Quasi-linear Venttsel problems with nonlocal boundary conditions on fractal domains

Let $\Omega\subseteq\mathbb{R\!}^{\,2}$ be an open domain with fractal boundary $\partial\Omega$. We define a proper, convex and lower semicontinuous functional on the space $\mathbb{X\!}^{\,2}(\Omega,\partial\Omega):=L^2(\Omega,dx)\times L^2(\partial\Omega,d\mu)$, and we characterize its subdifferential, which gives rise to nonlocal Venttsel' boundary conditions. Then we consider the associated nonlinear semigroup $T_p$ generated by the opposite of the subdifferential, and we prove that the corresponding abstract Cauchy problem is uniquely solvable. We prove that the (unique) strong solution solves a quasi-linear parabolic Venttsel' problem with a nonlocal term on the boundary $\partial\Omega$ of $\Omega$. Moreover, we study the properties of the nonlinear semigroup $T_p$ and we prove that it is order-preserving, Markovian and ultracontractive. At the end, we turn our attention to the elliptic Venttsel' problem, and we show existence, uniqueness and global boundedness of weak solutions.