Variational setting of a nonlinear interaction problem

A variational approach is here adopted to investigate a Nonlinear Interaction Problem. Specifically, the equilibrium problem of an elastic body whithin a surrounding media is studied. Accordingly, suitable functionals are introduced in connection with two different models, namely, in turn, first order and second order {\sl interaction potentials} are considered. The specific aim is to briefly analyze and compare some of the results obtained by Podio-Guidugli and Vergara Caffarelli and, subsequently, by Carillo, Podio-Guidugli and Vergara Caffarelli . Both studies concern an elastic body, in stationary conditions. To model the interaction between the body and the environment the energy functional is assumed to depend, further than on the elastic energy, on the surface interaction potential. The latter is suitably chosen precisely to take into account that the equilibrium depends on the deformation gradient. In particular, the case when such a dependence is on the first deformation gradient alone is comprised in Podio-Guidugli and Vergara Caffarelli \cite{PGlag}, and the case when the surface interaction potential depends also on the second deformation gradient has been investigated in Carillo, Podio-Guidugli and Vergara Caffarelli. Since the attention is focussed on the interaction which, mainly, takes place on the body surface, an admissibility condition on surface potentials follows.