On some recent advances in shape optimization

In this Note we give a short review on recent developements in shape optimi zation. We explain how the generic non-existence of solutions can be circum vent. Either one can impose some geometric restrictions on the class of adm issible domains to get existence (we then explain how to write the usual op timality conditions), or generalized designs are allowed which leads to rel axation by homogenization techniques (we thus obtain topology optimization methods). (C) 2001 Academie des sciences/Editions scientifiques et medicale s Elsevier SAS.