MATHEMATICAL METHODS IN THE APPLIED SCIENCES Math. Meth. Appl. Sci. 2002; 25:473–489 (DOI: 10.1002/mma.295) MOS subject classication: 49 Q 10; 76 S 05; 35 R 35 An optimal shape design formulation for inhomogeneous dam problems Abdelkrim Chakib 1;† , Touria Ghemires 2;‡ and Abdeljalil Nachaoui 1;∗;§ 1 CNRS UMR 6629=Universit e de Nantes; BP 92208; Nantes F-44322; France 2 Facult e des Sciences; D epartement de Math ematiques et Informatique; Universit e Mohamed V; B.P. 1014 - Rabat; Morocco Communicated by G. F. Roach SUMMARY In this paper, the ow problem of incompressible liquid through an inhomogeneous porous medium (say dam), with permeability allowing parametrization of the free boundary by a graph of continuous unidi- mensional function, is considered. We propose a new formulation on an optimal shape design problem. We show the existence of a solution of the optimal shape design problem. The nite element method is used to obtain numerical results which show the eciency of the proposed approach. Copyright ? 2002 John Wiley & Sons, Ltd. 1. INTRODUCTION This paper considers the ow problem of incompressible liquid through an inhomogeneous porous medium, under the impact of gravitation forces. This problem, known as the dam problem, is posed in the wet part of the dam with an unknown part of the boundary, which have to be determined from over-determined boundary data. This problem has re- ceived a great deal of attention. The rst results are due to Baiocchi [1], who considered the study of the problem in the case of homogeneous and isotropic dam, by using the so-called ‘Baiocchi transformation’ which permits to convert the problem into a variational inequal- ity of one unknown variable having a unique solution. Then the problem has been exten- sively studied by several authors in homogeneous and inhomogeneous case, using variational and quasivariational inequalities (see References [2–10]). The case of inhomogeneous dam complicates these approaches (see Reference [7]). ∗ Correspondence to: Abdeljalil Nachaoui, CNRS UMR 6629, Universit e de Nantes, BP 92208, Nantes F-44322, France † E-mail: chakib@math.univ-nantes.fr ‡ E-mail: ghemires@fsr.ac.ma § E-mail: nachaoui@math.univ-nantes.fr Contract=grant sponsor: LERMA-EMI, CNRS UMR6629; contract grant numbers: AI 95=844, AI 181 MA=99 Copyright ? 2002 John Wiley & Sons, Ltd. Received 26 September 2000