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**An introduction to the homogenization method in optimal design.**
*(English)*
Zbl 1040.49022

Cellina, Arrigo (ed.) et al., Optimal shape design. Lectures given at the joint CIM/CIME summer school, Tróia, Portugal, June 1–6, 1998. Berlin: Springer (ISBN 3-540-67971-5/pbk). Lect. Notes Math. 1740, 47-156 (2000).

This is a very interesting paper on the main ideas and history of the homogenization method in optimal control or optimal design problems. The paper represents a course of lectures given at the CIME/CIM Summer school in 1998 and as such it starts from simple counterexamples to the existence of an optimal control in problems governed by elliptic equations and ends with necessary optimality conditions and explicit relaxation formulae.

The paper contains (with proofs) most of the known results for the problems of minimizing weakly continuous functionals \(I(u)\) subject to the state equation \[ \begin{gathered}\text{div} \Biggl[ \sum^N_{k=1} \chi_{\Omega_k}(x) R_k(x)A_k R_k(x)\nabla u(x)\Biggr]= f(x)\quad \text{in }\Omega,\\ u\in H^1_0(\Omega), \end{gathered} \] where the role of controls play the characteristic functions \(\{\chi_{\Omega_1}; \dots;\chi_{\Omega_N}\}\) of all measurable partitions \(\{\Omega_1,\dots, \Omega_N\}\) of the reference domain \(\Omega\subset \mathbb{R}^n\) as well as the rotations \(R_k(\cdot)\in \text{SO}(n)\), \(k= 1,\dots, N\). Here, the matrices \(A_k\), \(k=1,\dots, N\), are fixed and can be treated as characteristics of given materials placed in \(\Omega_k\), \(k=1,\dots,N\), respectively for optimal design problems.

The paper gives a nice insight on how the homogenization ideas started and came in force in optimal design problems.

For the entire collection see [Zbl 0954.00031].

The paper contains (with proofs) most of the known results for the problems of minimizing weakly continuous functionals \(I(u)\) subject to the state equation \[ \begin{gathered}\text{div} \Biggl[ \sum^N_{k=1} \chi_{\Omega_k}(x) R_k(x)A_k R_k(x)\nabla u(x)\Biggr]= f(x)\quad \text{in }\Omega,\\ u\in H^1_0(\Omega), \end{gathered} \] where the role of controls play the characteristic functions \(\{\chi_{\Omega_1}; \dots;\chi_{\Omega_N}\}\) of all measurable partitions \(\{\Omega_1,\dots, \Omega_N\}\) of the reference domain \(\Omega\subset \mathbb{R}^n\) as well as the rotations \(R_k(\cdot)\in \text{SO}(n)\), \(k= 1,\dots, N\). Here, the matrices \(A_k\), \(k=1,\dots, N\), are fixed and can be treated as characteristics of given materials placed in \(\Omega_k\), \(k=1,\dots,N\), respectively for optimal design problems.

The paper gives a nice insight on how the homogenization ideas started and came in force in optimal design problems.

For the entire collection see [Zbl 0954.00031].

Reviewer: Uldis Raitums (Riga)

### MSC:

49J45 | Methods involving semicontinuity and convergence; relaxation |

49J20 | Existence theories for optimal control problems involving partial differential equations |

49K20 | Optimality conditions for problems involving partial differential equations |

35B27 | Homogenization in context of PDEs; PDEs in media with periodic structure |