A free boundary and an optimal shape problem of a floating body

The present article deals with a variational formulation of a floating body in a perfect fluid. Since the considered energy functional depends on both stream function and floating body, these two minimizations are realized in two separated steps. By the use of free boundary methods, existence and Lipschitz continuity of the minimizing stream function are proved. In addition, it is shown that the corresponding free surface has Hölder boundary. The dependence on the floating body leads to an optimal shape problem which is treated by the use of Hausdorff convergence. Under the postulate of a bounded density parameter of the domain, the existence of an optimal floating body is shown.