of the Dirichlet problem for the Laplacian is introduced and necessary conditions for optimality, both the shape derivative and the topological derivative, are given. A very interesting discussion on “shape continuity” is given in Section 4.5, through Mosco convergence of the associated functional spaces. This section is followed by discussions of continuity under geometric and topological (Šverák’s result) constraints. Necessary and sufficient conditions for the \(\gamma_p\)-convergence and stability in the sense of Keldysh are also included in this chapter.

In chapter 5 the existence of non-relaxed solutions is discussed for the problem \(\min\{F(\Omega) : | \Omega| \leq m\), \(\Omega\in {\mathcal A}\}\). This problem does not admit a solution in general, although it is shown that there is \(\gamma\)-compactness under geometrical conditions (convexity, uniform (flat) cone conditions, uniform capacity density, uniform Wiener conditions, …), and existence for costs \(F\) which are monotone decreasing with respect to the set inclusion by using the notion of weak \(\gamma\)-convergence. This is used to show existence of domains with minimal given function of the eigenvalues of the Laplacian and existence of domains with minimal capacity. Finally, the problem of optimal partitions and the optimal obstacle problem are treated.

The problems considered in chapter 6 are of the type \(\{\Phi(\lambda(\Omega)) : | \Omega| =c\), \(\Omega\in {\mathcal A}\}\), where \(\lambda(\Omega)\) denotes the sequence of eigenvalues of the Laplacian in \(\Omega\) with Dirichlet boundary condition. \({\mathcal A}\) will be taken as \(\{\Omega : \Omega\subset D\}\), where \(D\), the design region, will be either a bounded open set in \(\mathbb R^N\) or the whole \(\mathbb R^N\). Results from the previous chapter ensure the existence of at least one optimal region when \(D\) is bounded and \(\Phi\) lower semicontinuous and monotone decreasing (Corollary 6.2.1). In this chapter this corollary is extended to cover the cases of non-monotone functionals (§ 6.4) and unbounded design regions (§ 6.5). An interesting account on continuous Steiner symmetrization is given. Some open questions are stated at the end of the chapter.

Chapter 7 is devoted to the case of shape optimization problems governed by elliptic equations with homogeneous Neumann conditions on the free boundary. These include optimal cuttings of a membrane, image segmentation problems and the cantilever problem. Several additional difficulties arise precluding the development of a complete theory. An effort is made to treat completely the problem of optimal cutting, where the existence of an optimal cut is deduced in full generality.

The work is organized in seven chapters. Chapter 1 is an introductory one and includes some examples of shape optimization problems: the isoperimetric problem (both the boundaryless and the free boundary one), the Newton problem of optimal aerodynamical profiles, the optimal distribution of two different media in a given region, and the optimal shape of a thin insulator around a given conductor.

In chapter 2 the additional constraint of convexity is assumed on the competing domains: this situation often provides the extra compactness necessary to prove the existence of an optimal shape. A prototype for this class is the Newton problem, where the convexity in the competing bodies permits the existence of an optimal shape, together with some necessary conditions of optimality.

Chapter 3 treats optimal control problems in which an admissible shape plays the role of an admissible control, and the corresponding state variable is usually the solution of a partial differential equation on the control domain. The corresponding relaxation theory is developed, which provides a general way to construct relaxed solutions through \(\Gamma\)-convergence methods. Some applications when the state equation is an ordinary differential equation are given in Section 3.5.

In chapter 4 shape optimization problems associated to elliptic operators are discussed. After the introduction of the notion of capacity, an explicit example where no optimal solution exist is given. The relaxed form of the Dirichlet problem for the Laplacian is introduced and necessary conditions for optimality, both the shape derivative and the topological derivative, are given. A very interesting discussion on “shape continuity” is given in Section 4.5, through Mosco convergence of the associated functional spaces. This section is followed by discussions of continuity under geometric and topological (Šverák’s result) constraints. Necessary and sufficient conditions for the \(\gamma_p\)-convergence and stability in the sense of Keldysh are also included in this chapter.

In chapter 5 the existence of non-relaxed solutions is discussed for the problem \(\min\{F(\Omega) : | \Omega| \leq m\), \(\Omega\in {\mathcal A}\}\). This problem does not admit a solution in general, although it is shown that there is \(\gamma\)-compactness under geometrical conditions (convexity, uniform (flat) cone conditions, uniform capacity density, uniform Wiener conditions, …), and existence for costs \(F\) which are monotone decreasing with respect to the set inclusion by using the notion of weak \(\gamma\)-convergence. This is used to show existence of domains with minimal given function of the eigenvalues of the Laplacian and existence of domains with minimal capacity. Finally, the problem of optimal partitions and the optimal obstacle problem are treated.

The problems considered in chapter 6 are of the type \(\{\Phi(\lambda(\Omega)) : | \Omega| =c\), \(\Omega\in {\mathcal A}\}\), where \(\lambda(\Omega)\) denotes the sequence of eigenvalues of the Laplacian in \(\Omega\) with Dirichlet boundary condition. \({\mathcal A}\) will be taken as \(\{\Omega : \Omega\subset D\}\), where \(D\), the design region, will be either a bounded open set in \(\mathbb R^N\) or the whole \(\mathbb R^N\). Results from the previous chapter ensure the existence of at least one optimal region when \(D\) is bounded and \(\Phi\) lower semicontinuous and monotone decreasing (Corollary 6.2.1). In this chapter this corollary is extended to cover the cases of non-monotone functionals (§ 6.4) and unbounded design regions (§ 6.5). An interesting account on continuous Steiner symmetrization is given. Some open questions are stated at the end of the chapter.

Chapter 7 is devoted to the case of shape optimization problems governed by elliptic equations with homogeneous Neumann conditions on the free boundary. These include optimal cuttings of a membrane, image segmentation problems and the cantilever problem. Several additional difficulties arise precluding the development of a complete theory. An effort is made to treat completely the problem of optimal cutting, where the existence of an optimal cut is deduced in full generality.

Reviewer: Manuel Ritoré (Granada)

##### MSC:

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

49Q10 | Optimization of shapes other than minimal surfaces |

49Q15 | Geometric measure and integration theory, integral and normal currents in optimization |

74P10 | Optimization of other properties in solid mechanics |