The limit case in a nonlocal p-Laplacian equation with dynamical boundary conditions

In this paper we deal with the limit as p → ∞ for the nonlocal analogous to the p-Laplacian evolution with dynamic boundary conditions.Our main result states that in both the elliptic and the parabolic case.We are interested in smooth and singular kernels and show the existence and uniqueness of a limit solution.We obtain that the limit solution of the elliptic problem turns out to be also a viscosity solution of a corresponding problem.We prove that the natural energy functionals associated with this problem converge in the sense of Mosco convergence to a limit functional and therefore we obtain convergence of the solutions to the evolution problems for the parabolic case.For the limit problem, we provide examples of explicit solutions for some particular data.