  The purpose of this paper is three-fold. Firstly we attack a nonlinear
interface problem on an unbounded domain with nonmonotone set-valued
transmission conditions. The investigated problem involves a nonlinear monotone
partial differential equation in the interior domain and the Laplacian in the
exterior domain. Such a scalar interface problem models nonmonotone frictional
contact of elastic infinite media. The variational formulation of the interface
problem leads to a hemivariational inequality (HVI), which however lives on the
unbounded domain, and thus cannot analyzed in a reflexive Banach space setting.
By boundary integral methods we obtain another HVI that is amenable to
functional analytic methods using standard Sobolev spaces on the interior
domain and Sobolev spaces of fractional order on the coupling boundary.
Secondly broadening the scope of the paper, we consider extended real-valued
HVIs augmented by convex extended real-valued functions. Under a smallness
hypothesis, we provide existence and uniqueness results, also establish a
stability result with respect to the extended real-valued function as
parameter. Thirdly based on the latter stability result, we prove the existence
of optimal controls for four kinds of optimal control problems: distributed
control on the bounded domain, boundary control, simultaneous
distributed-boundary control governed by the interface problem, as well as
control of the obstacle driven by a related bilateral obstacle interface
problem.
