  The Rabinowitz-Floer homology of a Liouville domain W is the Floer homology
of the free period Hamiltonian action functional associated to a Hamiltonian
whose zero energy level is the boundary of W. It has been introduced by K.
Cieliebak and U. Frauenfelder. Together with A. Oancea, the same authors have
recently computed the Rabinowitz-Floer homology of the cotangent disk bundle
D^*M of a closed manifold M, by establishing a long exact sequence. The first
aim of this paper is to present a chain level construction of this exact
sequence. In fact, we show that this sequence is the long homology sequence
induced by a short exact sequence of chain complexes, which involves the Morse
chain complex and the Morse differential complex of the energy functional for
closed geodesics on M. These chain maps are defined by considering spaces of
solutions of the Rabinowitz-Floer equation on half-cylinders, with suitable
boundary conditions which couple them with the negative gradient flow of the
geodesic energy functional. The second aim is to generalize this construction
to the case of a fiberwise uniformly convex compact subset W of T^*M whose
interior part contains a Lagrangian graph. Equivalently, W is the energy
sublevel associated to an arbitrary Tonelli Lagrangian L on TM and to any
energy level which is larger than the strict Mane' critical value of L. In this
case, the energy functional for closed geodesics is replaced by the free period
Lagrangian action functional associated to a suitable calibration of L. An
important issue in our analysis is to extend the uniform estimates for the
solutions of the Rabinowitz-Floer equation to Hamiltonians which have quadratic
growth in the momenta. These uniform estimates are obtained by suitable
versions of the Aleksandrov maximum principle.
