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**Optimality conditions and homogenization.**
*(English)*
Zbl 0569.49015

Nonlinear variational problems, Int. Workshop, Elba/Italy 1983, Res. Notes Math. 127, 1-8 (1985).

[For the entire collection see Zbl 0561.00014.]

The authors deal with optimization problems where the variable appears as a domain and some partial differential equation is involved, related to the phenomenon of homogenization. A bounded open domain is considered and the problem consists to choose a subset w of given measure which will be optimal in some sense; no assumption of smoothness of w is imposed (the phenomenon corresponds to having w made of tiny pieces and this may be ruled out by some smoothness hypothesis). By the homogenization theory, a limit problem is described, using the weak star limit \(\theta\) of the characteristic function of w. The relaxed problem consisting in replacing a domain w, or its characteristic function, by \(\theta\) and the corresponding homogenized matrix, permits to obtain a necessary and sufficient condition for optimality.

The authors deal with optimization problems where the variable appears as a domain and some partial differential equation is involved, related to the phenomenon of homogenization. A bounded open domain is considered and the problem consists to choose a subset w of given measure which will be optimal in some sense; no assumption of smoothness of w is imposed (the phenomenon corresponds to having w made of tiny pieces and this may be ruled out by some smoothness hypothesis). By the homogenization theory, a limit problem is described, using the weak star limit \(\theta\) of the characteristic function of w. The relaxed problem consisting in replacing a domain w, or its characteristic function, by \(\theta\) and the corresponding homogenized matrix, permits to obtain a necessary and sufficient condition for optimality.

Reviewer: M.Codegone

### MSC:

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

35J25 | Boundary value problems for second-order elliptic equations |

49M20 | Numerical methods of relaxation type |

49J45 | Methods involving semicontinuity and convergence; relaxation |