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**Introduction to shape optimization problems and free boundary problems.**
*(English)*
Zbl 0765.76070

Shape optimization and free boundaries, Proc. NATO ASI, Sémin. Math. Supér., Montréal/Can. 1990, NATO ASI Ser., Ser. C 380, 397-457 (1992).

Summary: [For the entire collection see Zbl 0753.00022.]

We are concerned with existence results in shape optimization as well as with necessary conditions for optimality. In the first section we give existence results for a weak shape formulation of Bernoulli-like free boundary problems for stationary potential flows. In the second section it is shown how the bounded perimeter-constraint can apply to give an existence result for control in the transient wave equation. The third section deals with the very definition of shape derivatives and with results on the structure of the derivatives. The fourth section deals with the shape variational free boundary problem associated with the Stokes stationary fluid. It underlines that the free boundary condition cannot be achieved in such a linearized modelling. Also, we give existence and continuity results obtained by a penalty approach (via transmission “two-fluid” problems) which apply also to unilateral problems. Finally, the last section extends an existence result for eigenvalues of the Laplace operator.

We are concerned with existence results in shape optimization as well as with necessary conditions for optimality. In the first section we give existence results for a weak shape formulation of Bernoulli-like free boundary problems for stationary potential flows. In the second section it is shown how the bounded perimeter-constraint can apply to give an existence result for control in the transient wave equation. The third section deals with the very definition of shape derivatives and with results on the structure of the derivatives. The fourth section deals with the shape variational free boundary problem associated with the Stokes stationary fluid. It underlines that the free boundary condition cannot be achieved in such a linearized modelling. Also, we give existence and continuity results obtained by a penalty approach (via transmission “two-fluid” problems) which apply also to unilateral problems. Finally, the last section extends an existence result for eigenvalues of the Laplace operator.

### MSC:

76M30 | Variational methods applied to problems in fluid mechanics |

76D07 | Stokes and related (Oseen, etc.) flows |

35R35 | Free boundary problems for PDEs |