  In this paper, we introduce a new class of hemivariational inequalities,
called dynamic boundary hemivariational inequalities reflecting the fact that
the governing operator is also active on the boundary. In our context, it
concerns the Laplace operator with Wentzell (dynamic) boundary conditions
perturbed by a multivalued nonmonotone operator expressed in terms of Clarke
subdifferentials. We will show that one can reformulate the problem so that
standard techniques can be applied. We will use the well-established theory of
boundary hemivariational inequalities to prove that under growth and general
sign conditions, the dynamic boundary hemivariational inequality admits a weak
solution. Moreover, in the situation where the functionals are expressed in
terms of locally bounded integrands, a "filling in the gaps" procedure at the
discontinuity points is used to characterize the subdifferential on the product
space. Finally, we prove that, under a growth condition and eventually
smallness conditions, Faedo-Galerkin approximation sequence converges to a
desired solution.
