Shape optimization with Stokes constraints over the set of axisymmetric domains

Maïtine Bergounioux (1) , Yannick Privat (2)

1 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
2 IRMAR - Institut de Recherche Mathématique de Rennes

In this paper, we are interested in the study of shape optimizations problems with Stokes constraints within the class of axisymmetric domains represented by the graph of a function. Existence results with weak assumptions on the regularity of the graph are provided. We strongly use these assumptions to get some topological properties. We formulate the (shape) optimization problem using different constraints formulations: uniform bound constraints on the function and its derivative and/or volume (global) constraint. Writing the first order optimality conditions allows to provide quasi-explicit solutions in some particular cases and to give some hints for the treatment of the generic problem. Furthermore, we extend the (negative) result of [16] dealing with the non optimality of the cylinder.

Mots clés: Shape Optimization; Dissipated Energy; Stokes Problem; First order opti- mality conditions

Domaines: Equations aux dérivées partielles [math.AP]

Maïtine Bergounioux, Yannick Privat. Shape optimization with Stokes constraints over the set of axisymmetric domains. 2010. ⟨hal-00546165v1⟩