Convergence and density results for parabolic quasi-linear Venttsel' problems in fractal domains

Abstract

In this paper we study a quasi-linear evolution equation with nonlinear dynamical boundary conditions in a three dimensional fractal cylindrical domain \begin{document} $Q$ \end{document} , whose lateral boundary is a fractal surface \begin{document} $S$ \end{document} . We consider suitable approximating pre-fractal problems in the corresponding pre-fractal varying domains. After proving existence and uniqueness results via standard semigroup approach, we prove density results for the domains of energy functionals defined on \begin{document} $Q$ \end{document} and \begin{document} $S$ \end{document} . Then we prove that the pre-fractal solutions converge in a suitable sense to the limit fractal one via the Mosco convergence of the energy functionals.