  The goal of this paper is to prove the well-posedness of F. John's floating
body problem in the case of a fixed object and for unsteady waves, in
horizontal dimension $d=1$ and with a possibly emerging bottom. This problem
describes the interactions of waves with a partially immersed object using the
linearized Bernoulli equations. The fluid domain $\Omega$ therefore has corners
where the object meets the free surface, which consists of various connected
components. The energy space associated with this problem involves the space of
traces on these different connected components of all functions in the
Beppo-Levi space $\dot{H}^1(\Omega)$; we characterize this space, exhibiting
non local effects linking the different connected components. We prove the
well-posedness of the Laplace equation in corner domains, with mixed boundary
conditions and Dirichlet data in this trace space, and study several properties
of the associated Dirichlet-Neumann operator (self-adjointness, ellipticity
properties, etc.). This trace space being only semi-normed, we cannot use
standard semi-group theory to solve F. John's problem: one has to choose a
realization of the homogeneous space (i.e. choose an adequate representative in
the equivalence class) we are working with. When the fluid domain is bounded,
this realization is obtained by imposing a zero-mass condition; for unbounded
fluid domains, we have to choose a space-time realization which can be
interpreted as a particular choice of the Bernoulli constant. Well-posedness in
the energy space is then proved. Conditions for higher order regularity in
times are then derived, which yield some limited space regularity that can be
improved through smallness assumption on the contact angles. We finally show
that higher order regularity away from the contact points can be achieved
through weighted estimates.
