Abstract. We prove the following conjecture recently formulated by Jakobson, Nadirashvili and Polterovich [15]: on the Klein bottle K, the metric of revolution, 0 ≤ v < π, is the unique extremal metric of the first eigenvalue of the Laplacian viewed as a functional on the space of all Riemannian metrics of given area. The proof leads us to study a Hamiltonian dynamical system which turns out to be completely integrable by quadratures.

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In this paper, we consider the flow of fresh and saltwater in a saturated porous medium in order to describe the seawater intrusion. Starting from a formulation with constant densities respectively of fresh and saltwater, whose velocities are proportional to the gradient of pressure (Darcy's law), we consider the formal asymptotic limit as the aspect ratio between the thickness and the horizontal length of the porous medium tends to zero. In the limit of the regime defined by the Dupuit-Forchheimer condition, we identify reduced models of Boussinesq type both in the cases of unconfined and confined aquifers. MSC: 35R35, 35B40

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